Problem 23 · 2010 AMC 8
Hard
Geometry & Measurement
circle-radiipythagorean
Semicircles POQ and ROS pass through the center O. What is the ratio of the combined areas of the two semicircles to the area of circle O?
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Answer: B — 1/2.
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Hint 1 of 3
The two semicircles are identical, so together they're the same area as one full small circle — stop counting them separately.
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Hint 2 of 3
Use coordinates to nail the radii: the small radius is half of PQ = 1, and the big radius is the distance OQ = √(12+12) = √2 by the distance formula.
Still stuck? Show hint 3 →
Hint 3 of 3
Since you're forming a ratio, every π will cancel — only the squared radii matter.
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Approach: combine the semicircles, then ratio the squared radii
- Both semicircles have diameter PQ = RS = 2, so radius 1. Two of them glue into one full circle of radius 1, area π.
- The big circle's radius is the distance from center O(0,0) to Q(1,1): √(12+12) = √2, so its area is π(√2)2 = 2π.
- Ratio = π / 2π = 1/2 — the π cancels, as expected for a pure area ratio.
- Why this transfers: two matching semicircles always recombine into a whole circle, and area ratios reduce to (radius ratio)2. Here that's (1 vs √2)2, i.e. 1 : 2 — no π, no decimals.
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