2011 UNCO Math Contest II Problems/Problem 3
Problem
The two congruent rectangles shown have
dimensions 
in. by 
in. What is the area
of the shaded overlap region?
![[asy] filldraw((0,0)--(13,0)--(25,5)--(12,5)--cycle,blue); pair A1=(0,0),B1=(25,0),C1=(25,5),D1=(0,5); draw(A1--B1--C1--D1--cycle,black); pair P,R; P=unit(A1-D1)+D1; R=unit(C1-D1)+D1; draw(P--(P+R-D1)--R,black); pair A2=(0,0),B2=(25/13,-60/13),C2=(25,5),D2=(25-25/13,5+60/13); draw(A2--B2--C2--D2--cycle,black); P=unit(A2-D2)+D2; R=unit(C2-D2)+D2; draw(P--(P+R-D2)--R,black); [/asy]](http://latex.artofproblemsolving.com/1/4/8/148875dde5914a90f580d82c6db9acf9e96db728.png)
Solution
[asy] filldraw((0,0)--(13,0)--(25,5)--(12,5)--cycle,blue); pair A1=(0,0),B1=(25,0),C1=(25,5),D1=(0,5);
draw(A1--B1--C1--D1--cycle,black); pair P,R; P=unit(A1-D1)+D1; R=unit(C1-D1)+D1; draw(P--(P+R-D1)--R,black);
pair A2=(0,0),B2=(25/13,-60/13),C2=(25,5),D2=(25-25/13,5+60/13); draw(A2--B2--C2--D2--cycle,black);
P=unit(A2-D2)+D2; R=unit(C2-D2)+D2;
draw(P--(P+R-D2)--R,black);
label("A",(0,0),SW);
label("B",(0,5),NW);
label("M",(12,5),NW);
label("C",(25,5),NE);
label("D",(25,0),SE);
label("N",(13,0),SE);
label("O",(25/13,-60/13),S);
label("P",(25-25/13,5+60/13);
[/asy]
We notice that triangles 
and 
are congruent, since they share two angles and a side (a right angle, and the opposite angle, and the side 5) This means that if we label 
then 
and 
and so since and are congruent, then 
and so 
solving for we get 12, meaning that the area of the two triangles and 
(which are also congruent since the 
is parallel to 
) is 30, so the total area of them is 60, and so taking this away from rectangle 
we get the blue, so 
so the answer is 65.
See Also
| 2011 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
| All UNCO Math Contest Problems and Solutions | ||
