2004 AMC 10B Problems/Problem 16
Contents
Problem
Three circles of radius are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle?
Solution 1
The situation is shown in the picture below. The radius we seek is . Clearly . The point is clearly the center of the equilateral triangle , thus is of the altitude of this triangle. We get that . Therefore the radius we seek is .
Solution 2
Using Descartes' Circle Formula, we can assign curvatures to all the circles: , , , and (b/c it is the circle internally tangent to all the other circles, the radius of the bigger circle is negative). Then, we can solve:
See also
2004 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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All AMC 10 Problems and Solutions |
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