Problem 14 · 1999 AMC 8
Medium
Geometry & Measurement
pythagoreansymmetry
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Answer: D — 34.
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Hint 1 of 2
The slant sides AB and CD are equal, so the trapezoid is symmetric — drop a vertical from B and from C and the bottom splits into the middle (equal to the top, 8) plus two equal overhangs. The two overhangs share the leftover 16 − 8.
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Hint 2 of 2
Each slant side is now the hypotenuse of a right triangle with legs = height and overhang. Watch for a friendly Pythagorean triple.
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Approach: use symmetry to find the overhang, then a 3-4-5 triangle
- Because AB = CD, the figure is symmetric: dropping verticals from B and C carves a rectangle of width 8 in the middle, leaving 16 − 8 = 8 split evenly into two overhangs of 4 each.
- Each slant side is the hypotenuse of a right triangle with legs 3 (height) and 4 (overhang): √(3² + 4²) = 5 — a 3-4-5 triangle, no messy square root.
- Perimeter = 16 + 8 + 5 + 5 = 34. Two transferable habits: exploit symmetry to find the overhang for free, and recognize 3-4-5 (and 5-12-13) so you spot the hypotenuse on sight.
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