$A$ line drawn from the centre $O$ of a circle with radius $7 \, cm$ intersects the tangent at point $P$ on the circle at point $Q$,such that $PQ = 24 \, cm$. Find $OQ$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$. $A$ circle touches all the sides of $\Delta ABC$. If $AB + AC = 32$ and $BC = 24$,then find the radius of the incircle.

$AB$ is a diameter and $AC$ is a chord of a circle with centre $O$ such that $\angle BAC = 30^{\circ}$. The tangent at $C$ intersects the extended $AB$ at a point $D$. Prove that $BC = BD$.

Radii of two concentric circles are $13$ and $8$. $\overline{AB}$ is a diameter of the circle with the larger radius. $\overline{BD}$ touches the circle with the smaller radius at $D$. Find $AD$.

State whether the following statement is 'True' or 'False' and provide reasons for your answer: The length of a tangent from an external point to a circle is always greater than the radius of the circle.

$A$ tangent $\overline{PM}$ is drawn from a point $P$ outside $\odot(O, 8)$. $\overline{OP}$ intersects the circle at $N$. If $NP = 2$,then $PM = \ldots$