Question

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

Answer 1

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.