Difference between revisions of "2010 AMC 8 Problems/Problem 19"

Latest revision as of 08:04, 10 March 2023

Problem

The two circles pictured have the same center $C$ . Chord $\overline{AD}$ is tangent to the inner circle at $B$ , $AC$ is $10$ , and chord $\overline{AD}$ has length $16$ . 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$", (4.07,-0.24),NE*lsf); clip((-3.1,-7.72)--(-3.1,4.77)--(11.74,4.77)--(11.74,-7.72)--cycle); [/asy]$

$\textbf{(A)}\ 36 \pi \qquad\textbf{(B)}\ 49 \pi\qquad\textbf{(C)}\ 64 \pi\qquad\textbf{(D)}\ 81 \pi\qquad\textbf{(E)}\ 100 \pi$

Solution

Since $\triangle ACD$ is isosceles, $CB$ bisects $AD$ . Thus $AB=BD=8$ . From the Pythagorean Theorem, $CB=6$ . Thus the area between the two circles is $100\pi - 36\pi=64\pi$ $\boxed{\textbf{(C)}\ 64\pi}$

Note: The length $AC$ is necessary information, as this tells us the radius of the larger circle. The area of the annulus is $\pi(AC^2-BC^2)=\pi AB^2=64\pi$ .

Video Solution

https://youtu.be/Q6rnoQChiyU. Soo, DRMS, NM

Video Solution by OmegaLearn

https://youtu.be/51K3uCzntWs?t=3206

~ pi_is_3.14

Video by MathTalks

https://www.youtube.com/watch?v=KSYVsSJDX-0&feature=youtu.be

Video Solution by WhyMath

https://youtu.be/yjhitUYSAI0

~savannahsolver

@@ Line 32: / Line 32: @@
 https://www.youtube.com/watch?v=KSYVsSJDX-0&feature=youtu.be
+==Video Solution by WhyMath==
+https://youtu.be/yjhitUYSAI0
+~savannahsolver
 ==See Also==
 {{AMC8 box|year=2010|num-b=18|num-a=20}}
 {{MAA Notice}}

2010 AMC 8 (Problems • Answer Key • Resources)
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