1992 AHSME Problems/Problem 26
Problem
Semicircle has center and radius . Point is on and . Extend and to and , respectively, so that circular arcs and have and as their respective centers. Circular arc has center . The area of the shaded "smile" , is
Solution
The area of the entire outer shape is the area of sector , plus the area of sector , minus the area of triangle (since it is part of both sectors), plus the area of sector . We know , so the sector angles for and are degrees, and the radius of both of them is . The radius of is , and can be found using Pythagoras in triangle , giving and , so after doing all the calculations, the area of the entire outer shape is . To get the area of the smile, we need to subtract the area of semicircle , which is , so the answer is = .
See also
1992 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 25 |
Followed by Problem 27 | |
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