2004 AIME II Problems/Problem 13
Problem
Let be a convex pentagon with and Given that the ratio between the area of triangle and the area of triangle is where and are relatively prime positive integers, find
Solution
Let the intersection of and be . Since it follows that is a parallelogram, and so . Also, as , it follows that .
By the Law of Cosines, . Thus the length similarity ratio between and is .
Let and be the lengths of the altitudes in to respectively. Then, the ratio of the areas .
However, , with all three heights oriented in the same direction. Since , it follows that , and from the similarity ratio, . Hence , and the ratio of the areas is . The answer is .
Additional Trigonometry-Free Alternative
Instead of using the Law of Cosines, we can draw a line perpendicular to line BC down from point A until it intersects BC at a point . Since , we can use the triangle to deduce that , and . From here, we can use Pythagorean theorem to deduce that . Then, we can follow with the rest of the solution above.
See also
2004 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.