Question

A certain right triangle with integer side lengths has perimeter $72$. What is its area?

Answer 1

Answer: 216 square units

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Explanation:

A common pythagorean triple you may be familiar with is the 3-4-5 right triangle. This has two legs of 3 and 4, and a hypotenuse of 5. The perimeter is 3+4+5 = 7+5 = 12. Note how this is a factor of 72.

If we multiply the perimeter (12) by 6, then 12*6 = 72. So we have scaled the triangle by a factor of 6. Each length is 6 times longer

the side length 3 becomes 3*6 = 18

the side length 4 becomes 4*6 = 24

the side length 5 becomes 5*6 = 30

The new perimeter is 18+24+30 = 42+30 = 72

The last step is to find the area. The two legs of this triangle are the base and height

area = 0.5*base*height

area = 0.5*18*24

area = 9*24

area = 216

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Or you could find the area of the 3-4-5 right triangle to get

area = 0.5*base*height = 0.5*3*4 = 6

then multiply by 36 to get 6*36 = 216. The 36 is the square of the scale factor 6 we applied above. The new lengths are 6 times longer, so the new area is 6^2 = 36 times larger.