Difference between revisions of "2021 AMC 10A Problems/Problem 21"

Revision as of 14:53, 14 October 2023

Problem

Let $ABCDEF$ be an equiangular hexagon. The lines $AB, CD,$ and $EF$ determine a triangle with area $192\sqrt{3}$ , and the lines $BC, DE,$ and $FA$ determine a triangle with area $324\sqrt{3}$ . The perimeter of hexagon $ABCDEF$ can be expressed as $m +n\sqrt{p}$ , where $m, n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m + n + p$ ?

$\textbf{(A)} ~47\qquad\textbf{(B)} ~52\qquad\textbf{(C)} ~55\qquad\textbf{(D)} ~58\qquad\textbf{(E)} ~63$

Diagram

$[asy] /* Made by MRENTHUSIASM */ size(250); path P1, P2; P1 = scale(16sqrt(3))*polygon(3); P2 = shift(3,3)*scale(36)*rotate(180)*polygon(3); draw(P1, dashed+black); draw(P2, dashed+black); pair A, B, C, D, E, F; E = intersectionpoints(P1,P2)[0]; F = intersectionpoints(P1,P2)[1]; A = intersectionpoints(P1,P2)[2]; B = intersectionpoints(P1,P2)[3]; C = intersectionpoints(P1,P2)[4]; D = intersectionpoints(P1,P2)[5]; filldraw(A--B--C--D--E--F--cycle,yellow); dot("$E$",E,1.5*dir(0),linewidth(4)); dot("$F$",F,1.5*dir(60),linewidth(4)); dot("$A$",A,1.5*dir(120),linewidth(4)); dot("$B$",B,1.5*dir(180),linewidth(4)); dot("$C$",C,1.5*dir(-120),linewidth(4)); dot("$D$",D,1.5*dir(-60),linewidth(4)); dot(16sqrt(3)*dir(90)^^16sqrt(3)*dir(210)^^16sqrt(3)*dir(330),linewidth(4)); dot((3,3)+36*dir(30)^^(3,3)+36*dir(150)^^(3,3)+36*dir(270),linewidth(4)); [/asy]$ ~MRENTHUSIASM

Solution 1

Let $P,Q,R,X,Y,$ and $Z$ be the intersections $\overleftrightarrow{AB}\cap\overleftrightarrow{CD},\overleftrightarrow{CD}\cap\overleftrightarrow{EF},\overleftrightarrow{EF}\cap\overleftrightarrow{AB},\overleftrightarrow{BC}\cap\overleftrightarrow{DE},\overleftrightarrow{DE}\cap\overleftrightarrow{FA},$ and $\overleftrightarrow{FA}\cap\overleftrightarrow{BC},$ respectively.

The sum of the interior angles of any hexagon is $720^\circ.$ Since hexagon $ABCDEF$ is equiangular, each of its interior angles is $720^\circ\div6=120^\circ.$ By angle chasing, we conclude that the interior angles of $\triangle PBC,\triangle QDE,\triangle RFA,\triangle XCD,\triangle YEF,$ and $\triangle ZAB$ are all $60^\circ.$ Therefore, these triangles are all equilateral triangles, from which $\triangle PQR$ and $\triangle XYZ$ are both equilateral triangles.

We are given that $\begin{alignat*}{8} [PQR]&=\frac{\sqrt{3}}{4}\cdot PQ^2&&=192\sqrt3, \\ [XYZ]&=\frac{\sqrt{3}}{4}\cdot YZ^2&&=324\sqrt3, \end{alignat*}$ so we get $PQ=16\sqrt3$ and $YZ=36,$ respectively.

By equilateral triangles and segment addition, we find the perimeter of hexagon $ABCDEF:$ $\begin{align*} AB+BC+CD+DE+EF+FA&=AZ+PC+CD+DQ+YF+FA \\ &=(YF+FA+AZ)+(PC+CD+DQ) \\ &=YZ+PQ \\ &=36+16\sqrt{3}. \end{align*}$ Finally, the answer is $36+16+3=\boxed{\textbf{(C)} ~55}.$

~sugar_rush ~MRENTHUSIASM

Solution 2

Let the length $AB=x, BC=y.$ Then, we have $\begin{align*} (y+2x)^2\cdot\frac{\sqrt 3}{4}&=324\sqrt3, \\ (x+2y)^2\cdot\frac{\sqrt 3}{4}&=192\sqrt3. \end{align*}$ We get $\begin{align*} y+2x&=36, \\ x+2y&=16\sqrt3. \end{align*}$ We want $3x+3y,$ and it follows that $3x+3y=(y+2x)+(x+2y)=36+16\sqrt3.$ Finally, the answer is $36+16+3=\boxed{\textbf{(C)} ~55}.$

~mathboy282

Video Solution by OmegaLearn (Angle Chasing and Equilateral Triangles)

https://youtu.be/ptBwDcmDaLA

~ pi_is_3.14

Video Solution by TheBeautyofMath

https://youtu.be/8qcbZ8c7fHg

~IceMatrix

Video Solution by MRENTHUSIASM (English & Chinese)

https://www.youtube.com/watch?v=0n8EAu2VAiM

~MRENTHUSIASM

@@ Line 58: / Line 58: @@
 Let the length <math>AB=x, BC=y.</math> Then, we have
 <cmath>\begin{align*}
-(y+2x)^2\frac{\sqrt 3}{4}&=324\sqrt3, \\
+(y+2x)^2\cdot\frac{\sqrt 3}{4}&=324\sqrt3, \\
-(x+2y)^2\frac{\sqrt 3}{4}&=192\sqrt3.
+(x+2y)^2\cdot\frac{\sqrt 3}{4}&=192\sqrt3.
 \end{align*}</cmath>
 We get
 <cmath>\begin{align*}
-y+2x&=36,
+y+2x&=36, \\
 x+2y&=16\sqrt3.
 \end{align*}</cmath>
-We want <math>3x+3y,</math> so we have:
+We want <math>3x+3y,</math> and it follows that <cmath>3x+3y=(y+2x)+(x+2y)=36+16\sqrt3.</cmath>
-<cmath>3x+3y=(y+2x)+(x+2y)=36+16\sqrt3.</cmath>
+Finally, the answer is <math>36+16+3=\boxed{\textbf{(C)} ~55}.</math>
-Our answer is thus 55.
 ~mathboy282

2021 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
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