Area and perimeter of similar triangles

What is true about the ratio of the area of similar triangles?

Answer: The ratio of the areas is the square of the similarity ratio (aka the scale factor).

It's easiest to see that this is true if you look at some specific examples of real similar triangles.

A=12⋅24

           A = (1/2)*4*12 = 24                A = (1/2)*8*24 = 96

Notice: 2496=14

Therefore, if you know the similarity ratio, all that you have to do is square it to determine ratio of the triangle's areas.

What about the perimeter of similar triangles?

Answer: The ratio of the perimeters is equal to the similarity ratio (aka the scale factor).

Let's look at the two similar triangles below to see this rule in action.

The ratio of the perimeter's is exactly the same as the similarity ratio!

Practice Problems

Problem 1)

△ABC ~ △XYZ and have a scale factor (or similarity ratio) of 32.

What is the ratio of their areas?

Problem 2)

△ABC ~ △XYZ. The ratio of their perimeters is 115, what is their similarity ratio and the ratio of their areas?

What is the ratio of their areas?

Problem 3)

△ABC ~ △XYZ. The ratio of their areas is 3617, what is their similarity ratioand the ratio of their perimeters?

What is the ratio of their areas?

Problem 4)

△HIJ ~ △XYZ. The ratio of their areas is 2516, if XY has a length of 40, what is the length of HI?