Problem 18 · 2020 AMC 8
Medium
Geometry & Measurement
pythagorean-triplearea
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Answer: A — Area 240.
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Hint 1 of 2
The height DC is hidden — but any corner on the arc sits exactly one radius from the center. So draw the line from the center O to a top corner C: that segment is a known length (the radius).
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Hint 2 of 2
FE = 9 + 16 + 9 = 34, so the radius is 17. By symmetry the center O sits at the midpoint of DA, so OD = 8. Now OC = 17 is the hypotenuse of right triangle ODC — Pythagoras gives the height DC.
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Approach: draw a radius to a corner on the arc
- The whole diameter is FE = 9 + 16 + 9 = 34, so the radius is 17 and the center O is the midpoint of FE.
- By symmetry O is also the midpoint of the rectangle's base DA, so OD = 16 ÷ 2 = 8. The key move: C lies on the arc, so OC = radius = 17.
- Right triangle ODC has legs OD = 8 and DC (the rectangle's height) with hypotenuse OC = 17: DC2 = 172 − 82 = 289 − 64 = 225, so DC = 15 (the 8–15–17 triple).
- Area = DA · DC = 16 · 15 = 240.
- Why this transfers: whenever a point sits on a circle, the radius to it is a free known length — drawing it turns a vague distance into the hypotenuse of a right triangle. Recognizing 8–15–17 (a Pythagorean triple) then skips the square-root arithmetic.
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