$\overline{AB}$ is a diameter of $\odot(O, 15)$. $A$ tangent is drawn from $B$ to $\odot(O, 9)$ which touches $\odot(O, 9)$ at $D$. $\overrightarrow{BD}$ intersects $\odot(O, 15)$ at $C$. Find $AC$.

Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.

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.

Write 'True' or 'False' and give reasons for your answer.
In the figure,$PQL$ and $PRM$ are tangents to the circle with center $O$ at the points $Q$ and $R$ respectively,and $S$ is a point on the circle such that $\angle SQL = 50^{\circ}$ and $\angle SRM = 60^{\circ}$. Then $\angle QSR$ is equal to $40^{\circ}$.

In $\Delta PQR$,$\angle Q$ is a right angle. If $PQ = 8$ and $QR = 15$,then the radius of a circle touching all the three sides of $\Delta PQR$ is $\ldots \ldots.$

In $\Delta ABC$,if $AB = 7, BC = 24, AC = 25$,then the diameter of a circle touching all the three sides of the triangle is ............