Problem 18 · 2012 Math Kangaroo
Medium
Geometry & Measurement
perimetersubstitution
Three equally sized equilateral triangles are cut from the vertices of a large equilateral triangle of side length 6 cm. The three little triangles together have the same perimeter as the remaining grey hexagon. What is the side-length of one side of one small triangle?
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Answer: D — 1.5 cm
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Hint 1 of 2
Let the small triangle's side be x and write both perimeters in terms of x.
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Hint 2 of 2
The hexagon's six sides are the three cut edges and the three leftover pieces of the big sides.
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Approach: match the two perimeters
- With small side x, the three little triangles have total perimeter 3 × 3x = 9x.
- The hexagon's sides are three edges of length x plus three leftover pieces of length 6 − 2x, total 3x + 3(6−2x) = 18 − 3x.
- Setting 9x = 18 − 3x gives 12x = 18, so x = 1.5 cm.
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