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

Latest revision as of 19:33, 19 August 2024

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 1

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 $\frac{AC}{AB} = \frac{1}{2}$ and the angle between the hypotenuse and the shorter side is $60^\circ$ , 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: $\frac{\sin(60-x)}{a}=\frac{\sin(60+x)}{2a}\implies 2\sin(60-x)=\sin(60+x)$

By the sine addition formula( $\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)$ ): $2(\sin(60)\cos(-x)+\cos(60)\sin(-x))=\sin(60)\cos(x)+\cos(60)\sin(x)$

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 $90, \text{or} \textbf{(C)}$

-vsamc

Solution 3 (Similar Triangles)

Notice that $\angle AEB=\angle AFC = 120^{\circ}$ and $\angle ACF=\angle BAE$ . Hence, triangle AEB is similar to triangle CFA. Since $AB=2AC$ , $AE=2CF=2FE$ , as triangle CFE is equilateral. Therefore, $AF=FE=FC$ , and since $\angle AFC=120^{\circ}$ , $x=30$ . Thus, the measure of $\angle ACE$ equals to $\angle FCE+\angle ACF=90^{\circ}, \text{or} \textbf{(C)}$ -HarryW

Solution 4

Notice that $\triangle ADF \sim \triangle CDA$ (by AA Similarity.) Since the corresponding angles of a pair of similar triangles are congruent, we have $\angle DAC = \angle DFA = 60^\circ.$ Since $AB = 2 \cdot AC$ and $\angle DFA = 60^\circ,$ we have that $\triangle ABC$ is congruent by SAS to a $30-60-90$ right triangle, which gives the answer $\boxed{\textbf{(C)}}$ . ~clever14710owl

Video Solution by the Beauty of Math

https://youtu.be/kU70k1-ONgM?t=785

Video Solution by OmegaLearn

https://youtu.be/O_o_-yjGrOU?t=58

~ pi_is_3.14

@@ Line 4: / Line 4: @@
 <math>\textbf{(A)}\ 60^\circ \qquad \textbf{(B)}\ 75^\circ \qquad \textbf{(C)}\ 90^\circ \qquad \textbf{(D)}\ 105^\circ \qquad \textbf{(E)}\ 120^\circ</math>
-== Solution ==
+== Solution 1 ==
+<center>[[File:AMC 2010 12A Problem 8.png]]</center>
 Let <math>\angle BAE = \angle ACD = x</math>.
@@ Line 12: / Line 16: @@
   \angle BAC &= \angle EAC + \angle BAE = 60^\circ - x + x = 60^\circ\end{align*}</cmath>
-Since <math>\frac{AC}{AB} = \frac{1}{2}</math>, triangle <math>ABC</math> is a <math>30-60-90</math> triangle, so <math>\angle BCA = \boxed{90^\circ\ \textbf{(C)}}</math>
+Since <math>\frac{AC}{AB} = \frac{1}{2}</math> and the angle between the hypotenuse and the shorter side is <math>60^\circ</math>, triangle <math>ABC</math> is a <math>30-60-90</math> triangle, so <math>\angle BCA = \boxed{90^\circ\,\textbf{(C)}}</math>.
+== Solution 2(Trig and Angle Chasing) ==
+Let <cmath>AB=2a, AC=a</cmath>
+Let <cmath>\angle BAE=\angle ACD=x</cmath>
+Because <math>\triangle CFE</math> is equilateral, we get <math>\angle FCE=60</math>, so <math>\angle ACB=60+x</math>
+Because <math>\triangle CFE</math> is equilateral, we get <math>\angle CFE=60</math>.
+Angles <math>AFD</math> and <math>CFE</math> are vertical, so <math>\angle AFD=60</math>.
+By triangle <math>ADF</math>, we have <math>\angle ADF=120-x</math>, and because of line <math>AB</math>, we have <math>\angle BDC=60+x</math>.
+Because Of line <math>BC</math>, we have <math>\angle AEB=120</math>, and by line <math>CD</math>, we have <math>\angle DFE=120</math>.
+By quadrilateral <math>BDFE</math>, we have <math>\angle ABC=60-x</math>.
+By the Law of Sines:
+<cmath>\frac{\sin(60-x)}{a}=\frac{\sin(60+x)}{2a}\implies 2\sin(60-x)=\sin(60+x)</cmath>
+By the sine addition formula(<math>\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)</math>):
+<cmath>2(\sin(60)\cos(-x)+\cos(60)\sin(-x))=\sin(60)\cos(x)+\cos(60)\sin(x)</cmath>
+Because cosine is an even function, and sine is an odd function, we have <cmath>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)</cmath>
+We know that <math>\sin(60)=\frac{\sqrt{3}}{2}</math>, and <math>\cos(60)=\frac{1}{2}</math>, hence
+<cmath>\frac{\sqrt{3}}{2}\cos(x)=\frac{3}{2}\sin(x)\implies \tan(x)=\frac{\sqrt{3}}{3}</cmath>
+The only value of <math>x</math> that satisfies <math>60+x<180</math>(because <math>60+x</math> is an angle of the triangle) is <math>x=30^{\circ}</math>. We seek to find <math>\angle ACB</math>, which as we found before is <math>60+x</math>, which is <math>90</math>. The answer is <math>90, \text{or} \textbf{(C)}</math>
+-vsamc
+== Solution 3 (Similar Triangles) ==
+Notice that <math>\angle AEB=\angle AFC = 120^{\circ}</math> and <math>\angle ACF=\angle BAE</math>. Hence, triangle AEB is similar to triangle CFA. Since <math>AB=2AC</math>, <math>AE=2CF=2FE</math>, as triangle CFE is equilateral. Therefore, <math>AF=FE=FC</math>, and since <math>\angle AFC=120^{\circ}</math>, <math>x=30</math>. Thus, the measure of <math>\angle ACE</math> equals to <math>\angle FCE+\angle ACF=90^{\circ}, \text{or} \textbf{(C)}</math>
+-HarryW
+== Solution 4 ==
+Notice that <math>\triangle ADF \sim \triangle CDA</math> (by AA Similarity.) Since the corresponding angles of a pair of similar triangles are congruent, we have <math>\angle DAC = \angle DFA = 60^\circ.</math> Since <math>AB = 2 \cdot AC</math> and <math>\angle DFA = 60^\circ,</math> we have that <math>\triangle ABC</math> is congruent by SAS to a <math>30-60-90</math> right triangle, which gives the answer <math>\boxed{\textbf{(C)}}</math>.
+~clever14710owl
+==Video Solution by the Beauty of Math==
+https://youtu.be/kU70k1-ONgM?t=785
+== Video Solution by OmegaLearn ==
+https://youtu.be/O_o_-yjGrOU?t=58
+~ pi_is_3.14
 == See also ==
+{{AMC10 box|year=2010|ab=A|num-b=13|num-a=15}}
 {{AMC12 box|year=2010|num-b=7|num-a=9|ab=A}}
 [[Category:Introductory Geometry Problems]]
 {{MAA Notice}}

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