If the tangent at a point $P$ to a circle with centre $O$ cuts a line through $O$ at $Q$ such that $PQ=24 \mathrm{cm}$ and $OQ=25 \mathrm{cm}$. Find the radius of the circle.

$O$ is the centre of a circle of radius $8 \mathrm{cm}$. The tangent at a point $A$ on the circle cuts a line through $O$ at $B$ such that $AB=15 \mathrm{cm}$. If $OB=x \mathrm{cm}$, then find the value of $x$.

$PQ$ and $PT$ are tangents to a circle with centre $O$ and radius $5 \mathrm{cm}$. If $PQ=12 \mathrm{cm}$, then prove that the perimeter of the quadrilateral $PQOT$ is $34 \mathrm{cm}$.

In the figure, $\mathrm{AB}$ is a common tangent to the given circles, which touch externally at $\mathrm{P}$. If $\mathrm{AP}=3.2 \mathrm{cm}$, find the length of $\mathrm{AB}$.

In the figure, $\mathrm{CP}$ and $\mathrm{CQ}$ are tangents to a circle with centre $\mathrm{O}$. $\mathrm{ARB}$ is another tangent touching the circle at $\mathrm{R}$. If $\mathrm{CP}=11 \mathrm{cm}$ and $\mathrm{BC}=7 \mathrm{cm}$, then find the length of $\mathrm{BR}$.

In the given figure, $\mathrm{O}$ is the centre of the circle, $\mathrm{PT}$ is the tangent and $\mathrm{PLM}$ is the secant passing through centre $\mathrm{O}$ and $\mathrm{PT}=8 \mathrm{cm}$  and $\mathrm{PL}=4 \mathrm{cm}$. If the radius of the circle is $x \mathrm{cm}$, find the value of $x$.

With the vertices of a $∆\mathrm{ABC}$ as centre, three circles are described each touching the other two externally. If the sides of triangle are $4 \mathrm{cm}, 6 \mathrm{cm}$ and $8 \mathrm{cm}$, find the radii of the circles.
 

In the fig., from a point $\mathrm{O}$ tangents $\mathrm{OT}, \mathrm{OS}, \mathrm{OQ}$ are drawn. State whether $\mathrm{OS}=\mathrm{OT}$ or not and why?