In the given figure $AB$ is a chord of length $16 \mathrm{c}\mathrm{m}$ of a circle of radius $10 \mathrm{c}\mathrm{m}.$ The tangents at $A$ and $B$ intersect at a point $P.$ Find the length of $PA.$

In the given figure, $\mathrm{PA}$ and $\mathrm{PB}$ are tangents from an external point $\mathrm{P}$ to a circle with centre $\mathrm{O}.$ $\mathrm{LN}$ touches the circle at $M\mathit{.}$ Prove that $\mathrm{PL}+\mathrm{LM}=\mathrm{PN}+\mathrm{MN}.$

$\mathrm{BDC}$ is a tangent to the given circle at point $D$ such that $\mathrm{BD}= 30 \mathrm{cm}$ and $CD = 7 \mathrm{cm}.$ The other tangents $BE$ and $CF$ are drawn respectively from $B$ and $C$ to the circle and meet when produced at $A$ making $BAC$ a right-angled triangle. Calculate $AF$.

In the given figure, tangents $\mathrm{PQ}$ and $\mathrm{PR}$ are drawn from an external point to a circle with centre $\mathrm{O}$, such that $\mathrm{\angle }\mathrm{R}\mathrm{P}\mathrm{Q}=30°, \mathrm{\angle }\mathrm{R}\mathrm{P}\mathrm{Q}=30°$. A chord $RS$ is drawn parallel to the tangent $\mathrm{PQ}$. Find $\mathrm{\angle }\mathrm{R}\mathrm{Q}\mathrm{S}$.

From an external point $P$, tangents $PA=PB$ are drawn to a circle with centre $O.$ If $\mathrm{\angle }\mathrm{P}\mathrm{A}\mathrm{B}=50°$, then find $\mathrm{\angle }\mathrm{A}\mathrm{O}\mathrm{B}$.