Problem 15 · 1989 AJHSME
Hard
Geometry & Measurement
trapezoid-areaparallelogram
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Answer: D — 64.
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Hint 1 of 3
BEDC has two horizontal sides (BC on top, ED on the bottom) and the vertical BE joining them at right angles. What standard shape is that?
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Hint 2 of 3
It's a right trapezoid: BC and ED are the two parallel sides and BE is the height that's perpendicular to both. Use ½ × (sum of parallel sides) × height.
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Hint 3 of 3
You're missing BC, but the figure is a parallelogram — so BC equals the opposite side AD = 10. BE = 8 is the height.
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Approach: right-trapezoid area directly
- Spot the shape: BEDC has BC parallel to ED (both horizontal) with BE perpendicular to both, so it's a right trapezoid. Its height is the perpendicular side BE = 8.
- Find the missing parallel side: in parallelogram ABCD, BC equals its opposite side AD = 10. So the parallel sides are BC = 10 and ED = 6.
- Area = ½(10 + 6)(8) = ½ × 16 × 8 = 64.
Another way — whole parallelogram minus the corner triangle:
- The full parallelogram has base AD = 10 and height BE = 8, so its area is 10 × 8 = 80.
- The unshaded part is right triangle ABE. Since AD = 10 and ED = 6, the leg AE = 10 − 6 = 4, and the other leg is BE = 8, so its area is ½ × 4 × 8 = 16.
- Shaded BEDC = 80 − 16 = 64. Two routes, same answer — a reassuring check.
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