Problem 157

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Problem 157: (2025 AMC 12a, 8)

Pentagon $ABCDE$ is inscribed in a circle, and $\angle BEC = \angle CED = 30^\circ$. Let line $AC$ and line $BD$ intersect at point $F$, and suppose that $AB = 9$ and $AD = 24$. What is $BF$? $\textbf{(A) } \frac{57}{11} \qquad$ $\textbf{(B) } \frac{59}{11} \qquad$ $\textbf{(C) } \frac{60}{11} \qquad$ $\textbf{(D) } \frac{61}{11} \qquad$ $\textbf{(E) } \frac{63}{11}$

Solution

Exercise 157: Solution

Pentagon $ABCDE$ is inscribed in a circle, and $\angle BEC=\angle CED=45^\circ$. Let line $AC$ and line $BD$ intersect at point $F$, and suppose that $AB=6$ and $AD=8$. What is $BF$?

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