Problem 30 · 2012 Math Kangaroo
Hard
Geometry & Measurement
perimeterarea-decomposition
Three lines dissect a big triangle into four triangles and three quadrilaterals. The sum of the perimeters of the three quadrilaterals is 25 cm. The sum of the perimeters of the four triangles is 20 cm. The perimeter of the big triangle is 19 cm. How big is the sum of the lengths of the three dissecting lines?
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Answer: C — 13 cm
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Hint 1 of 2
Add up the perimeters of all seven pieces and see what gets counted twice.
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Hint 2 of 2
Each interior cut-line is shared by two pieces, so it is counted twice in that total.
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Approach: double-counting the cut lines
- Summing all seven small perimeters gives 20 + 25 = 45.
- In that sum the big triangle's boundary (19) is counted once and every interior cut-line is counted twice.
- So 45 = 19 + 2 × (total cut length), giving total cut length = (45 − 19)/2 = 13 cm.
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