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# Right triangle

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A certain right triangle with integer side lengths has perimeter 160. What is its area?

Apr 25, 2022

#1
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Use the formula A = (a)(b)/2 to find its area, but you haven't given me the sides of the triangle and only the perimeter so I can only guess what the length of each side is.

Apr 26, 2022
#2
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A certain right triangle with integer side lengths has perimeter 160. What is its area?

It's a right triangle,  Its sides are integers.  So the sides must be a Pythagorean Triple.

No Pythagorean Triple totals 160 per https://en.wikipedia.org/wiki/Pythagorean_Triple

So this triangle doesn't exist.

.

Apr 26, 2022
#3
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Actually, it does exist.

The triple can also be a multiple of a primitive Pythagorean Theorem.

For example, the triple $$(8, 15, 17)$$ has a perimeter of 40. We can scale this up by 4, to have the triple $$(32, 60,68)$$

The bases of this triangle are 32 and 60, so the area is: $$\Large {32 \times 60 \over 2} = \text{____}$$

Apr 26, 2022