Draw a circle and two lines parallel to a given line drawn outside the circle such that one is a tangent and the other, a secant to the circle.

Calculate the length of tangent from a point $15 \mathrm{cm}$ away from the centre of a circle of radius $9 \mathrm{cm}$.

Express your answer in centimetre(cm).

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

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 $\mathrm{AC}$.

In a right triangle $ABC$, a circle with a side $AB$ as diameter is drawn to intersect the hypotenuse $AC$ in $P$. Prove that the tangent to the circle at $P$ bisects the side $BC$.

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