Difference between revisions of "2004 AMC 12A Problems/Problem 19"
(→Solution) |
m (→Solution 1: Changed diagram.) |
||
Line 41: | Line 41: | ||
label("\(B\)", B,W); | label("\(B\)", B,W); | ||
label("\(C\)", C,E); | label("\(C\)", C,E); | ||
− | label("\(E\)", E, | + | label("\(E\)", E,SE); |
label("\(1\)",(-.4,.7)); | label("\(1\)",(-.4,.7)); | ||
label("\(1\)",(0,0.5),W); | label("\(1\)",(0,0.5),W); |
Revision as of 15:06, 15 July 2012
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
Solution 1
Note that since is the center of the larger circle of radius . Using the Pythagorean Theorem on ,
Now using the Pythagorean Theorem on ,
Substituting ,
Solution 2
We can apply Descartes' Circle Formula.
The four circles have curvatures , and .
We have
Simplifying, we get
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 |