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 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 $BC$.

.

Draw a tangent to a given circle with centre $\mathrm{O}$ from a point $\text{'}\mathrm{R}\text{'}$ outside the circle. How many tangents can be drawn to the circle from that point?

[Hint: The distance of two points to the point of contact is the same.]

The area of sector$=$_____${\mathrm{cm}}^2$, If the radius is $7 \mathrm{cm}$ with the given angle: ${60}^0$.

If the area of sector, with the radius $7 \mathrm{cm}$ and the angle $30°$ is $k {\mathrm{cm}}^2$, then find the value of $k$. (write the $k$ value to $2$ decimal place)