Given two similar triangles with a scale factor of a : b, show the ratio of their perimeters is also a : b?

1 Answer
Sep 17, 2016

Suppose two triangles triangleA and triangleB are similar with a scale factor of S = a/b. Then, for each side bar(A_i), i = 1, 2, 3 of triangle A, triangle B has a corresponding side bar(B_i) such that A_i = SB_i.

As the perimeter of a triangle is equal to the sum of the lengths of its sides, we have the perimeter P_A of triangleA as

P_A = A_1+A_2 + A_3

=SB_1+SB_2+SB_3

=S(B_1+B_2+B_3)

=SP_B

Thus the ratio of the perimeters is the same as the ratio of the sides, that is, of the scale factor between the triangles.