AMC 12 Mock Exam 1 Problem 10

Problem 10 of 25

Circles $\omega_1$ and $\omega_2$ have radii $3$ and $12$ , respectively, and are externally tangent to each other. Both circles are tangent to a line $\ell$ and lie on the same side of $\ell$ . Circle $\omega_3$ is tangent to $\ell$ and externally tangent to both $\omega_1$ and $\omega_2$ , and it lies in the region bounded by $\ell$ , $\omega_1$ , and $\omega_2$ , as shown (the figure is not necessarily drawn to scale). What is the radius of $\omega_3$ ?

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Let $\ell$ be the $x$ -axis. If a circle of radius $r$ is tangent to $\ell$ at $(p,0)$ and a circle of radius $s$ is tangent to $\ell$ at $(q,0)$ , and the two circles are externally tangent, then the distance between centers is $r+s$ , so $(p-q)^2+(r-s)^2=(r+s)^2$ , giving $|p-q|=2\sqrt{rs}$ .

Apply this to the three pairs. Let $\omega_3$ have radius $r$ and touch $\ell$ at $x_3$ , and place $\omega_1$ at $x=0$ . Then $\omega_2$ touches at $2\sqrt{3\cdot12}=12$ , $\omega_3$ touches at $2\sqrt{3r}$ (distance from $\omega_1$ ) and at $12-2\sqrt{12r}=12-4\sqrt{3r}$ (distance from $\omega_2$ , which is to the left of $\omega_2$ ). Therefore $2\sqrt{3r}=12-4\sqrt{3r}\implies 6\sqrt{3r}=12\implies \sqrt{3r}=2\implies r=\boxed{\tfrac43}.$

Tagged: tangent circles and a line

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