Difference between revisions of "1967 AHSME Problems/Problem 33"
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[[Category: Intermediate Geometry Problems]] | [[Category: Intermediate Geometry Problems]] | ||
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Latest revision as of 00:40, 16 August 2023
Problem
In this diagram semi-circles are constructed on diameters , , and , so that they are mutually tangent. If , then the ratio of the shaded area to the area of a circle with as radius is:
Solution
To make the problem much simpler while staying in the constraints of the problem, position point halfway between and . Then, call . The area of the shaded region is then Because the area of the circle with as radius is . Our ratio is then
Which corresponds with answer
See also
1967 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 32 |
Followed by Problem 34 | |
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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 | ||
All AHSME Problems and Solutions |
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