HMMT-Feb 2014 geo, Problem 1

Let $O_{1}$ and $O_{2}$ be concentric circles with radii 4 and 6 , respectively. A chord $A B$ is drawn in $O_{1}$ with length 2. Extend $A B$ to intersect $O_{2}$ in points $C$ and $D$. Find $C D$.

Solution: Let $O$ be the common center of $O_{1}$ and $O_{2}$, and let $M$ be the midpoint of $A B$. Then $O M \perp A B$, so by the Pythagorean Theorem, $O M=\sqrt{4^{2}-1^{2}}=\sqrt{15}$. Thus $C D=2 C M=$ $2 \sqrt{6^{2}-15}=2 \sqrt{21}$. $\fbox{\sqrt{2 \sqrt{21}}}$.

Answer: \sqrt{2 \sqrt{21}}