2011 AIME II Problems/Problem 4
Problem 4
In triangle , . The angle bisector of $\ang A$ (Error compiling LaTeX. ! Undefined control sequence.) intersects at point , and point is the midpoint of . Let be the point of the intersection of and . The ratio of to can be expressed in the form , where and are relatively prime positive integers. Find .
Solution
Solution 1
Let be on such that . It follows that , so by the Angle Bisector Theorem. Similarly, we see by the midline theorem that . Thus, and .
Solution 2
Assign mass points as follows: by Angle-Bisector Theorem, , so we assign . Since , then , and .
Solution 3
By Menelaus' Theorem on with transversal , [/hide]
See also
2011 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |