# 2010 AMC 12A Problems/Problem 8

## Problem

Triangle $ABC$ has $AB=2 \cdot AC$. Let $D$ and $E$ be on $\overline{AB}$ and $\overline{BC}$, respectively, such that $\angle BAE = \angle ACD$. Let $F$ be the intersection of segments $AE$ and $CD$, and suppose that $\triangle CFE$ is equilateral. What is $\angle ACB$? $\textbf{(A)}\ 60^\circ \qquad \textbf{(B)}\ 75^\circ \qquad \textbf{(C)}\ 90^\circ \qquad \textbf{(D)}\ 105^\circ \qquad \textbf{(E)}\ 120^\circ$

## Solution Let $\angle BAE = \angle ACD = x$. \begin{align*}\angle BCD &= \angle AEC = 60^\circ\\ \angle EAC + \angle FCA + \angle ECF + \angle AEC &= \angle EAC + x + 60^\circ + 60^\circ = 180^\circ\\ \angle EAC &= 60^\circ - x\\ \angle BAC &= \angle EAC + \angle BAE = 60^\circ - x + x = 60^\circ\end{align*}

Since $\frac{AC}{AB} = \frac{1}{2}$, triangle $ABC$ is a $30-60-90$ triangle, so $\angle BCA = \boxed{90^\circ\,\textbf{(C)}}$.

## Solution 2(Trig and Angle Chasing)

Let $AB=2a, AC=a$. Let $\angle BAE=\angle ACD=x$. Because $\triangle CFE$ is equilateral, we get $\angle FCE=60$, so $\angle ACB=60+x$. Because $\triangle CFE$ is equilateral, we get $\angle CFE=60$. Angles $AFD$ and $CFE$ are vertical, so $\angle AFD=60$. By triangle $ADF$, we have $\angle ADF=120-x$, and because of line $AB$, we have $\angle BDC=60+x$. Because Of line $BC$, we have $\angle AEB=120$, and by line $CD$, we have $\angle DFE=120$. By quadrilateral $BDFE$, we have $\angle ABC=60-x$.

By the Law of Sines, we have $\frac{\sin(60-x)}{a}=\frac{\sin(60+x)}{2a}\implies$\sin(60-x)=\frac{\sin(60+x)}{2}\implies 2\sin(60-x)=\sin(60+x) $. By the sine addition formula(which states$\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b) $by the way), we have$2(\sin(60)\cos(-x)+\cos(60)\sin(-x))=\sin(60)\cos(x)+\cos(60)\sin(x) $. Because cosine is an even function, and sine is an odd function, we have$2\sin(60)\cos(x)-2\cos(60)\sin(x)=\sin(60)\cos(x)+\cos(60)\sin(x) \implies \sin(60)\cos(x)=3\cos(60)\sin(x) $. We know that$\sin(60)=\frac{\sqrt{3}}{2} $, and$\cos(60)=\frac{1}{2} $, hence$\frac{\sqrt{3}}{2}\cos(x)=\frac{3}{2}\sin(x)\implies $\tan(x)=\frac{\sqrt(3)}{3}$. The only value of $x$ that satisfies $60+x<180$(because $60+x$ is an angle of the triangle) $is$x=30^{\circ} $. We seek to find$\angle ACB $, which as we found before is$60+x $, which is$90\$. The answer is (C)

-vsamc

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