2010 AMC 8 - Problem 19 - Area between the two circles
Source:
May 27, 2011
geometryinvariantPythagorean Theorem
Problem Statement
The two circles pictured have the same center . Chord is tangent to the inner circle at , is , and chord has length . What is the area between the two circles?[asy]
unitsize(45);
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((2,0.15)--(1.85,0.15)--(1.85,0)--(2,0)--cycle); draw(circle((2,1),2.24)); draw(circle((2,1),1)); draw((0,0)--(4,0)); draw((0,0)--(2,1)); draw((2,1)--(2,0)); draw((2,1)--(4,0));
dot((0,0),ds); label("", (-0.19,-0.23),NE*lsf); dot((2,0),ds); label("", (1.97,-0.31),NE*lsf); dot((2,1),ds); label("", (1.96,1.09),NE*lsf); dot((4,0),ds); label("", (4.07,-0.24),NE*lsf); clip((-3.1,-7.72)--(-3.1,4.77)--(11.74,4.77)--(11.74,-7.72)--cycle);
[/asy]