2011 UNCO Math Contest II Problems/Problem 3

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Problem

The two congruent rectangles shown have dimensions $5$ in. by $25$ 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]

Solution

$65$ [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 $AON$ and $NDC$ 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 $ON$ $x$ then $NC=25-x$ and $AN^2=5^2+x^2$ and so since $AON$ and $NDC$ are congruent, then $AN=NC$ and so $(25-x)^2=5^2+x^2$ solving for $x$ we get 12, meaning that the area of the two triangles $AON$ and $MPC$ (which are also congruent since the $AD$ is parallel to $BC$) is 30, so the total area of them is 60, and so taking this away from rectangle $AOCP$ we get the blue, so $125-60=65$ so the answer is 65.

See Also

2011 UNCO Math Contest II (ProblemsAnswer KeyResources)
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