Difference between revisions of "2004 AMC 12A Problems/Problem 19"
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== Problem 19 == | == Problem 19 == | ||
[[Circle]]s <math>A, B</math> and <math>C</math> are externally tangent to each other, and internally tangent to circle <math>D</math>. Circles <math>B</math> and <math>C</math> are congruent. Circle <math>A</math> has radius <math>1</math> and passes through the center of <math>D</math>. What is the [[radius]] of circle <math>B</math>? | [[Circle]]s <math>A, B</math> and <math>C</math> are externally tangent to each other, and internally tangent to circle <math>D</math>. Circles <math>B</math> and <math>C</math> are congruent. Circle <math>A</math> has radius <math>1</math> and passes through the center of <math>D</math>. What is the [[radius]] of circle <math>B</math>? | ||
+ | |||
+ | <center><asy> | ||
+ | unitsize(20mm); | ||
+ | pair A=(-1,0),B=(2/3,8/9),C=(2/3,-8/9),D=(0,0); | ||
+ | |||
+ | draw(Circle(D,2)); | ||
+ | draw(Circle(A,1)); | ||
+ | draw(Circle(B,8/9)); | ||
+ | draw(Circle(C,8/9)); | ||
+ | |||
+ | label("\(A\)", A); | ||
+ | label("\(B\)", B); | ||
+ | label("\(C\)", C); | ||
+ | label("D", (-1.2,1.8)); | ||
+ | </asy></center> | ||
<math>\text {(A)} \frac23 \qquad \text {(B)} \frac {\sqrt3}{2} \qquad \text {(C)}\frac78 \qquad \text {(D)}\frac89 \qquad \text {(E)}\frac {1 + \sqrt3}{3}</math> | <math>\text {(A)} \frac23 \qquad \text {(B)} \frac {\sqrt3}{2} \qquad \text {(C)}\frac78 \qquad \text {(D)}\frac89 \qquad \text {(E)}\frac {1 + \sqrt3}{3}</math> |
Revision as of 20:46, 26 December 2011
Problem 19
Circles and are externally tangent to each other, and internally tangent to circle . Circles and are congruent. Circle has radius and passes through the center of . What is the radius of circle ?
Solution
Note that since is the center of the larger circle of radius . Using the Pythagorean Theorem on ,
Now using the Pythagorean Theorem on ,
Substituting ,
See Also
2004 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |