The Postmaster realizes that Lencho, a poor farmer, is steadfast in his belief that God will help him. To fulfill his resolution, he decides to collect money from his colleagues, despite the fact that the amount is beyond his own means. He does this to support Lencho’s faith and provide him with help.
In the adjoining figure, $\triangle CAB$ is a right triangle, right angled at A and $AD \perp BC$. Prove that $\triangle ADB \sim \triangle CDA$. Further, if $BC = 10$ cm and $CD = 2$ cm, find the length of AD. 
In the adjoining figure, TP and TQ are tangents drawn to a circle with centre O. If $\angle OPQ = 15^\circ$ and $\angle PTQ = \theta$, then find the value of $\sin 2\theta$. 