Problem 17 · 2009 Math Kangaroo
Stretch
Geometry & Measurement
pythagorean-triplesquare-area
The centres of the four circles shown are at the corners of the square. The two big circles touch each other and also touch the two little circles. By what factor must you multiply the radius of the little circles to obtain the radius of the big circles?
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Answer: E — \(1+\sqrt{2}\)
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Hint 1 of 2
The two big circles touch along the square's diagonal; the big and small circles touch along a side.
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Hint 2 of 2
Write both touching conditions in terms of the side length, then take the ratio R/r.
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Approach: relate radii through the diagonal and the side
- Let the square have side s, big radius R, small radius r.
- Two big circles at opposite corners touch across the diagonal: 2R = s√2, so R = s/√2.
- A big and a small circle touch along a side: R + r = s, so r = s − s/√2.
- Then R/r = (1/√2)/(1 − 1/√2) = 1/(√2 − 1) = 1 + √2.
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