Problem 45

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Problem 45: (2022 AMC 10a, 15)

Quadrilateral $ABCD$ with side lengths $AB=7$, $BC=24$, $CD=20,$. $DA=15$ is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form $\frac{a\pi-b}{c},$ where $a,b,$ and $c$ are positive integers such that $a$ and $c$ have no common prime factor. What is $a+b+c?$ $\textbf{(A) } 260 \qquad$ $ \textbf{(B) } 855 \qquad$ $ \textbf{(C) } 1235 \qquad$ $ \textbf{(D) } 1565 \qquad$ $ \textbf{(E) } 1997$

Solution

Exercise 45: Solution

Quadrilateral $ABCD$ with side lengths $AB=33, BC=56, CD=39, DA=52$ is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written as $\frac{a\pi-b}c$, where $a,b,c$ are positive integers with $\gcd(a,c)=1$. What is $a+b+c$?

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