$\overline{PA}$ is a tangent to $\odot(O, r)$ drawn from a point $P$ outside a circle. If $OP = 10$ and $AP = 8$,then the diameter of the circle is equal to $\ldots \ldots \ldots \ldots .$

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.,$BO = 2BC$.

$\overline{PA}$ is a tangent to $\odot(O, r)$ drawn from a point $P$ outside a circle. If $m\angle AOP = 40^\circ$,then $m\angle OPA = \ldots$ (in $^\circ$)

$P$ is a point in the exterior of $\odot(O, r)$ and the tangents from $P$ to the circle touch the circle at $X$ and $Y$. Find $m \angle XPO$,if $m \angle XOY = 100^\circ$. (in $^\circ$)

$\overline{ PA }$ and $\overline{ PB }$ are the tangents to $\odot( O , r)$ drawn from a point $P$ outside a circle. If $m \angle APB = 65^{\circ}$,then $m \angle AOB = \ldots \ldots \ldots . .$ (in $^{\circ}$)

If $\odot(O, 5)$ touches all the sides of a square,then the perimeter of the square is .... .