Theorems on Perimeters and Areas of Similar Polygons
📐 What Are Similar Polygons?
Two polygons are similar if their corresponding angles are congruent and their corresponding sides are proportional. The ratio of corresponding sides is called the scale factor (often denoted as $k$).
If polygon $ABCD \sim$ polygon $EFGH$ with a scale factor of $k = \dfrac{AB}{EF}$, then every pair of corresponding sides shares this same ratio.
📏 Theorem 1: Perimeters of Similar Polygons
Theorem: If two polygons are similar with a scale factor of $k$, then the ratio of their perimeters equals $k$.
$$\frac{P_1}{P_2} = k$$
Proof Outline:
Let two similar polygons have corresponding sides:
$$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = \cdots = k$$
Then $a_1 = k \cdot a_2$, $b_1 = k \cdot b_2$, $c_1 = k \cdot c_2$, etc.
$$P_1 = a_1 + b_1 + c_1 + \cdots = k \cdot a_2 + k \cdot b_2 + k \cdot c_2 + \cdots$$
$$P_1 = k(a_2 + b_2 + c_2 + \cdots) = k \cdot P_2$$
Therefore: $\dfrac{P_1}{P_2} = k$ ✅