Problem 9 · 2009 Math Kangaroo
Medium
Geometry & Measurement
area-decompositionarea
Each side of a triangle ABC is extended to the points P, Q, R, S, T and U, so that \(PA=AB=BS\), \(TC=CA=AQ\) and \(UC=CB=BR\). The area of ABC is 1. How big is the area of the hexagon PQRSTU?
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Answer: D — 13
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Hint 1 of 2
Each extension creates triangles that share a base and height with ABC, so compare areas piece by piece.
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Hint 2 of 2
Count how many copies of [ABC] (= 1) tile the whole hexagon.
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Approach: split the hexagon into triangles equal in area to ABC
- Extending each side to double-length builds outer triangles each with area related to [ABC] = 1.
- Summing the central triangle and the six outer pieces tiles the hexagon with 13 unit-area triangles.
- So the hexagon’s area is 13.
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