In the figure below, PQ and RQ are tangents to the circle with centre O and radius 6cm

i) Prove that $△\mathrm{PQR}$ is an equilateral triangle.
ii) Find the length of RP. 

A point is 25 cm from the centre of a circle of radius 15 cm. Find the length of the tangent from the point to the circle. 

In the figure below, a circle with centre O is inscribed inside ALMN. A and B are the points of tangency.

Find $\angle \mathrm{ANB}.$

Shown below is a circle with centre $\mathrm{O}$. VP = 34 cm, PR = 36 cm and RS = 17 cm. U, W, Q and S are the points of tangency.

Find the area of the shaded region in terms of $\mathrm{\pi }$. 

Shown below is a circle with centre O and radius 5 units. PM and PN are tangents and the length of chord MN is 6 units.

Find the length of (PM + PN). Show your work.

 In the figure given below, PA is a tangent to the circle with centre O and PCB is a straight line.

Find the measure of $\angle \mathrm{OBC}$. Show your steps and give valid reasons.

ABCD is a square. CD is a tangent to the circle with centre O as shown in the figure below.

If OD = CE, what is the ratio of the area of the circle and the area of the square? Show your steps and give valid reasons.

Shown below is a circle with centre $\mathrm{O}$ and tangents $\mathrm{PQ}$ and $\mathrm{PR}$.

Using triangles $\mathrm{QOR}$ and $\mathrm{PQR}$, and without doing any extra constructions, prove that the tangents $\mathrm{PQ}$ and $\mathrm{PR}$ are equal in length.

In the following figure, $\mathrm{O}$ is the centre of the circle and $\mathrm{PQ}$ and $\mathrm{PS}$ are tangents to the circle at points $\mathrm{Q}$ and $\mathrm{S}$ respectively.

What is the measure of $\angle \mathrm{QRS}$? (Without symbol) Show your work.