Parallel Lines
Source: 2006 AIME A Problem 7
March 9, 2006
ratiogeometrytrapezoidAMCAIMErectangletrigonometry
Problem Statement
An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region to the area of shaded region is . Find the ratio of shaded region to the area of shaded region .[asy]
defaultpen(linewidth(0.7)+fontsize(10));
for(int i=0; i<4; i=i+1) {
fill((2*i,0)--(2*i+1,0)--(2*i+1,6)--(2*i,6)--cycle, mediumgray);
}
pair A=(1/3,4), B=A+7.5*dir(-17), C=A+7*dir(10);
draw(B--A--C);
fill((7.3,0)--(7.8,0)--(7.8,6)--(7.3,6)--cycle, white);
clip(B--A--C--cycle);
for(int i=0; i<9; i=i+1) {
draw((i,1)--(i,6));
}
label("", A+0.2*dir(-17), S);
label("", A+2.3*dir(-17), S);
label("", A+4.4*dir(-17), S);
label("", A+6.5*dir(-17), S);[/asy]