2018 AMC 10A Problems/Problem 15
Contents
Problem
Two circles of radius are externally tangent to each other and are internally tangent to a circle of radius at points and , as shown in the diagram. The distance can be written in the form , where and are relatively prime positive integers. What is ?
Solution 1
Let the center of the surrounding circle be . The circle that is tangent at point will have point as the center. Similarly, the circle that is tangent at point will have point as the center. Connect , , , and . Now observe that is similar to . Writing out the ratios, we get Therefore, our answer is .
Solution 2
Let the center of the large circle be . Let the common tangent of the two smaller circles be . Draw the two radii of the large circle, and and the two radii of the smaller circles to point . Draw ray and . This sets us up with similar triangles, which we can solve. The length of is equal to by Pythagorean Theorem, the length of the hypotenuse is , and the other leg is . Using similar triangles, is , and therefore half of is . Doubling gives , which results in . Nice
See Also
2018 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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