Geometry / Unit 4: Transformations
4.4 Dilations
Notes
A dilation stretches or shrinks a figure from a center by a scale factor $k$.
- Centered at the origin: $(x, y) \to (kx, ky)$.
- $k > 1$ is an enlargement; $0 < k < 1$ is a reduction.
- A dilation keeps angle measures and shape, but multiplies lengths by $k$. It is not a rigid motion (unless $k = 1$).
- A dilation maps a line not through the center to a parallel line.
- Example: $k = 3$ sends $(2, -1)$ to $(6, -3)$.
Practice
Dilate $(4, -6)$ by a scale factor of $\displaystyle \frac{1}{2}$ centered at the origin.
Multiply each coordinate by $\displaystyle \frac{1}{2}$.
A segment of length 9 is dilated by a scale factor of 4. What is the image length?
Lengths are multiplied by the scale factor: $9 \cdot 4 = 36$.
$A(3, 2)$ maps to $A'(12, 8)$ under a dilation centered at the origin. What is the scale factor?
$12 \div 3 = 4$ and $8 \div 2 = 4$.
Ready for more? Take the Unit 4 test.