Dilations and Scale Factor

By the end of this lesson, you will be able to perform dilations centered at the origin and other centers, identify the scale factor from a dilation, and understand why dilations preserve shape but not size (similarity, not congruence).

The Big Idea

A dilation scales a figure by a scale factor $k$ about a center point. Unlike the three rigid motions (translation, rotation, reflection), a dilation changes size — it is not an isometry when $k \neq 1$. Dilations produce similar figures, not congruent ones. They are the transformation behind maps, zoom functions, scale models, and similar triangles.

Definition

A dilation with center $C$ and scale factor $k$ maps every point $P$ to $P'$ such that:

$$CP' = k \cdot CP$$

• Points move along rays from the center
• If $k > 1$: enlargement
• If $0 < k < 1$: reduction
• If $k = 1$: identity (no change)
• If $k < 0$: dilation with rotation by 180° (rare at this level)

Dilation About the Origin

If the center is the origin and scale factor is $k$:

$$(x, y) \to (kx, ky)$$

Example 1 — Dilation About the Origin

Dilate $\triangle ABC$ ($A(2, 3)$, $B(4, 1)$, $C(6, 5)$) by scale factor $k = 2$.

$$A' = (4, 6), \quad B' = (8, 2), \quad C' = (12, 10)$$

All distances doubled; angles unchanged; triangles are similar with ratio 2:1.

Example 2 — Reduction

Dilate $P(9, -6)$ by scale factor $k = \frac{1}{3}$ about the origin.

$$P' = (3, -2)$$

Example 3 — Finding the Scale Factor

$\triangle ABC \sim \triangle A'B'C'$ (image from dilation). $AB = 6$ and $A'B' = 15$.

$$k = \frac{A'B'}{AB} = \frac{15}{6} = 2.5$$

Dilation About a Center Other Than the Origin

Given center $C(a, b)$ and scale factor $k$:

1. Translate so $C$ is at origin: $(x - a, y - b)$
2. Apply dilation: $(k(x-a), k(y-b))$
3. Translate back: $(k(x-a) + a, k(y-b) + b)$

Example 4 — Dilation About Other Center

Dilate $P(5, 4)$ by $k = 3$ about center $C(2, 1)$.

Step 1: $P - C = (3, 3)$

Step 2: $k \cdot (3, 3) = (9, 9)$

Step 3: $(9 + 2, 9 + 1) = (11, 10) = P'$

Properties of Dilations

1. Not an isometry (unless $k = 1$): distances change by factor $k$
2. all angles are unchanged