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.

  • $\begingroup$ Please provide some insight about your thoughts sofar. $\endgroup$ – laurent Dec 13 '16 at 12:36
  • $\begingroup$ im not sure..but i think we have to join all the points to the center...and go for the angles..?? im not sure.. and the ratio is 1:3..made a mistake while typing it.. @laurent $\endgroup$ – Shashank gb Dec 13 '16 at 12:41
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    $\begingroup$ Make a nice drawing and remember Thales' theorem. $\endgroup$ – laurent Dec 13 '16 at 12:47

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 $R=18$.

  • $\begingroup$ ok... thanks :) it was very helpful $\endgroup$ – Shashank gb Dec 13 '16 at 12:51
  • $\begingroup$ you are welcome! $\endgroup$ – Arnaldo Dec 13 '16 at 12:52
  • $\begingroup$ Is the solution clear? $\endgroup$ – Arnaldo Dec 13 '16 at 13:08

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