1965 AHSME Problems/Problem 30
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[hide]Problem
Let of right triangle be the diameter of a circle intersecting hypotenuse in . At a tangent is drawn cutting leg in . This information is not sufficient to prove that
Solution 1
We will prove every result except for .
By Thales' Theorem, and so . and are both tangents to the same circle, and hence equal. Let . Then , and so . We also have , which implies . This means that , so indeed bisects . We also know that , hence . And as .
Since all of the results except for are true, our answer is .
Solution 2
It's easy to verify that always equals . Since changes depending on the sidelengths of the triangle, we cannot be certain that . Hence our answer is .
See Also
1965 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 29 |
Followed by Problem 31 | |
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