If $\square ABCD$ is a cyclic quadrilateral and also a rectangle,and if $AB = 5$ and $BC = 12$,then $AC = \ldots$

$\overline{PA}$ is a tangent to $\odot(O, 8)$ drawn from a point $P$ outside the circle. If $m\angle AOP = 45^\circ$,then $AP = \ldots$

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.

If from an external point $B$ of a circle with centre $O$,two tangents $BC$ and $BD$ are drawn such that $\angle DBC = 120^{\circ}$,prove that $BC + BD = BO$,i.e.,$BO = 2BC$.

If a circle touches the side $BC$ of a triangle $ABC$ at $P$ and extended sides $AB$ and $AC$ at $Q$ and $R$ respectively,prove that $AQ = \frac{1}{2}(BC + CA + AB)$.

$P$ is in the exterior of $\odot (O, 9)$. $A$ tangent from $P$ touches the circle at $T$. If $PT = 40$,then $OP = \ldots$