Problem 19 · 2000 AMC 8
Hard
Geometry & Measurement
area-decompositionrearrangement
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Answer: C — 50 square units.
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Hint 1 of 2
Before reaching for Ο, notice every arc has the SAME radius 5. The two quarter-circles scooped OUT of the bottom and the semicircle bulging UP on top are built from identical-radius pieces β so the curves might just cancel.
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Hint 2 of 2
Two quarter-circles of radius 5 add up to one semicircle of radius 5. That's exactly the area added by the top bump. So removed area = added area, and the region equals a plain rectangle β the Ο's vanish.
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Approach: the curves cancel β equal area cut out equals area added
- All arcs have radius 5. Frame the figure with the rectangle whose width is the straight chord BD = 2Β·5 = 10 (B and D sit one radius either side of center) and whose height is 5.
- The bottom of the region dips inward along two quarter-circles (arcs AB and AD); together those two quarters make exactly one semicircle of radius 5 of *missing* area. The top of the region bulges out along semicircle BCD β exactly one semicircle of radius 5 of *extra* area.
- Missing = extra, so they cancel perfectly: the region has the same area as the 10 Γ 5 rectangle = 50.
- You'll see it again: when curved bites and curved bulges share the same radius, slide the bulge into the bite β equal curves cancel and you're left with straight-sided area. If you ever see Ο in your answer here, you forgot to cancel (the trap choices 25Ο, 10+5Ο are there for exactly that).
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