Problem 12 · 2010 Math Kangaroo
Stretch
Geometry & Measurement
pythagorean-triplearea
The chord AB touches the smaller of the two concentric circles. The length AB = 16. How big is the area of the grey part?
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Answer: C — \(64\pi\)
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Hint 1 of 2
The chord just touches the inner circle, so it is tangent there.
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Hint 2 of 2
Drop the radius to the tangent point; it splits the chord in half at a right angle.
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Approach: annulus area depends only on the chord
- The grey ring has area π(R² − r²), where r is the inner radius.
- The chord is tangent to the inner circle, so the perpendicular from the centre has length r and bisects the chord into halves of 8.
- By Pythagoras R² = r² + 8², so R² − r² = 64.
- The grey area is π·64 = 64π, independent of the radii.
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