Difference between revisions of "2015 AMC 10B Problems/Problem 22"

(Problem)
(Solution)
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==Solution==
 
==Solution==
  
Triangle <math>AFG</math> is isosceles, so <math>AG=AF=1</math>. Using the symmetry of pentagon <math>FGHIJ</math>, notice that <math>\triangle IGF \cong \triangle JHG \cong \triangle DIJ \cong AFG</math>. Therefore, <math>JH=AF=1</math>.
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Triangle <math>AFG</math> is isosceles, so <math>AG=AF=1</math>. <math>FJ = FG</math> since <math>\triangle FGJ</math> is also isosceles. Using the symmetry of pentagon <math>FGHIJ</math>, notice that <math>\triangle JHG \cong \triangle AFG</math>. Therefore, <math>JH=AF=1</math>.
  
Since <math>\triangle AJH \sim \triangle AFG</math>, <math>\frac{JH}{AF+FJ}=\frac{1}{1+FG}=\frac{FG}{FI}=\frac{FG}1</math>. From this, we get <math>FG=\frac{\sqrt{5} -1}{2}</math>.
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Since <math>\triangle AJH \sim \triangle AFG</math>,  
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<cmath>\frac{JH}{AF+FJ}=\frac{FG}{FA}</cmath>.
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<cmath>\frac{1}{1+FG} = \frac{FG}1</cmath>
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<cmath>1 = FG^2 + FG</cmath>
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<cmath>FG^2+FG-1 = 0</cmath>
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<cmath>FG = \frac{-1 \pm \sqrt{5} }{2}</cmath>
  
Since <math>\triangle DIJ \cong \triangle AFG</math>, <math>DJ=DI=AF=1</math>. Since <math>\triangle AFG \sim ADC</math>, <math> \frac{AF}{AF+FJ+JD}=\frac1{2+FG} = \frac{FG}{CD}=\frac{\frac{\sqrt{5} - 1}{2}}{CD}</math>. Thus, <math>CD = (\sqrt{5} - 1) + (\frac{\sqrt{5} - 1}{2})^2 = \frac{\sqrt{5} +1}{2}</math>
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However, <math>FG=\frac{-1 + \sqrt{5}}{2}</math> since <math>FG</math> must be greater than 0.
  
Therefore, <math>FG+JH+CD=\frac{\sqrt5-1}2+1+\frac{\sqrt5+1}2=\boxed{\mathbf{(D)}\ 1+\sqrt{5}\ }</math>
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Notice that <math>CD = AE = AJ = AF + FJ = 1 + \frac{-1 + \sqrt{5}}{2} = \frac{1 + \sqrt{5}}{2}</math>.
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Therefore, <math>FG+JH+CD=\frac{-1+\sqrt5}2+1+\frac{1+\sqrt5}2=\boxed{\mathbf{(D)}\ 1+\sqrt{5}\ }</math>
  
 
==See Also==
 
==See Also==

Revision as of 17:17, 8 October 2017

Problem

In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$. What is $FG + JH + CD$? [asy] pair A=(cos(pi/5)-sin(pi/10),cos(pi/10)+sin(pi/5)), B=(2*cos(pi/5)-sin(pi/10),cos(pi/10)), C=(1,0), D=(0,0), E1=(-sin(pi/10),cos(pi/10)); //(0,0) is a convenient point //E1 to prevent conflict with direction E(ast) pair F=intersectionpoints(D--A,E1--B)[0], G=intersectionpoints(A--C,E1--B)[0], H=intersectionpoints(B--D,A--C)[0], I=intersectionpoints(C--E1,D--B)[0], J=intersectionpoints(E1--C,D--A)[0]; draw(A--B--C--D--E1--A); draw(A--D--B--E1--C--A); draw(F--I--G--J--H--F); label("$A$",A,N); label("$B$",B,E); label("$C$",C,SE); label("$D$",D,SW); label("$E$",E1,W); label("$F$",F,NW); label("$G$",G,NE); label("$H$",H,E); label("$I$",I,S); label("$J$",J,W); [/asy] $\textbf{(A) } 3 \qquad\textbf{(B) } 12-4\sqrt5 \qquad\textbf{(C) } \dfrac{5+2\sqrt5}{3} \qquad\textbf{(D) } 1+\sqrt5 \qquad\textbf{(E) } \dfrac{11+11\sqrt5}{10}$

Solution

Triangle $AFG$ is isosceles, so $AG=AF=1$. $FJ = FG$ since $\triangle FGJ$ is also isosceles. Using the symmetry of pentagon $FGHIJ$, notice that $\triangle JHG \cong \triangle AFG$. Therefore, $JH=AF=1$.

Since $\triangle AJH \sim \triangle AFG$, \[\frac{JH}{AF+FJ}=\frac{FG}{FA}\]. \[\frac{1}{1+FG} = \frac{FG}1\] \[1 = FG^2 + FG\] \[FG^2+FG-1 = 0\] \[FG = \frac{-1 \pm \sqrt{5} }{2}\]

However, $FG=\frac{-1 + \sqrt{5}}{2}$ since $FG$ must be greater than 0.

Notice that $CD = AE = AJ = AF + FJ = 1 + \frac{-1 + \sqrt{5}}{2} = \frac{1 + \sqrt{5}}{2}$.

Therefore, $FG+JH+CD=\frac{-1+\sqrt5}2+1+\frac{1+\sqrt5}2=\boxed{\mathbf{(D)}\ 1+\sqrt{5}\ }$

See Also

2015 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 21
Followed by
Problem 23
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All AMC 10 Problems and Solutions

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