1994 AHSME Problems/Problem 2
Contents
Problem
A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the area of the fourth rectangle?
![[asy] draw((0,0)--(10,0)--(10,7)--(0,7)--cycle); draw((0,5)--(10,5)); draw((3,0)--(3,7)); label("6", (1.5,6)); label("?", (1.5,2.5)); label("14", (6.5,6)); label("35", (6.5,2.5)); [/asy]](http://latex.artofproblemsolving.com/3/1/f/31f1dc6fc961ef17ec97066f3f86050dca58b393.png)
Solution
![[asy] pair A=(0,0),B=(10,0),C=(10,7),D=(0,7),EE=(0,5),F=(10,5),G=(3,0),H=(3,7); path BG=shift(0,-0.5)*(B--G); path BF=shift(0.5,0)*(B--F); path FC=shift(0.5,0)*(F--C); path DH=shift(0,0.5)*(D--H); draw(A--B--C--D--cycle); draw(EE--F); draw(G--H); draw(BG,L=Label("$7$",position=MidPoint,align=(0,-1)),arrow=Arrows(),bar=Bars,red); draw(BF,L=Label("$5$",position=MidPoint,align=(1,0)),arrow=Arrows(),bar=Bars,red); draw(FC,L=Label("$2$",position=MidPoint,align=(1,0)),arrow=Arrows(),bar=Bars,red); draw(DH,L=Label("$3$",position=MidPoint,align=(0,1)),arrow=Arrows(),bar=Bars,red); label("$6$", (1.5,6)); label("$15$", (1.5,2.5),blue); label("$14$", (6.5,6)); label("$35$", (6.5,2.5)); [/asy]](http://latex.artofproblemsolving.com/9/b/1/9b1be94397c1f8fdf513d00d24423fdae84c4bb8.png)
We can easily see the dimensions of each small rectangle. So the area of the last rectangle is 
.
--Solution by TheMaskedMagician
Solution 2
![[asy] pair A=(0,0),B=(10,0),C=(10,7),D=(0,7),EE=(0,5),F=(10,5),G=(3,0),H=(3,7); path CH=C--H; path BF=B--F; path FC=F--C; path DH=D--H; draw(A--B--C--D--cycle); draw(EE--F); draw(G--H); draw(CH,L=Label("$b$",position=MidPoint,align=(0,1))); draw(BF,L=Label("$y$",position=MidPoint,align=(1,0))); draw(FC,L=Label("$x$",position=MidPoint,align=(1,0))); draw(DH,L=Label("$a$",position=MidPoint,align=(0,1))); label("$6$", (1.5,6)); label("$14$", (6.5,6)); label("$35$", (6.5,2.5)); [/asy]](http://latex.artofproblemsolving.com/9/9/8/9982a8f60a563a78e0f60ee91aabfe795ab4c96c.png)
In case its not immediately obvious from inspection what the dimensions of the small rectangles are, we can work it out. We know 
and 
and 
and we want to know 
. We can compute it as follows 
and the answer is 
See Also
| 1994 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
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| All AHSME Problems and Solutions | ||
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