Problem: 2004 AMC 10 #21
Source:
February 1, 2007
Problem Statement
Two distinct lines pass through the center of three concentric circles of radii , , and . The area of the shaded region in the diagram is of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: radians is degrees.)[asy]
defaultpen(linewidth(0.8));
pair O=origin;
fill(O--Arc(O, 2, 20, 160)--cycle, mediumgray);
fill(O--Arc(O, 1, 20, 160)--cycle, white);
fill(O--Arc(O, 2, 200, 340)--cycle, mediumgray);
fill(O--Arc(O, 1, 200, 340)--cycle, white);
fill(O--Arc(O, 3, 160, 200)--cycle, mediumgray);
fill(O--Arc(O, 2, 160, 200)--cycle, white);
fill(O--Arc(O, 1, 160, 200)--cycle, mediumgray);
fill(O--Arc(O, 3, -20, 20)--cycle, mediumgray);
fill(O--Arc(O, 2, -20, 20)--cycle, white);
fill(O--Arc(O, 1, -20, 20)--cycle, mediumgray);
draw(Circle(origin, 1));draw(Circle(origin, 2));draw(Circle(origin, 3));
draw(5*dir(200)--5*dir(20)^^5*dir(160)--5*dir(-20));[/asy]