Problem 21 · 2016 Math Kangaroo
Stretch
Geometry & Measurement
area-decomposition
In a square of area 36 there are grey parts as shown in the diagram. The sum of the areas of all the grey parts is 27. How long are the distances a, b, c and d together?
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Answer: D — 9
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Hint 1 of 2
Every grey triangle shares the same corner of the square and has its base (one of a, b, c, d) on an edge.
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Hint 2 of 2
From that corner each base is the full side away, so every triangle has height 6; the grey total is 3(a + b + c + d).
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Approach: grey area in terms of the four base segments
- The square has side 6 (area 36). Every grey triangle has its apex at the same corner, with base a, b, c or d lying on an edge a full side (6) away.
- So the total grey area is (1/2)(6)(a + b + c + d) = 3(a + b + c + d) = 27.
- Therefore a + b + c + d = 9.
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