2001 AIME I Problems/Problem 4
Contents
Problem
In triangle , angles and measure degrees and degrees, respectively. The bisector of angle intersects at , and . The area of triangle can be written in the form , where , , and are positive integers, and is not divisible by the square of any prime. Find .
Solution 1
Let be the foot of the altitude from to . By simple angle-chasing, we find that , and thus . Now is a right triangle and is a right triangle, so . The area of
and the answer is .
Solution 2
Since has a measure of , and thus has sines and cosines that are easy to compute, we attempt to find and , and use the formula that
By angle chasing, we find that is a triangle with and . Thus .
Switching to the lower triangle , , and , with .
Using the Law of Sines on :
We now plug in , and into the formula for the area:
Thus the answer is
See also
2001 AIME I (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 |