2022 AMC 10A Problems/Problem 15
Problem
Quadrilateral with side lengths is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form where and are positive integers such that and have no common prime factor. What is
Solution
DIAGRAM IN PROGRESS.
WILL BE DONE TOMORROW, WAIT FOR ME THANKS.
Opposite angles of every cyclic quadrilateral are supplementary, so
We claim that so by the Pythagorean Theorem. We can prove the claim by contradiction: Suppose that is false. Then, we can apply the Hinge Theorem to and We get in one triangle, and in the other triangle, arriving at a contradiction. Therefore, the claim must be true.
By the Inscribed Angle Theorem, we conclude that is the diameter of the circle. So, the radius of the circle is
The area of the requested region is Therefore, the answer is
~MRENTHUSIASM
See Also
2022 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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All AMC 10 Problems and Solutions |
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