Unlike a translation, a dilation changes size. It is the coordinate version of a scale drawing.
Why it works
A dilation with center $O$ and scale factor $k$ moves each point $P$ along the ray from $O$ through $P$, so that its new distance from $O$ is $k$ times the old one. When the center is the origin, the ray from $(0, 0)$ through $(x, y)$ reaches $(kx, ky)$ at $k$ times the distance. When the center is somewhere else, measure from the center instead.
Key idea
Center at the origin: $(x, y) \to (kx, ky)$. Any center $C$: find how far $P$ is from $C$, multiply that by $k$, and add it back to $C$. If $k > 1$ the figure grows; if $0 < k < 1$ it shrinks. Lengths are multiplied by $k$; angles stay the same.
Worked examples
Worked example: Center at the origin
Dilate $(3, -2)$ by a scale factor of 3, center at the origin.
- Multiply both coordinates by 3.
Answer: $(9, -6)$
Worked example: A different center
Dilate $(5, 4)$ by a scale factor of $\frac{1}{2}$ with center $(1, 2)$.
- From the center to the point: $(5 - 1, 4 - 2) = (4, 2)$.
- Scale it: $\frac{1}{2} \cdot (4, 2) = (2, 1)$.
- Add back to the center: $(1 + 2, 2 + 1)$.
Answer: $(3, 3)$
Worked example: Lengths
A segment 8 cm long is dilated by a scale factor of $\frac{3}{4}$. How long is the image?
- Lengths are multiplied by $k$: $\frac{3}{4} \cdot 8 = 6$.
Answer: 6 cm
Common mistake
Multiplying the coordinates by $k$ when the center is not the origin. In Example 2 that gives $(2.5, 2)$, which is wrong, because $(kx, ky)$ assumes the center is $(0, 0)$. Always measure from the actual center.