Similar Polygons
📐 What Are Similar Polygons?
Two polygons are similar if and only if:
- Their corresponding angles are congruent (equal in measure), AND
- Their corresponding sides are proportional (have the same ratio).
We use the symbol $\sim$ to denote similarity. If polygon $ABCD \sim$ polygon $EFGH$, then the order of the vertices tells us which parts correspond.
⚠️ Key Insight: All congruent polygons are similar, but not all similar polygons are congruent. Similarity is about shape, while congruence is about both shape and size.
📏 Scale Factor
The scale factor (or ratio of similitude) is the common ratio between corresponding sides of two similar polygons.
If polygon $ABCD \sim$ polygon $EFGH$ with scale factor $k$, then:
$$\frac{AB}{EF} = \frac{BC}{FG} = \frac{CD}{GH} = \frac{DA}{HE} = k$$
Properties of Scale Factor $k$:
- If $k = 1$, the polygons are congruent
- If $k > 1$, the image is an enlargement
- If $0 < k < 1$, the image is a reduction
Important Relationships:
- Ratio of perimeters = $k$
- Ratio of areas = $k^2$
- Ratio of volumes (3D) = $k^3$