Problem 27 · 2023 Math Kangaroo
Stretch
Geometry & Measurement
pythagorean-triplearea-decomposition
Consider the two touching semicircles with radius 1 and their diameters AB and CD respectively that are parallel to each other. The extensions of the two diameters are also tangents to the respective other semicircle (see diagram). How big is the square of the length AD?
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Answer: B — \(8 + 4\sqrt{3}\)
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Hint 1 of 2
Set coordinates: A and B on the lower line, C and D on the upper line, each diameter of length 2.
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Hint 2 of 2
Use that each diameter's extension is tangent to the other semicircle to fix the offset, then apply the distance formula for AD.
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Approach: coordinates and the tangency condition
- Each line is tangent to the other semicircle, so the distance between the two parallel lines equals the radius: the vertical gap is 1.
- The semicircles touch, so the distance between their centres is 1 + 1 = 2; with a vertical gap of 1, the horizontal offset of the centres is √(2² − 1²) = √3.
- Put A = (−1, 0) and the far end D = (√3 + 1, 1); then AD² = (√3 + 2)² + 1² = 3 + 4√3 + 4 + 1.
- So AD² = 8 + 4√3.
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