Integrated Math 2 / Unit 6: Similarity
6.1 Dilations
Notes
A dilation enlarges or shrinks a figure from a center by a scale factor $k$.
- If $k > 1$ the image is larger (enlargement). If $0 < k < 1$ it is smaller (reduction).
- Dilation about the origin: $(x, y) \to (kx, ky)$.
- Angle measures stay the same and lines stay parallel to their images. Lengths change by the factor $k$.
- Example: scale factor $\displaystyle \frac{1}{2}$ takes $(4, -6)$ to $(2, -3)$.
Practice
A dilation about the origin with scale factor $\displaystyle \frac{1}{2}$ is applied to $(4, -6)$. What is the image?
Multiply both coordinates by $\displaystyle \frac{1}{2}$.
A segment of length 6 is dilated to a segment of length 15. What is the scale factor?
$\displaystyle k = \frac{\text{image}}{\text{original}} = \frac{15}{6} = 2.5$.
Which property does NOT change under a dilation?
A dilation multiplies lengths by $k$ but keeps all angle measures.
Ready for more? Take the Unit 6 test.