Two tangents $\mathrm{TP}$ and $\mathrm{TQ}$ are drawn to a circle with centre $\mathrm{O}$ from an external point $\mathrm{T}$. Prove that $\angle \mathrm{PTQ}=2\angle \mathrm{OPQ}$.

A tangent $PT$ is drawn from an external point $P$ to a circle of radius $3\sqrt{2} \mathrm{cm}$ such that the distance of the point $P$ from $O$ is $6 \mathrm{cm}$ as shown in figure. The value of $\angle TPO$ is

In the fig.,$AP=2 \mathrm{cm}$, $BQ=3 \mathrm{cm}$ and $RC=4 \mathrm{cm}$ then the perimeter of $∆ABC$, is:

In the figure, $\mathrm{ABC}$ is a right-angled triangle, right-angled at $\mathrm{A}$. If $\mathrm{AB}=3 \mathrm{cm}, \mathrm{BC}=5 \mathrm{cm}$ and $\mathrm{AC}=4 \mathrm{cm}$ then the radius of the circle is:

$PQ$ and $PR$ are tangents to a circle with centre $O$ and radius $5 \mathrm{cm}$. If $OP=13 \mathrm{cm}$,then perimeter of quadrilateral $PQOR$ is:

In the fig., if $PA=15 \mathrm{cm}$, then perimeter of $△PQR$ is

Two circles touch externally at $A$. A common tangent $PQ$ to two circles touches the circles at $P$ and $Q$. Then the measure of $\angle PAQ$ is

In the figure, if $PA$ and $PB$ are tangents to the circle with centre $O$ such that $\angle APB=50°$, then $\angle OAB$ is equal to

In Fig., shows two concentric circles with centre $\mathrm{O}$. $\mathrm{AB} \mathrm{and} \mathrm{APQ}$ are tangents to the inner circle from point $\mathrm{A}$ lying on the outer circle. If $\mathrm{AB}=7.5 \mathrm{cm}$ then $\mathrm{AQ}$ is equal to