Problem 26 · 2014 Math Kangaroo
Stretch
Geometry & Measurement
symmetry
Two regular polygons with side length 1 lie on opposite sides of the common edge AB. One of them is the 15-sided polygon \(ABC_1D_1E_1\ldots\) and the other is the \(n\)-sided polygon \(ABC_2D_2E_2\ldots\). For which value of \(n\) is the distance from \(C_1\) to \(C_2\) exactly 1?
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Answer: A — 10
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Hint 1 of 2
Place the shared edge AB and find the second vertices C₁ and C₂ of each polygon.
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Hint 2 of 2
Their separation depends on the polygons' interior angles; test which n makes C₁C₂ = 1.
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Approach: locate the two C-vertices and set their distance to 1
- Put A=(0,0), B=(1,0). For each regular polygon the next vertex C is found from its interior angle (a unit step from B).
- The 15-gon fixes C₁; the n-gon (on the other side) fixes C₂.
- Trying values, n = 10 makes the distance C₁C₂ exactly 1.
- So n = 10.
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