$A$ circle touches all the sides of a quadrilateral $ABCD$. If $AB = 8$,$BC = 10$,and $CD = 7$,then find the length of $AD$.

The incircle of $\Delta ABC$ touches the sides $\overline{AB}$, $\overline{BC}$ and $\overline{CA}$ at points $P$, $Q$ and $R$ respectively. If $AB = 14$, $BC = 11$ and $CA = 7$, find the lengths of $AP$, $BQ$ and $RC$.

Write 'True' or 'False' and give reasons for your answer.
The tangent to the circumcircle of an isosceles triangle $ABC$ at $A$,in which $AB = AC$,is parallel to $BC$.

Fill in the blanks:
$(i)$ $A$ tangent to a circle intersects it in $.........$ point$(s)$.
$(ii)$ $A$ line intersecting a circle in two points is called a $.........$

For $\Delta ABC$,$a=5$,$b=12$ and $c=13$. The radius of a circle touching all the three sides of $\Delta ABC$ is ..... .

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