Difference between revisions of "2010 AMC 8 Problems/Problem 19"
(Created page with "== Problem == The two circles pictured have the same center <math>C</math>. Chord <math>\overline{AD}</math> is tangent to the inner circle at <math>B</math>, <math>AC</math> is ...") |
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== Solution == | == Solution == | ||
Since <math>\triangle ACD</math> is isosceles, <math>CB</math> bisects <math>AD</math>. Thus <math>AB=BD=8</math>. From the Pythagorean Theorem, <math>CB=6</math>. Thus the area between the two circles is | Since <math>\triangle ACD</math> is isosceles, <math>CB</math> bisects <math>AD</math>. Thus <math>AB=BD=8</math>. From the Pythagorean Theorem, <math>CB=6</math>. Thus the area between the two circles is | ||
| − | <math>100\pi - 36\pi=64\pi</math> <math>\ | + | <math>100\pi - 36\pi=64\pi</math> <math>\boxed{\textbf{(C)}\ 64\pi}</math> |
Revision as of 12:43, 11 March 2012
Problem
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("$A$", (-0.19,-0.23),NE*lsf); dot((2,0),ds); label("$B$", (1.97,-0.31),NE*lsf); dot((2,1),ds); label("$C$", (1.96,1.09),NE*lsf); dot((4,0),ds); label("$D$", (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]](http://latex.artofproblemsolving.com/2/c/f/2cfa82378d1bafff2108201dd94835ed7a9aab8d.png)
Solution
Since 
is isosceles, 
bisects 
. Thus 
. From the Pythagorean Theorem, 
. Thus the area between the two circles is
