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| − | Circle <math>C_1</math> and <math>C_2</math> each have radius <math>1</math>, and the distance between their centers is <math>\frac{1}{2}</math>. Circle <math>C_3</math> is the largest circle internally tangent to both <math>C_1</math> and <math>C_2</math>. Circle <math>C_4</math> is internally tangent to both <math>C_1</math> and <math>C_2</math> and externally tangent to <math>C_3</math>. What is the radius of <math>C_4</math>?
| + | #redirect[[2023 AMC 12A Problems/Problem 18]] |
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| − | <math>\textbf{(A) } \frac{1}{14} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{10} \qquad \textbf{(D) } \frac{3}{28} \qquad \textbf{(E) } \frac{1}{9}</math>
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| − | ==Solution==
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| − | Connect the centers of <math>C_1</math> and <math>C_4</math>, and the centers of <math>C_3</math> and <math>C_4</math>. Let the radius of <math>C_4</math> be <math>r</math>. Then, from the auxillary lines, we get <math>(\frac{1}{4})^2 + (\frac{3}{4}+r)^2 = (1-r)^2</math>. Solving, we get <math>r = \boxed{\frac{3}{28}}</math>
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| − | -andliu766
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| − | == Video Solution 1 by OmegaLearn ==
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| − | https://youtu.be/jcHeJXs9Sdw
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| − | ==See Also==
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| − | {{AMC10 box|year=2023|ab=A|num-b=21|num-a=23}}
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| − | {{MAA Notice}}
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