# Radius of the smaller circle

Two circles $$(C_1$$ with radius $$r_1$$, and $$C_2$$ with radius $$r_2$$) are having the same centre. $$P_1$$ and $$P_2$$ are two distinct points lying on the circumference of $$C_1$$. The length of line $$P_1P_2$$ is $$l$$ units and it is tangent to $$C_2$$.

How can I express $$r_2$$ in terms of $$r_1$$ and $$l$$?

If $$P_1P_2$$ is tangent to the innermost circle, then the distance from the center to $$P_1P_2$$ is $$r_2.$$ You know that the distance from the center to either $$P_1$$ or $$P_2$$ is $$r_1.$$ You only need to use the Pythagorean theorem to find the value you need.
• Then, $r_2=\frac{1}{2}\sqrt{4r_1^2-l^2}$. Am I right? – Hussain-Alqatari Oct 30 '18 at 9:43