Problem 14 · 2014 AMC 8
Medium
Geometry & Measurement
area-formulapythagorean-triple
Rectangle ABCD and right triangle DCE have the same area. They are joined to form a trapezoid, as shown. What is DE?
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Answer: B — DE = 13.
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Hint 1 of 2
"Same area" is the bridge: compute the rectangle's area, then that number is the triangle's area too. The shared side DC = 5 is one leg, so the area equation hands you the other leg CE.
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Hint 2 of 2
With both legs known, DE is the hypotenuse — watch for a familiar Pythagorean triple before reaching for the calculator.
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Approach: transfer the area across, then Pythagoras
- Rectangle area = 5 × 6 = 30. The triangle has the same area, so ▵DCE = 30.
- ▵DCE is right-angled at C with legs DC = 5 and CE: (1/2)(5)(CE) = 30 ⇒ CE = 12.
- DE = √(52 + 122) = √169 = 13. (Recognize the 5-12-13 triple and skip the square root.)
- Why this transfers: "equal areas" (or equal perimeters, equal anything) is a free equation — set the two expressions equal and an unknown pops out. And memorizing 3-4-5, 5-12-13, 8-15-17 turns many right-triangle problems into instant recall.
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