Similarity of Arbitrary Figures
Big Idea: Similarity isn't just for triangles! Any two figures can be similar — circles, pentagons, irregular blobs — as long as one is a perfectly scaled copy of the other.
Formal Definition
Definition: Two figures are similar if one can be obtained from the other by a sequence of similarity transformations:
- Translations (slides)
- Rotations (turns)
- Reflections (flips)
- Dilations (scaling up or down)
We write $F_1 \sim F_2$ and say "Figure 1 is similar to Figure 2."
The Scale Factor
When two figures are similar, every length in one figure is multiplied by the same constant $k$ called the scale factor (or ratio of similitude).
Key Relationships for Similar Figures with Scale Factor $k$
| Property | Relationship |
|---|---|
| Corresponding lengths | $\dfrac{l_2}{l_1} = k$ |
| Corresponding perimeters | $\dfrac{P_2}{P_1} = k$ |
| Corresponding areas | $\dfrac{A_2}{A_1} = k^2$ |
| Corresponding angles | All corresponding angles are equal |
Why Does Area Scale by $k^2$?
Think of it this way: area is measured in square units. When you scale each dimension by $k$, the area scales by $k \times k = k^2$. For example, if you double every length ($k = 2$), the area becomes $2^2 = 4$ times as large.
Similarity Beyond Triangles
For triangles, we have convenient shortcuts (AA, SAS~, SSS~). For arbitrary figures, we need the full definition:
- All corresponding angles must be equal.
- All corresponding sides (or distances) must be proportional with the same ratio $k$.
⚠️ Caution: For quadrilaterals and more complex figures, equal angles alone are not enough