Problem 8 · 2007 AMC 8
Easy
Geometry & Measurement
right-triangle-area
In trapezoid ABCD, AD is perpendicular to DC, AD = AB = 3, and DC = 6. In addition, E is on DC, and BE is parallel to AD. Find the area of ▵BEC.
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Answer: B — 4.5.
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Hint 1 of 2
Drop BE straight down and the shape splits into a square ABED and the triangle. The square hands you both legs of the triangle for free.
Still stuck? Show hint 2 →
Hint 2 of 2
Slice an awkward shape into a familiar one: a parallel-and-perpendicular setup usually hides a rectangle (here a square) you can read lengths off of.
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Approach: carve off the square to expose a right triangle
- Since BE ∥ AD and AD ⊥ DC, segment BE is also ⊥ DC. With AB ∥ DC too, ABED is a rectangle — in fact a square, because AD = AB = 3. So BE = 3 and DE = 3.
- That leaves EC = DC − DE = 6 − 3 = 3, and ▵BEC is right-angled at E with legs 3 and 3.
- Area = (1/2)(3)(3) = 4.5.
- Worth keeping: when one side is parallel to a perpendicular height, that height is just the rectangle's width — no Pythagoras needed.
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