Problem 17 · 2010 Math Kangaroo
Stretch
Counting & Probability
careful-counting
If we connect three corner points of a regular 14-sided polygon then a triangle is created. How many of those triangles are right-angled?
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Answer: B — 84
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Hint 1 of 2
A triangle inscribed in a circle is right-angled exactly when one side is a diameter.
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Hint 2 of 2
Count the diameters of the 14-gon, then the third vertex.
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Approach: right angle in a circle means a diameter side
- The 14 vertices lie on a circle; a right angle occurs only when one side is a diameter.
- Opposite vertices give 7 diameters.
- Each diameter pairs with any of the other 12 vertices: 7×12 = 84 right triangles.
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