Problem 16 · 2002 AMC 8
Hard
Geometry & Measurement
areapythagorean-triplesquare-area
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Answer: E — X + Y = Z.
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Hint 1 of 2
Each outer triangle is right *isosceles*, so its two legs equal the 3-4-5 side it's built on. That makes its area a clean ½·(side)² — area depends only on which side it sits on.
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Hint 2 of 2
So X, Y, Z scale like 3², 4², 5². You already know a famous fact linking those three squares.
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Approach: each isosceles right triangle's area is half the square on its side
- A right isosceles triangle with legs s has area ½·s². Since each one sits on a 3-4-5 side, X = ½·3² = 4.5, Y = ½·4² = 8, Z = ½·5² = 12.5.
- Now the Pythagorean fact 3² + 4² = 5² is hiding here: halve every term and you get 4.5 + 8 = 12.5, i.e. X + Y = Z, choice E.
- *The big idea worth keeping:* build *any* matching shape on the three sides of a right triangle — triangles here, but squares, semicircles, anything similar — and the two smaller areas always sum to the largest. It's the Pythagorean theorem dressed in areas, because every such area is (some fixed constant)·(side)².
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