Prove that the tangents to a circle at the endpoints of a diameter are parallel.

From a point $\mathrm{Q}$, the length of the tangent to a circle is $24 \mathrm{cm}$ and the distance of $\mathrm{Q}$ from the centre is $25 \mathrm{cm}$. The radius of the circle is

If $\mathrm{AP}$ and $\mathrm{AQ}$ are the two tangents a circle with centre $\mathrm{O}$ so that $\angle \mathrm{POQ}=110°,$  then $\angle \mathrm{PAQ}$ is equal to

If tangents $PA$ and $PB$ from a point $P$ to a circle with centre $O$ are inclined to each other at angle of $80°,$ then $\angle POA$ is equal to

In the figure, $\mathrm{XY}$ and ${\mathrm{X}}^{\text{'}}{\mathrm{Y}}^{\text{'}}$ are two parallel tangents to a circle with centre $\mathrm{O}$ and another tangent $\mathrm{AB}$ with point of contact $\mathrm{C}$ intersecting $\mathrm{XY}$ at $\mathrm{A}$ and ${\mathrm{X}}^{\text{'}}{\mathrm{Y}}^{\text{'}}$ at $\mathrm{B}$ then $\angle \mathrm{AOB}=$

Two concentric circles of radii $5 \mathrm{cm}$ and $3 \mathrm{cm}$ are drawn. Find the length of the chord of the larger circle which touches the smaller circle.

Express your answer in centimetres(cm).

A triangle $\mathrm{ABC}$ is drawn to circumscribe a circle of radius $3 \mathrm{cm}$ such that the segments $\mathrm{BD}$ and $\mathrm{DC}$ into which $\mathrm{BC}$ is divided by the point of contact $\mathrm{D}$ are of length $9 \mathrm{cm}$ and $3 \mathrm{cm}$ respectively (See adjacent figure). Find the sides $\mathrm{AB}$ and $\mathrm{AC}$.