8.6 Similarity Transformations
Notes
A similarity transformation is a dilation, possibly combined with rigid motions. Two figures are similar if one can be mapped onto the other this way.
- A dilation with scale factor $k$ from center $C$ sends $P$ to $P'$ on ray $\overrightarrow{CP}$ with $CP' = k \cdot CP$.
- Dilations keep angles; lengths become $k$ times as long.
- Example: $\triangle ABC$ dilated by $k = 3$ from the origin, then reflected, is similar to the original with scale factor 3.
Practice
A dilation from center $C$ with scale factor 2.5 maps $P$ to $P'$. If $CP = 4$, find $CP'$.
$CP' = 2.5 \cdot 4 = 10$.
Which sequence maps a figure to a similar but NOT congruent figure?
Only the dilation changes the size.
$\triangle ABC$ with $A(2, 4)$ is dilated by scale factor $\displaystyle \frac{1}{2}$ from the origin, then translated by $(x, y) \to (x + 3, y)$. Where is $A$'s final image?
Dilate: $(1, 2)$. Translate: $(4, 2)$.
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