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Difference between revisions of "2010 AMC 12A Problems/Problem 8"

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<cmath>\sin{\angle ACB}=\frac{AB\cdot\sin{\angle ABC}}{AC}</cmath>
 
<cmath>\sin{\angle ACB}=\frac{AB\cdot\sin{\angle ABC}}{AC}</cmath>
 
<cmath>\sin{\angle ACB}=2\cdot\sin{\angle ABC}</cmath>
 
<cmath>\sin{\angle ACB}=2\cdot\sin{\angle ABC}</cmath>
 +
Now it's easy to see that the equality holds
  
 
== See also ==
 
== See also ==

Revision as of 13:25, 17 November 2019

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

AMC 2010 12A Problem 8.png


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

Applying the Law of Sines on $\bigtriangleup ABC$, we have \[\frac{\sin{\angle ABC}}{AC}=\frac{\sin{\angle ACB}}{AB}\] \[\sin{\angle ACB}=\frac{AB\cdot\sin{\angle ABC}}{AC}\] \[\sin{\angle ACB}=2\cdot\sin{\angle ABC}\] Now it's easy to see that the equality holds

See also

2010 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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