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

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

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

Prove that the parallelogram circumscribing a circle is a rhombus.

A triangle $\mathrm{ABC}$ is drawn to circumscribe a circle of radius $3 \mathrm{cm}$ such that $\mathrm{BD}=9 \mathrm{cm}$ and $\mathrm{DC}=3 \mathrm{cm}$. Find the sides $\mathrm{AB}$ and $\mathrm{AC}$.