Problem 8 · 2009 Math Kangaroo
Medium
Geometry & Measurement
areasquare-area
The square in the diagram has side length 1. The radius of the small circle would then be of length
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Answer: E — \((\sqrt{2}-1)^2\)
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Hint 1 of 2
The small circle sits in the corner left over after the square fills the quarter region.
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Hint 2 of 2
Work along the diagonal from the centre: the leftover gap from the square’s far corner out to the big circle holds the small circle.
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Approach: measure the leftover corner along the diagonal
- With the square of side 1 placed in the corner, the diagonal distance from the centre to its far corner is √2.
- The big circle’s radius is √2, so the gap beyond the square along the diagonal is √2 − 1.
- Fitting the small circle into that gap gives a radius of (√2 − 1)².
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