2019 AMC 10A Problems/Problem 13

Problem

Let $\Delta ABC$ be an isosceles triangle with $BC = AC$ and $\angle ACB = 40^{\circ}$ . Contruct the circle with diameter $\overline{BC}$ , and let $D$ and $E$ be the other intersection points of the circle with the sides $\overline{AC}$ and $\overline{AB}$ , respectively. Let $F$ be the intersection of the diagonals of the quadrilateral $BCDE$ . What is the degree measure of $\angle BFC ?$

$\textbf{(A) } 90 \qquad\textbf{(B) } 100 \qquad\textbf{(C) } 105 \qquad\textbf{(D) } 110 \qquad\textbf{(E) } 120$

Solution

Drawing it out, we see $\angle BDC$ and $\angle BEC$ are right angles, as they are inscribed in a semicircle. Noting that they subtend to the same arc of $180^{\circ}$ , we can find $\angle ECB=20$ and $\angle DBC=50$ by the triangle angle sum. Then we take triangle $BFC$ , and find $\angle BFC=180-50-20=\boxed{(\text{D})110}.$

2019 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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All AMC 10 Problems and Solutions

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2019 AMC 10A Problems/Problem 13

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