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Difference between revisions of "2024 AMC 12B Problems/Problem 19"

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==Problem 19==
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Equilateral <math>\triangle ABC</math> with side length <math>14</math> is rotated about its center by angle <math>\theta</math>, where <math>0 < \theta < 60^{\circ}</math>, to form <math>\triangle DEF</math>. See the figure. The area of hexagon <math>ADBECF</math> is <math>91\sqrt{3}</math>. What is <math>\tan\theta</math>?
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<asy>
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// Credit to shihan for this diagram.
  
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defaultpen(fontsize(13)); size(200);
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pair O=(0,0),A=dir(225),B=dir(-15),C=dir(105),D=rotate(38.21,O)*A,E=rotate(38.21,O)*B,F=rotate(38.21,O)*C;
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draw(A--B--C--A,gray+0.4);draw(D--E--F--D,gray+0.4); draw(A--D--B--E--C--F--A,black+0.9); dot(O); dot("$A$",A,dir(A)); dot("$B$",B,dir(B)); dot("$C$",C,dir(C)); dot("$D$",D,dir(D)); dot("$E$",E,dir(E)); dot("$F$",F,dir(F));
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</asy>
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<math>\textbf{(A)}~\frac{3}{4}\qquad\textbf{(B)}~\frac{5\sqrt{3}}{11}\qquad\textbf{(C)}~\frac{4}{5}\qquad\textbf{(D)}~\frac{11}{13}\qquad\textbf{(E)}~\frac{7\sqrt{3}}{13}</math>
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==Solution #1==
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let O be circumcenter of the equilateral triangle
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OF = <math>\frac{14\sqrt{3}}{3} </math>
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2(Area(OFC) + Area (OCE)) = <cmath>  OF^2 * sin(\theta) + OF^2 * sin(120 - \theta)    </cmath>
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<cmath> = \frac{14^2 * 3}{9} (  sin(\theta)  +  sin(120 - \theta) )  </cmath>
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<cmath> = \frac{196}{3}  (  sin(\theta)  +  sin(120 - \theta) )  </cmath>
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<cmath> = 2 * {\frac{1}{3}  } * Area  (ABCDEF) = 2* \frac{91\sqrt{3}}{3} </cmath>
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  <cmath> sin(\theta)  +  sin(120 - \theta) = \frac{13\sqrt{3}}{14}  </cmath>
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  <cmath> sin(\theta)  +  \frac{ \sqrt{3}}{2}cos(  \theta) +\frac{ \sqrt{1}}{2}sin(  \theta) = \frac{13\sqrt{3}}{14}  </cmath>
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<cmath>    \sqrt{3} sin(  \theta) + cos(  \theta) = \frac{13 }{14}  </cmath>
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<cmath>  cos(  \theta)  = \frac{13 }{14}  - \sqrt{3} sin(  \theta) </cmath>
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<cmath>  \frac{169 }{49}  - \frac{26\sqrt{3} }{7} sin(  \theta)  + 4 sin(  \theta)^2 =0 </cmath>
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<cmath>    sin(  \theta)  = \frac{5\sqrt{3} }{14}  or \frac{4\sqrt{3} }{7}  </cmath>
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<math>\frac{4\sqrt{3} }{7} </math> is invalid given <math>\theta</math><60
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<cmath>  cos(  \theta)  = \frac{11 }{14}  </cmath>
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<cmath>  tan(  \theta)  = \frac{5\sqrt{3} }{11} </cmath>
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<math>\boxed{B -34} </math>. 
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~[https://artofproblemsolving.com/wiki/index.php/User:Cyantist luckuso]
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==See also==
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{{AMC12 box|year=2024|ab=B|num-b=18|num-a=20}}
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{{MAA Notice}}

Revision as of 02:59, 14 November 2024

Problem 19

Equilateral $\triangle ABC$ with side length $14$ is rotated about its center by angle $\theta$, where $0 < \theta < 60^{\circ}$, to form $\triangle DEF$. See the figure. The area of hexagon $ADBECF$ is $91\sqrt{3}$. What is $\tan\theta$? [asy] // Credit to shihan for this diagram. defaultpen(fontsize(13)); size(200); pair O=(0,0),A=dir(225),B=dir(-15),C=dir(105),D=rotate(38.21,O)*A,E=rotate(38.21,O)*B,F=rotate(38.21,O)*C; draw(A--B--C--A,gray+0.4);draw(D--E--F--D,gray+0.4); draw(A--D--B--E--C--F--A,black+0.9); dot(O); dot("$A$",A,dir(A)); dot("$B$",B,dir(B)); dot("$C$",C,dir(C)); dot("$D$",D,dir(D)); dot("$E$",E,dir(E)); dot("$F$",F,dir(F)); [/asy]

$\textbf{(A)}~\frac{3}{4}\qquad\textbf{(B)}~\frac{5\sqrt{3}}{11}\qquad\textbf{(C)}~\frac{4}{5}\qquad\textbf{(D)}~\frac{11}{13}\qquad\textbf{(E)}~\frac{7\sqrt{3}}{13}$

Solution #1

let O be circumcenter of the equilateral triangle

OF = $\frac{14\sqrt{3}}{3}$

2(Area(OFC) + Area (OCE)) = \[OF^2 * sin(\theta) + OF^2 * sin(120 - \theta)\] 

\[= \frac{14^2 * 3}{9} (   sin(\theta)  +  sin(120 - \theta) )\]
\[= \frac{196}{3}  (   sin(\theta)  +  sin(120 - \theta) )\]
\[= 2 * {\frac{1}{3}  } * Area  (ABCDEF) = 2* \frac{91\sqrt{3}}{3}\]

 \[sin(\theta)  +  sin(120 - \theta) = \frac{13\sqrt{3}}{14}\]
 \[sin(\theta)  +   \frac{ \sqrt{3}}{2}cos(  \theta) +\frac{ \sqrt{1}}{2}sin(  \theta) = \frac{13\sqrt{3}}{14}\]
\[\sqrt{3} sin(  \theta) + cos(  \theta) = \frac{13 }{14}\]
\[cos(  \theta)  = \frac{13 }{14}  - \sqrt{3} sin(  \theta)\]
\[\frac{169 }{49}  - \frac{26\sqrt{3} }{7} sin(  \theta)  + 4 sin(  \theta)^2 =0\]
\[sin(  \theta)  = \frac{5\sqrt{3} }{14}  or \frac{4\sqrt{3} }{7}\]

$\frac{4\sqrt{3} }{7}$ is invalid given $\theta$<60 \[cos( \theta) = \frac{11 }{14}\] \[tan( \theta) = \frac{5\sqrt{3} }{11}\]

$\boxed{B -34}$.


~luckuso

See also

2024 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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