Problem 6 · 2003 AMC 8
Medium
Geometry & Measurement
pythagorean-triplesquare-areaarea
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Answer: B — 30.
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Hint 1 of 2
Each square is built on a side of the triangle, so the side of a square IS a side of the triangle — square-root each area.
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Hint 2 of 2
5, 12, 13 is a famous Pythagorean triple, so the triangle has a right angle and the two shorter sides are its legs.
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Approach: areas give the side lengths; recognize the right triangle
- A square's side length is the square root of its area, and each square shares a side with the triangle. So the triangle's sides are √169 = 13, √144 = 12, and √25 = 5.
- Check 5² + 12² = 25 + 144 = 169 = 13²: the sides fit the Pythagorean rule, so the angle between the two short sides is a right angle. That makes 5 and 12 the perpendicular legs — the base and height.
- Area of a right triangle = ½ × leg × leg = ½ × 5 × 12 = 30.
- Worth keeping: 5-12-13 and 3-4-5 (and its scalings) are the triples to memorize — spotting one instantly tells you a triangle is right-angled, no slow Pythagorean computation needed.
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