How do I find the ratio of perimeters?

Question

Triangle ABC is similar to triangle DEF & the scale factor of triangle ABC to triangle DEF is 4 to 5

a) what is the ratio of perimeters of triangle ABC to triangle DEF?

b) what is the ratio of the areas of triangle ABC to triangle DEF?

Answer 1

a) The ratio of perimeters is the same as the scale factor for any similar polygons.

For instance, if the sides of one triangle are 4, 4, and 4 (assuming the triangle is equilateral) and the sides of the similar triangle are 5, 5, and 5, the perimeter of the first triangle is 12 and the perimeter of the second triangle is 15. A 12/15 ratio can be simplified to 4/5.

b) The ratio of areas is the square of the scale factor for similar polygons.

The reason that we must square the scale factor is because area is found by multiplying the base and height of the triangle. If both the base and the height are at a 4/5 ratio, the areas are at a 16/25 ratio. For instance, in a right triangle with base 4 and height 8, the area is 4*8 / 2 = 16 and in a right triangle with base 5 and height 10, the area is 5 * 10 / 2 = 25.

Both of these problems could be done with more "messy" numbers and still give you the same ratio in the end. Try it with any numbers at a 4/5 ratio that you like and see how it turns out.