Problem 15 · 2015 Math Kangaroo
Medium
Geometry & Measurement
areaarea-decomposition
The square ABCD has area 80. The points E, F, G and H are on the sides of the square and AE = BF = CG = DH. How big is the area of the grey part, if AE = 3 × EB?
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Answer: B — 25
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Hint 1 of 3
AE = 3·EB places E (and likewise F, G, H) three-quarters of the way along each side.
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Hint 2 of 3
Put the square on coordinates with side s where s² = 80, then read the grey region's corners.
Still stuck? Show hint 3 →
Hint 3 of 3
Use the shoelace area on the grey polygon's vertices — the side length squared (80) cancels into a clean number.
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Approach: place coordinates, then find the grey area by proportion
- Let the square have side s with s² = 80; since AE = 3·EB, each of E, F, G, H sits 3/4 of the way along its side (so AE = 3s/4, EB = s/4).
- By the equal spacing the figure is symmetric under a quarter-turn, so the grey region is a fixed fraction of the whole square.
- Working out that fraction with coordinates gives 5/16 of the square, and 5/16 × 80 = 25.
- Grey area = 25.
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