Question

Circles $C_1$ and $C_2$ 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 is externally tangent to $C_3.$ What is the radius of $C_4?$

$\dfrac{1}{14}$

$\dfrac{1}{12}$

$\dfrac{1}{10}$

$\dfrac{3}{28}$

$\dfrac{1}{9}$

Solution:

Put the centers of $C_1, C_2$ at $\left(\pm\frac14, 0\right).$ By symmetry the largest circle inside both sits at the origin with radius $r_3,$ where $1 - r_3 = \frac14,$ so $r_3 = \frac34.$ Let $C_4$ be centered at $(0, y)$ with radius $r.$ Internal tangency to $C_1$ gives $\sqrt{\frac1{16} + y^2} = 1 - r,$ and external tangency to $C_3$ gives $y = \frac34 + r.$ Substitute the second into the first: $\frac1{16} + \left(\frac34 + r\right)^2 = (1 - r)^2.$ This collapses to $\frac72 r = \frac38,$ so $r = \frac{3}{28}.$ Therefore, the answer is D.

2023 AMC 10A Problem 22

Solution:

Problem 22 in Other Years