2023 AMC 10A Problems/Problem 22

Circle $C_1$ and $C_2$ each have radius $1$ , and the distance between their centers is $\frac{1}{2}$ . Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2$ . Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3$ . What is the radius of $C_4$ ?

$\textbf{(A) } \frac{1}{14} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{10} \qquad \textbf{(D) } \frac{3}{28} \qquad \textbf{(E) } \frac{1}{9}$

Solution

Connect the centers of $C_1$ and $C_4$ , and the centers of $C_3$ and $C_4$ . Let the radius of $C_4$ be $r$ . Then, from the auxillary lines, we get $(\frac{1}{4})^2 + (\frac{3}{4}+r)^2 = (1-r)^2$ . Solving, we get $r = \boxed{\frac{3}{28}}$

Video Solution 1 by OmegaLearn

https://youtu.be/jcHeJXs9Sdw

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2023 AMC 10A Problems/Problem 22

Solution

Video Solution 1 by OmegaLearn