2009 AIME II Problems/Problem 10
Four lighthouses are located at points , , , and . The lighthouse at is kilometers from the lighthouse at , the lighthouse at is kilometers from the lighthouse at , and the lighthouse at is kilometers from the lighthouse at . To an observer at , the angle determined by the lights at and and the angle determined by the lights at and are equal. To an observer at , the angle determined by the lights at and and the angle determined by the lights at and are equal. The number of kilometers from to is given by , where , , and are relatively prime positive integers, and is not divisible by the square of any prime. Find + + .
Solution 1
Let be the intersection of and . By the Angle Bisector Theorem, = , so = and = , and + = = , so = , and = . Let be the foot of the altitude from to . It can be seen that triangle is similar to triangle , and triangle is similar to triangle . If = , then = , = , and = . Since + = = , = , and = (by the pythagorean theorem on triangle we sum and ). The answer is + + = .
Solution 2
Extend and to intersect at . Note that since and by ASA congruency we have . Therefore .
By the angle bisector theorem, and . Now we apply Stewart's theorem to find :
and our final answer is .
See Also
2009 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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