Difference between revisions of "2010 AMC 12A Problems/Problem 3"
Line 4: | Line 4: | ||
<center><asy> | <center><asy> | ||
unitsize(1mm); | unitsize(1mm); | ||
− | defaultpen(linewidth(.8pt)); | + | defaultpen(linewidth(.8pt)+fontsize(8pt)); |
draw((0,0)--(0,25)--(25,25)--(25,0)--cycle); | draw((0,0)--(0,25)--(25,25)--(25,0)--cycle); | ||
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draw((0,15)--(0,20)--(25,20)--(25,15)--cycle); | draw((0,15)--(0,20)--(25,20)--(25,15)--cycle); | ||
draw((25,15)--(25,20)--(50,20)--(50,15)--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></center> | </asy></center> | ||
+ | |||
<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 19:41, 12 February 2010
Problem 3
Rectangle , pictured below, shares
of its area with square
. Square
shares
of its area with rectangle
. What is
?
![[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]](http://latex.artofproblemsolving.com/5/4/3/54371c10b04c0bcf5a86491a02c9b7ecf851cb2d.png)
Solution
Let , let
, and let
.