Problem 26 · 2016 Math Kangaroo
Stretch
Geometry & Measurement
proportion
Two heights of a triangle have lengths 10 cm and 11 cm. Which of the following lengths cannot be the length of the third height?
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Answer: A — 5 cm
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Hint 1 of 2
Each height is inversely proportional to the side it falls on.
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Hint 2 of 2
The sides (proportional to 1/10, 1/11, 1/h) must satisfy the triangle inequality.
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Approach: triangle inequality on reciprocals
- Sides are proportional to 1/10, 1/11 and 1/h, and must form a triangle.
- This forces the third height to satisfy roughly 5.24 < h < 110.
- Only 5 cm falls outside that range, so 5 cm is impossible.
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