If an isosceles triangle $ABC$, in which $AB = AC = 6\, cm$, is inscribed in a circle of radius $9\, cm$, find the area of the triangle in $cm^{2}$.

Write 'True' or 'False' and give reasons for your answer.
In the figure,$BOA$ is a diameter of a circle and the tangent at a point $P$ meets $BA$ extended at $T$. If $\angle PBO = 30^{\circ}$,then $\angle PTA$ is equal to $30^{\circ}$.

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 $AQ = \frac{1}{2}(BC + CA + AB)$.

Two circles with centres $O$ and $O^{\prime}$ of radii $3\, cm$ and $4\, cm$,respectively,intersect at two points $P$ and $Q$ such that $OP$ and $O^{\prime}P$ are tangents to the two circles. Find the length of the common chord $PQ$ (in $cm$).

In $Fig.$,$AB$ and $CD$ are common tangents to two circles of unequal radii. Prove that $AB = CD$.

$A$ circle touches all the sides of $\square ABCD$. If the largest side of $\square ABCD$ is $\overline{AB}$,then the smallest side is: