2017 AMC 12B Problems/Problem 24
Problem
Quadrilateral has right angles at and , Triangle ~ Triangle , and . There is a point in the interior of such that Triangle ~ Triangle and the area of Triangle is times the area of Triangle . What is
Solution
Let , , and . Note that . By the Pythagorean Theorem, . Since , the ratios of side lengths must be equal. Since , and . Let F be a point on such that is an altitude of triangle . Note that , so and can be calculated. Solving for these lengths gives and . Since and form altitudes of triangles and , respectively, the areas of these triangles can be calculated. Additionally, the area of triangle can be calculated, as it is a right triangle. Solving for each of these yields: Therefore, the answer is
See Also
2017 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
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