A dilation with center $O$ and scale factor $k > 0$ sends each point $P$ to the point $P'$ on ray $\overrightarrow{OP}$ with $OP' = k \cdot OP$. The center stays put. In Transformations as Functions you saw that a dilation is not a rigid motion; here we pin down exactly what it does change and what it keeps.
Por qué funciona
Put the center at the origin. Then the dilation is $(x, y) \to (kx, ky)$. For two points $P(x_1, y_1)$ and $Q(x_2, y_2)$, the slope of $\overline{P'Q'}$ is $\frac{k y_2 - k y_1}{k x_2 - k x_1} = \frac{y_2 - y_1}{x_2 - x_1}$: the same slope, so the image is parallel to the original (or on the same line, if that line passes through the center). The length is $\sqrt{k^2 (x_2 - x_1)^2 + k^2 (y_2 - y_1)^2} = k \cdot PQ$. And since every side keeps its direction, every angle keeps its measure.
Idea clave
A dilation by $k$ multiplies every length by $k$, keeps every angle, and sends each line to a parallel line (a line through the center goes to itself). $k > 1$ enlarges; $0 < k < 1$ shrinks.
Ejemplo resuelto: Dilating about a center that is not the origin
Dilate $P(5, 3)$ about $C(1, 1)$ with scale factor $\frac{3}{2}$.
- From $C$ to $P$: $\langle 4, 2 \rangle$.
- Scale it: $\frac{3}{2}\langle 4, 2 \rangle = \langle 6, 3 \rangle$.
- $P' = C + \langle 6, 3 \rangle = (7, 4)$.
Respuesta: $P'(7, 4)$
Ejemplo resuelto: Finding the scale factor
A dilation maps $\triangle ABC$, with $AB = 12$, to $\triangle A'B'C'$ with $A'B' = 8$. What is $k$, and how long is $B'C'$ if $BC = 9$?
- $k = \frac{A'B'}{AB} = \frac{8}{12} = \frac{2}{3}$. It is less than 1, so the triangle shrinks.
- $B'C' = \frac{2}{3} \cdot 9 = 6$.
Respuesta: $k = \frac{2}{3}$ and $B'C' = 6$.
Ejemplo resuelto: What happens to angles
$\triangle ABC$ has $m\angle A = 35^\circ$. It is dilated with scale factor 4. Find $m\angle A'$.
- Dilations keep angle measures: each side of the angle stays parallel to its original.
Respuesta: $35^\circ$
Error común
Multiplying angles by $k$ too. A dilation stretches distances, not turns: doubling a triangle doubles its sides, but a $35^\circ$ angle is still $35^\circ$. Check: a photo enlargement does not make a corner of a picture any less square.