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

$A$ circle touches all four sides of $\square ABCD$. If $AB = 5, BC = 8$ and $CD = 6$,then $AD = .......$

Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines.

$P$ 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 $OP$,if $r = 12$ and $XP = 5$.

In the figure,$AB$ is a chord of the circle and $AOC$ is its diameter such that $\angle ACB = 50^{\circ}$. If $AT$ is the tangent to the circle at point $A$,then $\angle BAT$ is equal to (in $^{\circ}$)

Write 'True' or 'False' and give reasons for your answer.
If a number of circles pass through the end points $P$ and $Q$ of a line segment $PQ$,then their centres lie on the perpendicular bisector of $PQ$.