Dilation

Introduction

Learning Objectives

After studying dilation, students are expected to be able to:

1. Explain dilation as a geometric transformation in contextual problems.
2. Solve contextual problems involving dilation.

Material

Digital cameras often provide a zoom feature that enlarges or reduces an image. This idea is closely related to dilation in geometry.

Dilation is a transformation that changes the size of a figure, either enlarging or reducing it, without changing its shape.

If A is the original figure before dilation, then B, C, and D are images obtained from dilation with certain scale factors.

Two pieces of information are needed to perform a dilation:

Scale Factor

The scale factor determines the amount of enlargement or reduction. Its value may be positive, zero, or negative.

Center of Dilation

The center of dilation is a coordinate point. If it is not specified, the center is usually taken as $O(0,0)$.

Example 1

Triangle ABC has vertices A(1, 3), B(2, 3), and C(2, 1). Determine the image of the triangle after a dilation with scale factor 3 and center at the origin.

Solution

The steps are as follows.

1. Plot points A, B, and C on the Cartesian plane.
2. Multiply each coordinate by the scale factor 3:

Therefore, the image of triangle ABC is triangle $A'B'C'$ with vertices $A'(3,9)$, $B'(6,9)$, and $C'(6,3)$.

Example 2

Rectangle ABCD has vertices A(2, 3), B(5, 3), C(5, 1), and D(2, 1). Determine the coordinates of $A'B'C'D'$, the image of rectangle ABCD after a dilation with center P(1, 4) and scale factor 3.

Solution

1. Plot the center of dilation $P(1,4)$ and the rectangle ABCD on the coordinate plane.
2. Draw rays from point P through each vertex so that each image point follows the given scale factor.
3. The image points are:

4. Connect points $A'$, $B'$, $C'$, and $D'$ to form rectangle $A'B'C'D'$.

Conclusion

The image of point $(x,y)$ after dilation with scale factor $k$ and center $O(0,0)$ is:

Simulations

1. Slider-based image simulation

2. Triangle ABC simulation

3. Uzumaki Naruto simulation

4. Two-circle dilation simulation

Use a laptop for the best result.

Evaluation

After studying this topic, solve the following problems.

1. Point A(2, 3) is dilated with center O(0, 0) and scale factor 2. If $A'(x,y)$ is the image point, determine the coordinates of $A'$.
2. Look at the following figure.

Parallelogram $A'B'C'D'$ is the image of parallelogram ABCD under a dilation with center P and scale factor $k$. From the picture, determine:

• the coordinates of the center point P;
• the scale factor of the dilation.

Closing

That is the discussion of dilation as part of geometric transformations. I hope this material is useful.