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 $cm^{2}$.

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

$A$ 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 $OQ$.

At one end $A$ of a diameter $AB$ of a circle of radius $5 \, cm$,a tangent $XY$ is drawn to the circle. The length of the chord $CD$ parallel to $XY$ and at a distance of $8 \, cm$ from $A$ is (in $cm$):

$A$ line that intersects a circle at two distinct points is called a $\ldots \ldots \ldots \ldots$ of the circle.

$A$ tangent from $P$,a point in the exterior of a circle,touches the circle at $Q$. If $OP = 29$ and $PQ = 20$,then the diameter of the circle is: