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

In the given figure, $\mathrm{O}\mathrm{Q}:\mathrm{P}\mathrm{Q}=3:4$ and perimeter of $\mathrm{\Delta }\mathrm{P}\mathrm{O}\mathrm{Q}=60 \mathrm{c}\mathrm{m}.$ Determine $\mathrm{PQ}, \mathrm{QR}$ and $\mathrm{OP}$.

In 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 $\mathrm{C}\mathrm{P}=11 \mathrm{c}\mathrm{m}$ and $\mathrm{B}\mathrm{C}=7 \mathrm{c}\mathrm{m},$ then find the length of $\mathrm{BR}$.

In, figure, $\mathrm{CP}$ and $\mathrm{CQ}$ are tangents from an external point $\mathrm{C}$ to a circle with centre $\mathrm{O}$. $\mathrm{AB}$ is another tangent which touches the circle at $\mathrm{R}$. If $\mathrm{CP}=11 \mathrm{cm}$ and $\mathrm{BR}=4 \mathrm{cm}$, if the length of $\mathrm{BC}=k\mathit{ }\mathrm{cm}$, find $k$.

In figure, if $\mathrm{\angle }\mathrm{A}\mathrm{O}\mathrm{B}=125°$, then $\mathrm{\angle }\mathrm{DOC}$ is equal to