2010 AMC 8 Problems/Problem 17

Problem

The diagram shows an octagon consisting of unit squares. The portion below is a unit square and a triangle with base . If bisects the area of the octagon, what is the ratio ?

import graph; size(300); 
real lsf = 0.5; 
pen dp = linewidth(0.7) + fontsize(10); 
defaultpen(dp); 
pen ds = black; 
pen xdxdff = rgb(0.49,0.49,1);
draw((0,0)--(6,0),linewidth(1.2pt)); 
draw((0,0)--(0,1),linewidth(1.2pt)); 
draw((0,1)--(1,1),linewidth(1.2pt)); 
draw((1,1)--(1,2),linewidth(1.2pt)); 
draw((1,2)--(5,2),linewidth(1.2pt)); 
draw((5,2)--(5,1),linewidth(1.2pt)); 
draw((5,1)--(6,1),linewidth(1.2pt)); 
draw((6,1)--(6,0),linewidth(1.2pt)); 
draw((1,1)--(5,1),linewidth(1.2pt);
//+linetype("2pt 2pt"));
 draw((1,1)--(1,0),linewidth(1.2pt);
//+linetype("2pt 2pt")); 
draw((2,2)--(2,0),linewidth(1.2pt);
//+linetype("2pt 2pt")); 
draw((3,2)--(3,0),linewidth(1.2pt);
//+linetype("2pt 2pt")); 
draw((4,2)--(4,0),linewidth(1.2pt);
//+linetype("2pt 2pt")); 
draw((5,1)--(5,0),linewidth(1.2pt);
//+linetype("2pt 2pt")); 
draw((0,0)--(5,1.5),linewidth(1.2pt));
dot((0,0),ds); label("$P$", (-0.23,-0.26),NE*lsf); 
dot((0,1),ds); 
dot((1,1),ds); 
dot((1,2),ds); 
dot((5,2),ds); 
label("$X$", (5.14,2.02),NE*lsf); dot((5,1),ds); 
label("$Y$", (5.12,1.14),NE*lsf); dot((6,1),ds); 
dot((6,0),ds); dot((1,0),ds); dot((2,0),ds); dot((3,0),ds); 
dot((4,0),ds); dot((5,0),ds); dot((2,2),ds); dot((3,2),ds); 
dot((4,2),ds); dot((5,1.5),ds); 
label("$Q$", (5.14,1.51),NE*lsf); 
clip((-4.19,-5.52)--(-4.19,6.5)--(10.08,6.5)--(10.08,-5.52)--cycle);
 (Error making remote request. Unknown error_msg)

Solution

We see that half the area of the octagon is . We see that the triangle area is . That means that . Meaning,

See Also

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination