Prove that a diameter $\mathrm{AB}$ of a circle bisects all those chords which are parallel to the tangent at that point $\mathrm{A}$.

If $\mathrm{AB}, \mathrm{AC}, \mathrm{PQ}$are tangents in the given figure and $\mathrm{AB} = 5 \mathrm{cm}$, find the perimeter of $\mathrm{\Delta }\mathrm{A}\mathrm{P}\mathrm{Q}.$

In the figure, $\mathrm{PQ}$ is tangent at a point $\mathrm{R}$ of the circle with centre $\mathrm{O}$. If $\mathrm{\angle }\mathrm{T}\mathrm{R}\mathrm{Q}=30°,$ find $\mathrm{m}\mathrm{\angle }\mathrm{P}\mathrm{R}\mathrm{S}.$

In the given figure, $\mathrm{ABC}$ is a right triangle right-angled at $\mathrm{B}$ such that $\mathrm{BC}=6 \mathrm{cm}$ and $\mathrm{AB}=8 \mathrm{cm}.$ Find the radius of its incircle.