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2021 Fall AMC 12A Problems/Problem 14

Revision as of 01:09, 7 January 2022 by MRENTHUSIASM (talk | contribs) (→‎Solution 1 (Law of Cosines and Equilateral Triangle Area))

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 1

Isosceles triangles $ABF$, $CBD$, and $EDF$ are congruent by SAS congruence. 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 $\frac{1}{2}s \cdot s \cdot \sin{30}=\frac{1}{4}s^2$ by the fourth formula here.

By the Law of Cosines on triangle $ABF$, $BF^2=s^2+s^2-2s^2\cos{30^\circ}=2s^2-\sqrt{3}s^2$. 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 $6s=\boxed{\textbf{(E)} \: 12\sqrt{3}}$.

Solution 2

We will be referring to the following diagram.

[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,G=(1/2)*(C+E); draw(C--D--E--F--A--B--cycle,p); draw(C--E--A--C); draw(D--G); dot(A,q); dot(B,q); dot(C,q); dot(D,q); dot(E,q); dot(F,q); dot(G,q); label("$C$",C,2*S); label("$D$",D,2*N); label("$E$",E,2*S); label("$F$",F,2*dir(0)); label("$A$",A,2*N); label("$G$",G,2*S); label("$B$",B,2*W); [/asy]

Observe that \begin{align}6\sqrt3=[ACE]-3\cdot[DCE]\end{align}. Letting $x=CD,$ the perimeter will be $6x.$

We know that $\angle CDG=75^{\circ}$ and using such, we have \[CG=x\sin(75^{\circ})=\frac{\sqrt6+\sqrt2}{4}x\] and \[DG=x\cos(75^{\circ})=\frac{\sqrt6-\sqrt2}{4}x.\] Thus, we have \begin{align*}[ACE]&=\frac{\sqrt3}{4}\left(2\cdot CG\right)^2\\ &=\frac{\sqrt3}{4}(2+\sqrt3)x^2 \\ &=\frac{3+2\sqrt3}{4} x^2.\end{align*} Computing the area of $DCE,$ we have \begin{align*}[DCE]&=\frac12 \cdot 2\cdot CG\cdot DG \\ &=CG\cdot DG\\ &=\frac{x^2}{4}.\end{align*} Plugging back into $(1),$ we have \begin{align*}6\sqrt3&=\frac{3+2\sqrt3}{4} x^2 -\frac{3x^2}{4} \\ &=\frac{\sqrt3}{2}x^2\end{align*} which means $x=2\sqrt3$ and \[6x=\boxed{\textbf{(E)} \: 12\sqrt3}.\]

~ASAB

See Also

2021 Fall AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
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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