2005 AMC 8 Problem 23
Attempt Problem 23 of the 2005 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
23.
Isosceles right triangle encloses a semicircle of area The circle has its center on hypotenuse and is tangent to sides and What is the area of triangle
Answer: B
Small hint:
Reflect the triangle and semicircle across the hypotenuse.
Big hint:
This makes a circle inscribed in a square.
Solution:
Reflect the triangle and semicircle across hypotenuse . This forms a full circle inscribed in a square.
The semicircle has area , so the full circle has area and radius . The square side length is therefore .
The square area is , so the original triangle has half that area, .
Thus, B is the correct answer.
Problem 23 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
- 1988 AMC 8
- 1989 AMC 8
- 1990 AMC 8
- 1991 AMC 8
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- 1993 AMC 8
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