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Difference between revisions of "2021 Fall AMC 12A Problems/Problem 14"

(→‎Solution (Law of Cosines and Equilateral Triangle Area))
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==Solution (Law of Cosines and Equilateral Triangle Area)==
 
==Solution (Law of Cosines and Equilateral Triangle Area)==
  
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Isosceles triangles <math>ABE</math>, <math>CBD</math>, and <math>EDF</math> are identical by [[SAS_Similarity|SAS similarity]]. <math>BF=BD=DF</math> by [[CPCTC]], and triangle <math>BDF</math> is equilateral.  
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Isosceles triangles <math>ABF</math>, <math>CBD</math>, and <math>EDF</math> are congruent by [[SAS_Similarity|SAS similarity]]. By [[CPCTC]], <math>BF=BD=DF</math>, so triangle <math>BDF</math> is equilateral.  
  
 
Let the side length of the hexagon be <math>s</math>.  
 
Let the side length of the hexagon be <math>s</math>.  
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The area of each isosceles triangle is <math>\frac{1}{2}s^2\sin{30}=\frac{1}{4}s^2</math> by the fourth formula [[Area#Area_of_Triangle|here]].
 
The area of each isosceles triangle is <math>\frac{1}{2}s^2\sin{30}=\frac{1}{4}s^2</math> by the fourth formula [[Area#Area_of_Triangle|here]].
  
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By the [[Law of Cosines]], the square of the side length of equilateral triangle BDF is <math>2s^2-2x^2\cos{30}=(2-\sqrt{3})s^2</math>. Hence, the [[Area_of_an_equilateral_triangle|area of the equilateral triangle]] is <math>\frac{\sqrt{3}}{4}(2-\sqrt{3})s^2=\frac{\sqrt{3}}{2}s^2-\frac{3}{4}s^2</math>.  
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By the [[Law of Cosines]] on triangle <math>ABF</math>, <math>BF^2=2s^2-2s^2\cos{30^\circ}=2s^2-\sqrt{3}s^2</math>. Hence, the [[Area_of_an_equilateral_triangle|area of the equilateral triangle]] <math>BDF</math> is <math>\frac{\sqrt{3}}{4}\left(2s^2-\sqrt{3}s^2\right)=\frac{\sqrt{3}}{2}s^2-\frac{3}{4}s^2</math>.  
  
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So, the total area of the hexagon is the area of the equilateral triangle plus thrice the area of each isosceles triangle or <math>\frac{\sqrt{3}}{2}s^2-\frac{3}{4}s^2+3(\frac{1}{4}s^2)=\frac{\sqrt{3}}{2}s^2=6\sqrt{3}</math>. The perimeter is <math>6s=\boxed{12\sqrt{3}}</math>.
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The total area of the hexagon is thrice the area of each isosceles triangle plus the area of the equilateral triangle, or <math>3\left(\frac{1}{4}s^2\right)+\frac{\sqrt{3}}{2}s^2-\frac{3}{4}s^2=\frac{\sqrt{3}}{2}s^2=6\sqrt{3}</math>. Hence, <math>s=2\sqrt{3}</math> and the perimeter is <math>6s=\boxed{\textbf{(E)} \: 12\sqrt{3}}</math>.
  
 
{{AMC12 box|year=2021 Fall|ab=A|num-b=13|num-a=15}}
 
{{AMC12 box|year=2021 Fall|ab=A|num-b=13|num-a=15}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 21:08, 23 November 2021

Problem

In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\circ$. The enclosed area of the hexagon is $6\sqrt{3}$. What is the perimeter of the hexagon? [asy] size(10cm); pen p=black+linewidth(1),q=black+linewidth(5); pair C=(0,0),D=(cos(pi/12),sin(pi/12)),E=rotate(150,D)*C,F=rotate(-30,E)*D,A=rotate(150,F)*E,B=rotate(-30,A)*F; draw(C--D--E--F--A--B--cycle,p); dot(A,q); dot(B,q); dot(C,q); dot(D,q); dot(E,q); dot(F,q); label("$C$",C,2*S); label("$D$",D,2*S); label("$E$",E,2*S); label("$F$",F,2*dir(0)); label("$A$",A,2*N); label("$B$",B,2*W); [/asy] $\textbf{(A)} \: 4 \qquad \textbf{(B)} \: 4\sqrt3 \qquad \textbf{(C)} \: 12 \qquad \textbf{(D)} \: 18 \qquad \textbf{(E)} \: 12\sqrt3$

Solution (Law of Cosines and Equilateral Triangle Area)

Isosceles triangles $ABF$, $CBD$, and $EDF$ are congruent by SAS similarity. By CPCTC, $BF=BD=DF$, so triangle $BDF$ is equilateral.

Let the side length of the hexagon be $s$.

The area of each isosceles triangle is $\frac{1}{2}s^2\sin{30}=\frac{1}{4}s^2$ by the fourth formula here.

By the Law of Cosines on triangle $ABF$, $BF^2=2s^2-2s^2\cos{30^\circ}=2s^2-\sqrt{3}s^2$. Hence, the area of the equilateral triangle $BDF$ is $\frac{\sqrt{3}}{4}\left(2s^2-\sqrt{3}s^2\right)=\frac{\sqrt{3}}{2}s^2-\frac{3}{4}s^2$.

The total area of the hexagon is thrice the area of each isosceles triangle plus the area of the equilateral triangle, or $3\left(\frac{1}{4}s^2\right)+\frac{\sqrt{3}}{2}s^2-\frac{3}{4}s^2=\frac{\sqrt{3}}{2}s^2=6\sqrt{3}$. Hence, $s=2\sqrt{3}$ and the perimeter is $6s=\boxed{\textbf{(E)} \: 12\sqrt{3}}$.

2021 Fall AMC 12A (Problems • Answer Key • Resources)
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Problem 15
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