Difference between revisions of "2010 AMC 12A Problems/Problem 17"
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Revision as of 17:29, 11 February 2010
Problem
Equiangular hexagon has side lengths and . The area of is of the area of the hexagon. What is the sum of all possible values of ?
Solution
It is clear that is an equilateral triangle. From the Law of Cosines, we get that . Therefore, the area of is .
If we extend , and so that and meet at , and meet at , and and meet at , we find that hexagon is formed by taking equilateral triangle of side length and removing three equilateral triangles, , and , of side length . The area of is therefore .
Based on the initial conditions, . Simplifying, we get , so by Vieta's Formulas we know that the sum of the possible value of is .