Case Study on Tangents and Circles

Ramesh was fond of spinning/revolving a ball tied with a rope of length 2 m. While playing, the ball slipped from the rope and hit his brother Suresh lightly. Suresh threw the ball back to Ramesh that made an angle of $60^\circ$ with AX along the path XYZ which is a tangent to the imaginary circle as shown in fig. 1. i.e. $\angle AXZ = 60^\circ$.

Based on the above, answer the following questions:

If the length of rope used by Ramesh is 2 m, what is the diameter of the circle ?

Find the length XY shown in figure.

(a) Ramesh made the same circle on the ground using the rope he had and placed 4 sticks around it as shown in the figure with $AB = 3\text{ m}, AD = BC = 5\text{ m}$. Find length of CD. Justify your answer.

OR

(b) Suresh made another circle with radius 4 m and placed 3 sticks as shown in the figure. If $AD = 6\text{ m}, BD = 8\text{ m}$ and area of triangle ABC is $84\text{ sq. m.}$, find the lengths of AC and BC.