In Fig. $\displaystyle 1$, there are two concentric circles with centre O. If ARC and AQB are tangents to the smaller circle from the point A lying on the larger circle, find the length of AC , if $\displaystyle \mathrm{AQ}=5 \mathrm{~cm}$. Fig. - $\displaystyle 1$

\(\displaystyle A Q=A R\) (tangents drawn from external point to the circle) \[\therefore A R=5 \mathrm{~cm} \] Join OR \[\therefore O R \perp A C \text { (radius tangent) } \] Now \(\displaystyle A C\) is the chord of larger circle and we know that perpendicular from the centre bisects the chord \[\therefore A R=R C=5 \mathrm{~cm} \] \[\Rightarrow \mathrm{AC}=5+5=10 \mathrm{~cm} \]