Problem 13 · 2025 Math Kangaroo
Hard
Geometry & Measurement
areaarea-decomposition
In the diagram we see a quarter circle SP with centre O and radius r, as well as a triangle ORP. The two grey regions have the same area. How long is the segment OR?
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Answer: A — \(\dfrac{\pi r}{2}\)
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Hint 1 of 2
Equal grey areas means a shared region can be added to both without changing the equality.
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Hint 2 of 2
Adding the common piece turns "two greys equal" into "triangle ORP = quarter circle".
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Approach: add the shared region to both grey pieces
- Equal greys imply area(triangle ORP) = area(quarter circle) = πr²/4.
- Triangle ORP is right-angled at O with legs OR and OP = r, so ½·OR·r = πr²/4.
- Thus OR = πr/2.
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