Difference between revisions of "1951 AHSME Problems/Problem 32"
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<math> \textbf{(A)}\ \text{equal to }AB\qquad\textbf{(B)}\ \text{equal to }AB\sqrt{2}\qquad\textbf{(C)}\ \geq AB\sqrt{2}\qquad\textbf{(D)}\ \leq AB\sqrt{2} </math> <math> | <math> \textbf{(A)}\ \text{equal to }AB\qquad\textbf{(B)}\ \text{equal to }AB\sqrt{2}\qquad\textbf{(C)}\ \geq AB\sqrt{2}\qquad\textbf{(D)}\ \leq AB\sqrt{2} </math> <math> | ||
\textbf{(E)}\ AB^{2} </math> | \textbf{(E)}\ AB^{2} </math> | ||
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==Solution== | ==Solution== | ||
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[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
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Latest revision as of 12:26, 5 July 2013
Problem
If is inscribed in a semicircle whose diameter is , then must be
Solution
Because is the diameter of the semi-circle, it follows that . Now we can try to eliminate all the solutions except for one by giving counterexamples.
Set point anywhere on the perimeter of the semicircle except on . By triangle inequality, , so is wrong. Set point on the perimeter of the semicircle infinitesimally close to , and so almost equals , therefore is wrong. Because we proved that can be very close to in case , it follows that is wrong. Because we proved that can be very close to in case , it follows that is wrong. Therefore, the only possible case is
See Also
1951 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 31 |
Followed by Problem 33 | |
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All AHSME Problems and Solutions |
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