Problem 16 · 2024 Math Kangaroo
Hard
Geometry & Measurement
proportion
A point P is chosen inside an equilateral triangle ABC. Through P, segments of lengths 2 m, 3 m and 6 m are drawn parallel to the three sides, as shown. What is the perimeter of the triangle?
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Answer: C — 33 m
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Hint 1 of 2
The three lines through P cut the big triangle into three small equilateral triangles plus parallelograms.
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Hint 2 of 2
Lining up the three small equilateral triangles along one side shows their sides just add up to that whole side.
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Approach: the three side-parallel segments tile one side of the triangle
- Each segment parallel to a side is the side of a small equilateral triangle that sits in a corner of the big triangle.
- The three small triangles slide along one side of the big triangle and together cover it exactly, so the side length is \(2+3+6=11\) m.
- The perimeter is \(3\times 11=33\) m, answer C.
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