Difference between revisions of "1987 AJHSME Problems/Problem 22"
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{{AJHSME box|year=1987|num-b=21|num-a=23}} | {{AJHSME box|year=1987|num-b=21|num-a=23}} | ||
[[Category:Introductory Geometry Problems]] | [[Category:Introductory Geometry Problems]] | ||
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Latest revision as of 22:53, 4 July 2013
Problem
is a rectangle, is the center of the circle, and is on the circle. If and , then the area of the shaded region is between
Solution
The area of the shaded region is equal to the area of the quarter circle with the area of the rectangle taken away. The area of the rectangle is , so we just need the quarter circle.
Applying the Pythagorean Theorem to , we have Since is a rectangle,
Clearly is a radius of the circle, so the area of the whole circle is and the area of the quarter circle is .
Finally, the shaded region is so the answer is
See Also
1987 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.