2009 UNCO Math Contest II Problems/Problem 9
Contents
Problem
A square is divided into three pieces of
equal area by two parallel lines as shown.
If the distance between the two parallel
lines is 
what is the area of the square?
![[asy] draw((0,0)--(1,0)--(1,1)--(0,1)--cycle,black); draw((1,0)--(0,2/3),black); draw((1,1/3)--(0,1),black); [/asy]](http://latex.artofproblemsolving.com/5/9/4/594509287728bec7666a5a011722c3330803fba0.png)
Solution
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Let x be the length of a side. Then the square has area 
and each portion has area 
If x is the base of one of the triangles, then the height will be 
.
By the pythaogrean theorem, longer side of the parallelogram has length 
Thus sqrt(13)*x/3*8 = x^2/3. Solving this gives x = 8*sqrt(13).
Thus, the area of the square is 64*13 = 832.
Solution 2
![[asy] unitsize(135); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair A, B, C, D, E, F; A=(0, 0); B=(1, 0); C=(1, -1); D=(0,-1); E=(0,-1/3); F=(1,-2/3); draw(A--B--C--D--cycle); label("$A$",A,N); label("$B$",B,N); label("$C$",C,E); label("$D$",D,S); label("$E$",E,W); label("$F$",F,SE); draw(A--F); draw(C--E); draw(A--C); draw(E--F); [/asy]](http://latex.artofproblemsolving.com/7/3/3/7332665745d0c13c92b6188693abd3e776e1ff4b.png)
In 
ADC, 
See also
| 2009 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
| All UNCO Math Contest Problems and Solutions | ||
