πŸ‡ΊπŸ‡Έ AMC 8 ⇄ switch contest
1987 AJHSME

Problem 22

Problem 22 · 1987 AJHSME Stretch
Geometry & Measurement quarter-circle-minus-rectangle
Figure for AJHSME 1987 Problem 22
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Answer: D — between 7 and 8.
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Hint 1 of 3
B is the only labeled point ON the circle, and D is the center β€” so the segment DB is a radius. Notice DB is also the diagonal of the rectangle. What's its length?
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Hint 2 of 3
The shaded sliver is the quarter-circle in that corner with the rectangle punched out of it. Find the quarter-circle's area and subtract the rectangle.
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Hint 3 of 3
The question only asks 'between which two whole numbers,' so you can approximate Ο€ β‰ˆ 3.14 at the end rather than carrying it exactly.
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Approach: quarter-disk minus the rectangle inside it
  1. D is the center and B is on the circle, so the radius is DB β€” the rectangle's diagonal. By the Pythagorean theorem DBΒ² = ADΒ² + CDΒ² = 4Β² + 3Β² = 25, so the radius is 5.
  2. The shaded region is the quarter-disk at corner D with rectangle ABCD removed: (1⁄4)(Ο€ Β· 5Β²) βˆ’ (4 Γ— 3) = 25π⁄4 βˆ’ 12.
  3. Estimate: 25π⁄4 β‰ˆ 25 Γ— 3.14 Γ· 4 β‰ˆ 19.6, so the area β‰ˆ 19.6 βˆ’ 12 β‰ˆ 7.6 β€” between 7 and 8.
  4. Why this transfers: when an unknown length is the radius to a point on a circle, hunt for a right triangle whose hypotenuse reaches that point. Here the rectangle hands you a clean 3-4-5.
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