Difference between revisions of "2010 AMC 12A Problems/Problem 3"

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<center><asy>
 
<center><asy>
 
unitsize(1mm);
 
unitsize(1mm);
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defaultpen(linewidth(.8pt)+fontsize(8pt));
  
 
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draw((0,0)--(0,25)--(25,25)--(25,0)--cycle);
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draw((25,15)--(25,20)--(50,20)--(50,15)--cycle);
 
draw((25,15)--(25,20)--(50,20)--(50,15)--cycle);
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label("$A$",(0,20),W);
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label("$B$",(50,20),E);
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label("$C$",(50,15),E);
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label("$D$",(0,15),W);
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label("$E$",(0,25),NW);
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label("$F$",(25,25),NE);
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label("$G$",(25,0),SE);
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label("$H$",(0,0),SW);
 
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<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10</math>
 
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10</math>

Revision as of 18:41, 12 February 2010

Problem 3

Rectangle $ABCD$, pictured below, shares $50\%$ of its area with square $EFGH$. Square $EFGH$ shares $20\%$ of its area with rectangle $ABCD$. What is $\frac{AB}{AD}$?

[asy] unitsize(1mm); defaultpen(linewidth(.8pt)+fontsize(8pt));  draw((0,0)--(0,25)--(25,25)--(25,0)--cycle); fill((0,20)--(0,15)--(25,15)--(25,20)--cycle,gray); draw((0,15)--(0,20)--(25,20)--(25,15)--cycle); draw((25,15)--(25,20)--(50,20)--(50,15)--cycle);  label("$A$",(0,20),W); label("$B$",(50,20),E); label("$C$",(50,15),E); label("$D$",(0,15),W); label("$E$",(0,25),NW); label("$F$",(25,25),NE); label("$G$",(25,0),SE); label("$H$",(0,0),SW); [/asy]


$\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10$

Solution

Let $EF = FG = GF = HE = s$, let $AD = BC = h$, and let $AB = CD = x$.


$.2s^2 = hs$

$s = 5h$

$.5hx = hs$

$x = 2s = 10h$


$\frac{AB}{AD} = \frac{x}{h} = \boxed{10\ \textbf{(E)}}$