Difference between revisions of "2018 AIME I Problems/Problem 15"
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Revision as of 14:38, 22 December 2018
Problem 15
David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, , which can each be inscribed in a circle with radius . Let denote the measure of the acute angle made by the diagonals of quadrilateral , and define and similarly. Suppose that , , and . All three quadrilaterals have the same area , which can be written in the form , where and are relatively prime positive integers. Find .
Solution
Suppose our four sides lengths cut out arc lengths of , , , and , where . Then, we only have to consider which arc is opposite . These are our three cases, so Our first case involves quadrilateral with , , , and .
Then, by Law of Sines, and . Therefore,
so our answer is .
By S.B. LaTeX by willwin4sure
See Also
2018 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last question | |
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