As shown in the figure,$AB$,$AC$ and $\overleftrightarrow{PQ}$ are tangents to the circle. If $AB = 6$,then the perimeter of $\Delta APQ = \ldots \ldots \ldots$.

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

$\overleftrightarrow{PA}$ and $\overleftrightarrow{PB}$ are tangents to the circle $\odot(O, r)$ at $A$ and $B$ respectively. If $m\angle AOB = 100^\circ$,then $m\angle OPB = \dots$ (in $^\circ$)

$A$ line passing through the centre of the circle $O$ intersects the tangent to the circle at $Q$. $P$ is the point of contact. If the radius of the circle is $9$ and $PQ = 40$,then find $OQ$.

$AB$ 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 $BC = BD$.

$AB$ and $CD$ are common tangents to two circles. If the radii of the two circles are equal,prove that $AB = CD$.