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

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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>
 
<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>
  
==Solution #1==
+
==Solution 1==
let O be circumcenter of the equilateral triangle  
+
Let O be circumcenter of the equilateral triangle  
  
OF = <math>\frac{14\sqrt{3}}{3} </math>
+
Easily get OF = <math>\frac{14\sqrt{3}}{3} </math>
  
2(Area(OFC) + Area (OCE)) = <cmath>  OF^2 * sin(\theta) + OF^2 * sin(120 - \theta)    </cmath>
+
2(area(<math>\triangle</math>OFC) + area (<math>\triangle</math>OCE)) = <cmath>  OF^2 * sin(\theta) + OF^2 * sin(120 - \theta)    </cmath>
<cmath> = \frac{14^2 * 3}{9} (  sin(\theta)  +  sin(120 - \theta) )  </cmath>
+
<cmath> = \frac{14^2 * 3}{9} (  sin(\theta)  +  sin(120 - \theta) )  </cmath>
<cmath> = \frac{196}{3}  (  sin(\theta)  +  sin(120 - \theta) )  </cmath>
+
<cmath> = \frac{196}{3}  (  sin(\theta)  +  sin(120 - \theta) )  </cmath>
<cmath> = 2 * {\frac{1}{3}  } * Area (ABCDEF) = 2* \frac{91\sqrt{3}}{3} </cmath>
+
<cmath> = 2 * {\frac{1}{3}  } * area (ABCDEF) = 2* \frac{91\sqrt{3}}{3} </cmath>
  
  <cmath> sin(\theta)  +  sin(120 - \theta) = \frac{13\sqrt{3}}{14}  </cmath>
+
<cmath> sin(\theta)  +  sin(120 - \theta) = \frac{13\sqrt{3}}{14}  </cmath>
  <cmath> sin(\theta)  +  \frac{ \sqrt{3}}{2}cos(  \theta) +\frac{ \sqrt{1}}{2}sin(  \theta) = \frac{13\sqrt{3}}{14}  </cmath>
+
<cmath> sin(\theta)  +  \frac{ \sqrt{3}}{2}cos(  \theta) +\frac{ \sqrt{1}}{2}sin(  \theta) = \frac{13\sqrt{3}}{14}  </cmath>
<cmath>     \sqrt{3} sin(  \theta) + cos(  \theta) = \frac{13 }{14}  </cmath>
+
<cmath> \sqrt{3} sin(  \theta) + cos(  \theta) = \frac{13 }{7}  </cmath>
<cmath> cos(  \theta)  = \frac{13 }{14}  - \sqrt{3} sin(  \theta) </cmath>
+
<cmath> cos(  \theta)  = \frac{13 }{7}  - \sqrt{3} sin(  \theta) </cmath>
<cmath> \frac{169 }{49}  - \frac{26\sqrt{3} }{7} sin(  \theta)  + 4 sin(  \theta)^2 =0 </cmath>
+
<cmath> \frac{169 }{49}  - \frac{26\sqrt{3} }{7} sin(  \theta)  + 4 sin(  \theta)^2 =0 </cmath>
<cmath>   sin(  \theta)  = \frac{5\sqrt{3} }{14}  or \frac{4\sqrt{3} }{7}  </cmath>
+
<cmath> sin(  \theta)  = \frac{5\sqrt{3} }{14}  or \frac{4\sqrt{3} }{7}  </cmath>
<math>\frac{4\sqrt{3} }{7} </math> is invalid given <math>\theta</math><60
+
<math>\frac{4\sqrt{3} }{7} </math> is invalid given <math>\theta \leq 60^\circ </math> , <math>sin(\theta ) < sin( 60^\circ ) = \frac{\sqrt{3} }{2} = \frac{\sqrt{3} \cdot 3.5}{7} </math>
 
<cmath>  cos(  \theta)  = \frac{11 }{14}  </cmath>
 
<cmath>  cos(  \theta)  = \frac{11 }{14}  </cmath>
<cmath>  tan(  \theta)  = \frac{5\sqrt{3} }{11} </cmath>  
+
<cmath>  tan(  \theta)  = \frac{5\sqrt{3} }{11} \boxed{B } </cmath>  
  
<math>\boxed{B -34} </math>.   
+
~[https://artofproblemsolving.com/wiki/index.php/User:Cyantist luckuso]
 +
 
 +
==Solution #2==
 +
From <math>\triangle ABC</math>'s side lengths of 14, we get OF = OC = OE =<cmath>\frac{14\sqrt{3}}{3} </cmath>
 +
We let angle FOC = (<math>\theta</math>)
 +
And therefore angle EOC = 120 - (<math>\theta</math>)
 +
 
 +
The answer would be 3 * (Area <math>\triangle FOC</math> + Area <math>\triangle COE</math>)
 +
 
 +
Which area <math>\triangle FOC</math> = 0.5 * <math> (\frac{14\sqrt{3}}{3} )^2 * sin(\theta)</math>
 +
 
 +
And area <math>\triangle COE</math> = 0.5 * <math> (\frac{14\sqrt{3}}{3} )^2 * sin(120 - \theta)</math>
 +
 
