Introduction to Similar Figures
What Are Similar Figures?
Two figures are similar if they have the same shape but not necessarily the same size. Think of it like a photograph and an enlarged copy — the image looks the same, just bigger or smaller.
- Their corresponding angles are congruent (equal in measure), AND
- Their corresponding sides are proportional (have the same ratio).
If polygon $ABCD \sim$ polygon $EFGH$, then:
$$ \angle A \cong \angle E, \quad \angle B \cong \angle F, \quad \angle C \cong \angle G, \quad \angle D \cong \angle H $$
$$ \frac{AB}{EF} = \frac{BC}{FG} = \frac{CD}{GH} = \frac{DA}{HE} = k $$
The constant $k$ is called the scale factor (or ratio of similitude).
The Scale Factor
The scale factor tells you how much larger or smaller one figure is compared to the other.
If $k > 1$:
The second figure is an enlargement (bigger).
If $0 < k < 1$:
The second figure is a reduction (smaller).
If $k = 1$:
The figures are congruent (same size and shape). Congruence is a special case of similarity!
Key Properties of Similar Figures:
- Perimeter ratio = scale factor $k$
- Area ratio = $k^2$ (the square of the scale factor)
$$ \frac{\text{Perimeter}_1}{\text{Perimeter}_2} = k \qquad \frac{\text{Area}_1}{\text{Area}_2} = k^2 $$