Problem 2 · 2013 Math Kangaroo
Easy
Geometry & Measurement
symmetry
The regular octagon shown has sides of length 10. A circle touches all of the octagon's long diagonals (the inscribed star). What is the radius of this circle?
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Answer: C — 5
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Hint 1 of 2
The inscribed star is made of the long diagonals; the circle just touches each of them.
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Hint 2 of 2
The radius is the distance from the octagon's centre to those diagonals — find it from the side length.
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Approach: distance from centre to the long diagonals
- Set up the regular octagon with side 10; its diagonals form the inscribed star that the circle touches.
- By symmetry every such diagonal sits the same distance from the centre, and that distance is the circle's radius.
- Computing it for side 10 gives radius 5, so C.
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