Problem 19 · 2005 AMC 8
Medium
Geometry & Measurement
trapezoid-altitudespythagorean
What is the perimeter of trapezoid ABCD?
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Answer: A — 180.
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Hint 1 of 2
The only mystery side is the long bottom, AD. Drop a vertical from each top corner (B and C) to the base — that carves the trapezoid into a rectangle in the middle and a right triangle on each end.
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Hint 2 of 2
Each slanted leg becomes the hypotenuse of a right triangle whose other leg is the height 24. Look for friendly Pythagorean triples instead of grinding square roots.
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Approach: slice into a rectangle plus two right triangles
- Drop verticals from B and C down to base AD, with feet E and F. Both have height 24, and the middle piece EF equals the top BC = 50.
- Left triangle: hypotenuse 30, one leg 24 ⇒ the other leg is 18 (it's a 3-4-5 triple scaled by 6: 18-24-30). So AE = 18.
- Right triangle: hypotenuse 25, leg 24 ⇒ other leg 7 (the 7-24-25 triple). So FD = 7.
- Base AD = 18 + 50 + 7 = 75. Perimeter = 75 + 30 + 50 + 25 = 180.
- Why this transfers: any trapezoid splits into a rectangle plus two right triangles by dropping the two heights — and recognizing 18-24-30 and 7-24-25 as scaled triples turns the Pythagorean step into instant recall.
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