In figure given below, quadrilateral $ABCD$ is circumscribed; find the perimeter of quadrilateral $ABCD$.

In the adjoining figure, quadrilateral $ABCD$ is circumscribed. If the radius of incircle (centre $O$) is $10 \mathrm{cm}$ and $AD\perp DC$. Find the value of $x$.

In the given figure, $\mathrm{O}$ is the centre of the circle and $\mathrm{AB}$ is a tangent at $\mathrm{B}$. If  $\mathrm{AB}=15 \mathrm{cm}$ and $\mathrm{AC}=7.5 \mathrm{cm}$, then calculate the radius of the circle.

In the figure given below, from an external point $P$, tangents $PA$ and $PB$ are drawn to a circle. $CE$ is a tangent to the circle at $D$. If $AP=15 \mathrm{cm}$, find the perimeter of the triangle $PEC$.

If $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ are sides of a right triangle where $\mathrm{c}$ is the hypotenuse, prove that the radius of the circle which touches the sides of the triangle is $\mathrm{r}=\frac{\mathrm{a}+\mathrm{b}-\mathrm{c}}{2}$.

In the given figure, $PB$ is a tangent to a circle with centre $O$ at $B$. $AB$ is a chord of length $24 \mathrm{cm}$ at a distance of $5 \mathrm{cm}$ from the centre. If the length of the tangent is $20 \mathrm{cm}$, find the length of $OP$.

If the sides of a quadrilateral $\mathrm{ABCD}$ touch a circle, prove that $\mathrm{AB}+\mathrm{CD}=\mathrm{BC}+\mathrm{AD}.$

$\mathrm{ABC}$ is a triangle with $\mathrm{AB}=10 \mathrm{cm}, \mathrm{BC}=8 \mathrm{cm}$ and $\mathrm{AC}=6 \mathrm{cm}$. Three circles are drawn touching each other with the vertices as their centres. Find the radii of the three circles.