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Difference between revisions of "2022 AMC 12B Problems/Problem 19"

m (→‎Solution 2 (Also Law of Cosines, but with one less computation))
m (When we refer to the length of a line segment, we do not use overline. Fixed that.)
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== Problem ==
 
== Problem ==
  
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In <math>\triangle ABC</math> medians <math>\overline{\rm AD}</math> and <math>\overline{\rm BE}</math> intersect at <math>G</math> and <math>\triangle AGE</math> is equilateral. Then <math>\cos(C)</math> can be written as <math>\frac{m\sqrt{p}}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers and <math>p</math> is a positive integer not divisible by the square of any prime. What is <math>m+n+p</math>?
+
In <math>\triangle{ABC}</math> medians <math>\overline{AD}</math> and <math>\overline{BE}</math> intersect at <math>G</math> and <math>\triangle{AGE}</math> is equilateral. Then <math>\cos(C)</math> can be written as <math>\frac{m\sqrt p}n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers and <math>p</math> is a positive integer not divisible by the square of any prime. What is <math>m+n+p?</math>
  
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<math>\textbf{(A)}\ 44 \qquad
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<math>\textbf{(A) }44 \qquad \textbf{(B) }48 \qquad \textbf{(C) }52 \qquad \textbf{(D) }56 \qquad \textbf{(E) }60</math>
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\textbf{(B)}\ 48 \qquad
 
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\textbf{(C)}\ 52 \qquad
 
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\textbf{(D)}\ 56 \qquad
 
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\textbf{(E)}\ 60 \qquad</math>
 
  
 
== Diagram ==
 
== Diagram ==
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</asy>
 
</asy>
  
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==Solution 1: Law of Cosines==
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==Solution 1 (Law of Cosines)==
  
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Let <math>\overline{AG}=\overline{AE}=\overline{EG}=2x</math>. Since <math>E</math> is the midpoint of <math>\overline{AC}</math>, <math>\overline{EC}</math> must also be <math>2x</math>.
+
Let <math>AG=AE=EG=2x</math>. Since <math>E</math> is the midpoint of <math>\overline{AC}</math>, we must have <math>\EC=2x</math>.
  
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Since the centroid splits the median in a <math>2:1</math> ratio, <math>\overline{GD}</math> must be equal to <math>x</math> and <math>\overline{BG}</math> must be equal to <math>4x</math>.
+
Since the centroid splits the median in a <math>2:1</math> ratio, <math>GD=x</math> and <math>BG=4x</math>.
  
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Applying Law of Cosines on <math>\triangle{}ADC</math> and <math>\triangle{}AGB</math> yields <math>\overline{AB}=\sqrt{28}x</math> and <math>\overline{CD}=\overline{BD}=\sqrt{13}x</math>. Finally, applying Law of Cosines on <math>\triangle{}ABC</math> yields <math>\cos{C}=\frac{5}{2\sqrt{13}}=\frac{5\sqrt{13}}{26}</math>. The requested sum is <math>5+13+26=44</math>.
+
Applying Law of Cosines on <math>\triangle ADC</math> and <math>\triangle{}AGB</math> yields <math>AB=\sqrt{28}x</math> and <math>CD=BD=\sqrt{13}x</math>. Finally, applying Law of Cosines on <math>\triangle ABC</math> yields <math>\cos(C)=\frac{5}{2\sqrt{13}}=\frac{5\sqrt{13}}{26}</math>. The requested sum is <math>5+13+26=44</math>.
  
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==Solution 2 (Also Law of Cosines, but with one less computation) ==
+
==Solution 2 (Law of Cosines: One Fewer Step) ==
  
 
Let <math>AG = 1</math>. Since <math>\frac{BG}{GE}=2</math> (as <math>G</math> is the centroid), <math>BE = 3</math>. Also, <math>EC = 1</math> and <math>\angle{BEC} = 120^{\circ}</math>. By the law of cosines (applied on <math>\bigtriangleup BEC</math>), <math>BC = \sqrt{13}</math>.
 
Let <math>AG = 1</math>. Since <math>\frac{BG}{GE}=2</math> (as <math>G</math> is the centroid), <math>BE = 3</math>. Also, <math>EC = 1</math> and <math>\angle{BEC} = 120^{\circ}</math>. By the law of cosines (applied on <math>\bigtriangleup BEC</math>), <math>BC = \sqrt{13}</math>.

Revision as of 00:29, 10 January 2023

Problem

In $\triangle{ABC}$ medians $\overline{AD}$ and $\overline{BE}$ intersect at $G$ and $\triangle{AGE}$ is equilateral. Then $\cos(C)$ can be written as $\frac{m\sqrt p}n$, where $m$ and $n$ are relatively prime positive integers and $p$ is a positive integer not divisible by the square of any prime. What is $m+n+p?$

$\textbf{(A) }44 \qquad \textbf{(B) }48 \qquad \textbf{(C) }52 \qquad \textbf{(D) }56 \qquad \textbf{(E) }60$

Diagram

[asy] import geometry; unitsize(2cm); real arg(pair p) { return atan2(p.y, p.x) * 180/pi; } pair G=(0,0),E=(1,0),A=(1/2,sqrt(3)/2),D=1.5*G-0.5*A,C=2*E-A,B=2*D-C; pair t(pair p) { return rotate(-arg(dir(B--C)))*p; } path t(path p) { return rotate(-arg(dir(B--C)))*p; } void d(path p, pen q = black+linewidth(1.5)) { draw(t(p),q); } void o(pair p, pen q = 5+black) { dot(t(p),q); } void l(string s, pair p, pair d) { label(s, t(p),d); } d(A--B--C--cycle); d(A--D); d(B--E); o(A); o(B); o(C); o(D); o(E); o(G); l("$A$",A,N); l("$B$",B,SW); l("$C$",C,SE); l("$D$",D,S); l("$E$",E,NE); l("$G$",G,NW); [/asy]

Solution 1 (Law of Cosines)

Let $AG=AE=EG=2x$. Since $E$ is the midpoint of $\overline{AC}$, we must have $\EC=2x$ (Error compiling LaTeX. Unknown error_msg).

Since the centroid splits the median in a $2:1$ ratio, $GD=x$ and $BG=4x$.

Applying Law of Cosines on $\triangle ADC$ and $\triangle{}AGB$ yields $AB=\sqrt{28}x$ and $CD=BD=\sqrt{13}x$. Finally, applying Law of Cosines on $\triangle ABC$ yields $\cos(C)=\frac{5}{2\sqrt{13}}=\frac{5\sqrt{13}}{26}$. The requested sum is $5+13+26=44$.

Solution 2 (Law of Cosines: One Fewer Step)

Let $AG = 1$. Since $\frac{BG}{GE}=2$ (as $G$ is the centroid), $BE = 3$. Also, $EC = 1$ and $\angle{BEC} = 120^{\circ}$. By the law of cosines (applied on $\bigtriangleup BEC$), $BC = \sqrt{13}$.

Applying the law of cosines again on $\bigtriangleup BEC$ gives $\cos{\angle{C}} = \frac{1 + 13 - 9}{2\sqrt{13}} = \frac{5\sqrt{13}}{26}$, so the answer is $\fbox{\textbf{(A)}\ 44}$.

~ Bxiao31415

See Also

2022 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
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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