In the given figure, a circle touches all the four sides of a quadrilateral $\mathrm{ABCD}$, whose three sides are $\mathrm{AB}=6 \mathrm{cm}, \mathrm{BC}=7 \mathrm{cm}$ and $\mathrm{CD}=4 \mathrm{cm}$. Find $\mathrm{AD} \left(\mathrm{cm}\right)$.

In the given figure, $\mathrm{AB}$ is a diameter of the circle with centre $\mathrm{O}. \mathrm{AP}, \mathrm{BQ} \mathrm{and} \mathrm{PRQ}$ are tangents. Prove that $\angle \mathrm{POQ} = 90°$.

In the given figure, $\mathrm{AB}$ is a diameter of a circle with centre $\mathrm{O}$ and $\mathrm{CP}$ is the tangent to the circle at the point $\mathrm{C}$. If $\mathrm{CP}=15 \mathrm{cm}$ and $\mathrm{BP}=7.5 \mathrm{cm}$, then find the radius of the circle (in $\mathrm{cm})$

In the given figure, $\mathrm{AB}$ is a diameter of a circle with centre $\mathrm{O}$ and $\mathrm{CP}$ is the tangent to the circle at the point $\mathrm{C}$. If  $\mathrm{AP} = 20 \mathrm{cm} \mathrm{and} \mathrm{CP} = \mathrm{l}0 \mathrm{cm}$, then find the radius of the circle (in $\mathrm{cm}$).

In the given figure, $\mathrm{AB}$ is a diameter of a circle with centre $\mathrm{O}$ and $\mathrm{CP}$ is the tangent to the circle at the point $\mathrm{C}$. If $\mathrm{CP}=15 \mathrm{cm}$ and $\mathrm{BP}=7.5 \mathrm{cm}$, then find the radius $\left(\mathrm{cm}\right)$ of the circle.

In the given figure, $\mathrm{ABC}$ is a triangle in which $\mathrm{AB} = \mathrm{AC}$. A circle through $\mathrm{B}$ touches $\mathrm{AC} \mathrm{at} \mathrm{D}$ and intersects $\mathrm{AB} \mathrm{at} \mathrm{P}$. If $\mathrm{D}$ is the mid-point of $\mathrm{AC}$, then show that  $4\mathrm{AP}=\mathrm{AB}$.

In the given figure, $\mathrm{O}$ is the centre of a circle. Find the value of $\mathrm{b}$.