problem related to finding radius of circle

Question

The radii of $2$ concentric circles are in the ratio of $1:3$. $AC$ is the diameter of the big circle; $BC$ is a chord in the big circle which is tangent to the small circle, and the length of $AB$ is $12$ units. Find the radius of both the circles.

Answer 1

Let's call $O$ the center of both circle and $I$ the tangent point of $BC$.

We have that $OI$ is perpendicular to $BC$ and $AB$ is perpendicular to $BC$ (because $AC$ is a diameter). So the triangles $ABC$ and $OIC$ are similar.

$$\frac{OC}{AC}=\frac{OI}{AB} \Rightarrow \frac{R}{2R}=\frac{r}{12} \Rightarrow r=6$$

but $\frac{r}{R}=\frac{1}{3}$ so $R=18$.