2005 AIME I Problems/Problem 7
Problem
In quadrilateral and Given that where and are positive integers, find
Solution
Solution 1
Draw the perpendiculars from and to , labeling the intersection points as and . This forms 2 right triangles, so and . Also, if we draw the horizontal line extending from to a point on the line , we find another right triangle . . The Pythagorean theorem yields that , so . Therefore, , and .
Solution 2
Extend and to an intersection at point . We get an equilateral triangle . We denote the length of a side of as and solve for it using the Law of Cosines: This simplifies to ; the quadratic formula yields the (discard the negative result) same result of .
See also
2005 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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All AIME Problems and Solutions |