1988 AJHSME Problems/Problem 17
Revision as of 13:57, 20 April 2009 by 5849206328x (talk | contribs) (New page: ==Problem== The shaded region formed by the two intersecting perpendicular rectangles, in square units, is <asy> fill((0,0)--(6,0)--(6,-3.5)--(9,-3.5)--(9,0)--(10,0)--(10,2)--(9,2)--(9,4...)
Problem
The shaded region formed by the two intersecting perpendicular rectangles, in square units, is
![[asy] fill((0,0)--(6,0)--(6,-3.5)--(9,-3.5)--(9,0)--(10,0)--(10,2)--(9,2)--(9,4.5)--(6,4.5)--(6,2)--(0,2)--cycle,black); label("2",(0,.9),W); label("3",(7.3,4.5),N); draw((0,-3.3)--(0,-5.3),linewidth(1)); draw((0,-4.3)--(3.7,-4.3),linewidth(1)); label("10",(4.7,-3.7),S); draw((5.7,-4.3)--(10,-4.3),linewidth(1)); draw((10,-3.3)--(10,-5.3),linewidth(1)); draw((11,4.5)--(13,4.5),linewidth(1)); draw((12,4.5)--(12,2),linewidth(1)); label("8",(11.3,1),E); draw((12,0)--(12,-3.5),linewidth(1)); draw((11,-3.5)--(13,-3.5),linewidth(1)); [/asy]](http://latex.artofproblemsolving.com/a/d/d/addd68de53329242c875ed0d9bdfcc9127497b92.png)
Solution
Looking at the diagram, the shaded region is the union of two rectangles, with a small rectangle as overlap. If we just add the areas of the two rectangles, then we'll overcount the small rectangle, so we must subtract that area to get the desired area.
The areas of the two larger rectangles are 
and 
, and the area of the small rectangle is 
. The desired area is thus 
.
