DWG NO. 07.5 — Lesson 5 of 5

Dilations and Similarity Transformations

Unit 7: Similarity · ~15–30 min

Objective

Perform a dilation given a center and scale factor, and use dilations to show that two figures are similar.

What a dilation does

A dilation is a transformation that enlarges or reduces a figure from a fixed center of dilation, using a scale factor k. Every point moves directly along the line connecting it to the center, landing k times farther from (or closer to) the center than it started.

If k > 1, the dilation is an enlargement — the image is bigger than the original.
If 0 < k < 1, the dilation is a reduction — the image is smaller.
If k = 1, the image is congruent to (exactly the same size as) the original.

Unlike translations, reflections, and rotations, a dilation is not a rigid motion — it changes size. But it always preserves angle measures and the shape of the figure, which is exactly why the original figure and its dilated image are always similar.

Dilations on the coordinate plane

When the center of dilation is the origin, a dilation with scale factor k sends every point (x, y) to (kx, ky) — just multiply both coordinates by the scale factor.

Every vertex moves along a ray from center O, landing 3 times farther out.

Worked Example 1 · Dilating coordinates

ProblemTriangle with vertices A(2, 1), B(4, 1), C(4, 3) is dilated with center the origin and scale factor k = 2.5. Find the image coordinates.

1.Multiply each coordinate by 2.5: A′(5, 2.5), B′(10, 2.5), C′(10, 7.5).

A′(5, 2.5), B′(10, 2.5), C′(10, 7.5)

Worked Example 2 · Finding the scale factor

ProblemA dilation centered at the origin maps point P(6, 9) to P′(2, 3). Find the scale factor.

1.Since dilation multiplies coordinates by k, solve 6k = 2, giving k = 1/3 (check with the y-coordinate: 9(1/3) = 3, which matches).

k = 1/3 (a reduction)

Guided practice

1.Point Q(3, 5) is dilated from the origin with scale factor k = 4. Find Q′.
Show answer
Q′(12, 20).
2.A dilation centered at the origin maps R(8, 12) to R′(2, 3). Find the scale factor, and state whether it's an enlargement or reduction.
Show answer
8k = 2, so k = 1/4. Since 0 < k < 1, it's a reduction.
3.Triangle DEF has vertices D(0, 2), E(3, 2), F(3, 6). Dilate it from the origin with scale factor 0.5, and find the image vertices.
Show answer
D′(0, 1), E′(1.5, 1), F′(1.5, 3).
4.A rectangle has area 40 square units. After a dilation with scale factor k = 3, what is the area of the image? (Hint: think about how both dimensions change.)
Show answer
Both length and width scale by 3, so the area scales by 3² = 9. New area = 40 × 9 = 360 square units.
5.Explain why a dilation is not considered a rigid motion, and why the original figure and its image are still always similar.
Show answer
Rigid motions preserve distance, but a dilation multiplies every distance from the center by the scale factor, so lengths change (unless k = 1) — that's why it's not rigid. Even so, angle measures stay exactly the same and every length changes by the same constant ratio, which is precisely the definition of similarity.

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