Mock AIME 2 2006-2007 Problems/Problem 14
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Contents
Problem
In triangle , and . Given that , and intersect at and are an angle bisector, median, and altitude of the triangle, respectively, compute the length of
Solution
Let .
By the Angle Bisector Theorem, .
Let . Then by the Pythagorean Theorem, and . Subtracting the former equation from the latter to eliminate , we have so . Since , . We can solve these equations for and in terms of to find that and .
Now, by Ceva's Theorem, , so and . Plugging in the values we previously found,
so
and
which yields finally .
See Also
Mock AIME 2 2006-2007 (Problems, Source) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Problem Source
4everwise thought of this problem after reading the first chapter of Geometry Revisited.