Problem 25 · 2017 Math Kangaroo
Stretch
Geometry & Measurement
pythagorean-triplesquare-area
The sum of the three side lengths of a right-angled triangle equals 18. The sum of the squares of these three side lengths equals 128. How big is the area of the triangle?
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Answer: E — 9
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Hint 1 of 2
For a right triangle the sum of the two leg-squares equals the hypotenuse-square, so the squared-sum simplifies.
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Hint 2 of 2
Use the perimeter and the squared identity to find the legs' product, which gives the area.
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Approach: use the Pythagorean relation to find the legs' product
- With legs a, b and hypotenuse c: a^2 + b^2 = c^2, so a^2 + b^2 + c^2 = 2c^2 = 128, giving c = 8.
- Then a + b = 18 - 8 = 10, and (a + b)^2 = a^2 + b^2 + 2ab = 64 + 2ab = 100, so ab = 18.
- Area = ab/2 = 9.
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