If two polygons are similar, then the ratio of their perimeters is equal to the ratio of their corresponding side lengths.
We will prove this theorem for two similar rectangles.
Let's begin by writing the perimeter of each rectangle.
Perimeter of ABCD
Perimeter of PQRS
P_1 = 2 a+2 b ⇕ P_1 = 2( a+ b)
P_2 = 2 x + 2 y ⇕ P_2 = 2( x + y)
Since ABCD ~ PQRS, we know that the ratio between corresponding side lengths make a proportion. Let k be the scale factor.
PQ/AB = QR/BC = k
⇕
x/a = y/b = k
From the above, we can rewrite x in terms of a and y in terms of b.
x/a = k y/b = k
⇕
x = k a y = k b
Next, let's substitute the two equations above into the formula for P_2, the perimeter for PQRS.
Consider the expression in the right-hand side of the above equation. Note that the second factor is P_1, the perimeter of ABCD. Therefore, we can substitute P_1 for 2( a+ b).
Therefore, the ratio of the perimeters of the similar rectangles is equal to the ratio of their corresponding side lengths.
P_2/P_1 = PQ/AB = QR/BC = k
Let's summarize our work with a two-column proof.
Two-Column Proof
Given: & ABCD and PQRS are similar
& rectangles with scale factork
Prove: & P_2/P_1 = k
We will summarize our proof with a two-column table.
Statements
Reasons
1.
ABCD and PQRS are similar rectangles with scale factor k.