 +
Therefore the answer would be
 +
3 * 0.5 * (<math>\frac{14\sqrt{3}}{3} )^2 * (sin(\theta)+sin(120 - \theta)) = {91\sqrt{3}}</math>
 +
 
 +
Which <cmath> sin(\theta)+sin(120 - \theta) = \frac{91\sqrt{3}}{98} </cmath>
 +
 
 +
So <cmath> \frac{1}{2}cos(\theta)+\frac{\sqrt{3}}{2}sin(\theta) = \frac{91}{98} </cmath>
 +
 
 +
Therefore <cmath> sin(\theta + 30) = \frac{91}{98} </cmath>
 +
 
 +
And <cmath> cos (\theta + 30) = \frac{21\sqrt{3}}{98} </cmath>
  
 +
Which <cmath> tan(\theta + 30) = \frac{91}{21\sqrt{3}} = \frac{91\sqrt{3}}{63} </cmath>
  
~[https://artofproblemsolving.com/wiki/index.php/User:Cyantist luckuso]
+
<math>tan(\theta)</math> can be calculated using addition identity, which gives the answer of
 +
 
 +
<cmath>(B)\frac{5\sqrt{3}}{11}</cmath>
 +
 
 +
(I would really appreciate if someone can help me fix my code and format)
 +
 
 +
~mitsuihisashi14
 +
~[https://artofproblemsolving.com/wiki/index.php/User:Cyantist luckuso] (fixed Latex error )
  
 
==See also==
 
==See also==
 
{{AMC12 box|year=2024|ab=B|num-b=18|num-a=20}}
 
{{AMC12 box|year=2024|ab=B|num-b=18|num-a=20}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 01:13, 16 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

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

2(area($\triangle$OFC) + area ($\triangle$OCE)) = \[OF^2 * sin(\theta) + OF^2 * sin(120 - \theta)\] \[= \frac{14^2 * 3}{9} (   sin(\theta)  +  sin(120 - \theta) )\] \[= \frac{196}{3}  (   sin(\theta)  +  sin(120 - \theta) )\] \[= 2 * {\frac{1}{3}  } * area  (ABCDEF) = 2* \frac{91\sqrt{3}}{3}\]

\[sin(\theta)  +  sin(120 - \theta) = \frac{13\sqrt{3}}{14}\] \[sin(\theta)  +   \frac{ \sqrt{3}}{2}cos(  \theta) +\frac{ \sqrt{1}}{2}sin(  \theta) = \frac{13\sqrt{3}}{14}\] \[\sqrt{3} sin(  \theta) + cos(  \theta) = \frac{13 }{7}\] \[cos(  \theta)  = \frac{13 }{7}  - \sqrt{3} sin(  \theta)\] \[\frac{169 }{49}  - \frac{26\sqrt{3} }{7} sin(  \theta)  + 4 sin(  \theta)^2 =0\] \[sin(  \theta)  = \frac{5\sqrt{3} }{14}  or \frac{4\sqrt{3} }{7}\] $\frac{4\sqrt{3} }{7}$ is invalid given $\theta \leq 60^\circ$ , $sin(\theta ) < sin( 60^\circ ) = \frac{\sqrt{3} }{2} = \frac{\sqrt{3} \cdot 3.5}{7}$ \[cos(  \theta)  = \frac{11 }{14}\] \[tan(  \theta)  = \frac{5\sqrt{3} }{11} \boxed{B }\]

~luckuso

Solution #2

From $\triangle ABC$'s side lengths of 14, we get OF = OC = OE =\[\frac{14\sqrt{3}}{3}\] We let angle FOC = ($\theta$) And therefore angle EOC = 120 - ($\theta$)

The answer would be 3 * (Area $\triangle FOC$ + Area $\triangle COE$)

Which area $\triangle FOC$ = 0.5 * $(\frac{14\sqrt{3}}{3} )^2 * sin(\theta)$

And area $\triangle COE$ = 0.5 * $(\frac{14\sqrt{3}}{3} )^2 *  sin(120 - \theta)$

Therefore the answer would be 3 * 0.5 * ($\frac{14\sqrt{3}}{3} )^2 * (sin(\theta)+sin(120 - \theta)) = {91\sqrt{3}}$

Which \[sin(\theta)+sin(120 - \theta) = \frac{91\sqrt{3}}{98}\]

So \[\frac{1}{2}cos(\theta)+\frac{\sqrt{3}}{2}sin(\theta) = \frac{91}{98}\]

Therefore \[sin(\theta + 30) = \frac{91}{98}\]

And \[cos (\theta + 30) = \frac{21\sqrt{3}}{98}\]

Which \[tan(\theta + 30) = \frac{91}{21\sqrt{3}} = \frac{91\sqrt{3}}{63}\]

$tan(\theta)$ can be calculated using addition identity, which gives the answer of

\[(B)\frac{5\sqrt{3}}{11}\]

(I would really appreciate if someone can help me fix my code and format)

~mitsuihisashi14 ~luckuso (fixed Latex error )

See also

2024 AMC 12B (ProblemsAnswer KeyResources)
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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