Question

A person moving towards a house observes that a flag-staff on the top of the house subtends the greatest angle $\theta$ when his distance from the house is d. Prove the heights of the flag staff and the house are $2d\tan \theta$ and $d\tan ({45}^o-\frac{\theta }{2})$ respectively.

Answer 1

Let OP = y and PQ = x represent the height of the house and the flag staff and the direction of the person is from D to O. Clearly the flag staff PQ subtends the greatest angle at a point A at which a circle through P, Q touches DO.
Hence $\angle PAQ=\theta$ and AO = d.
Let $\angle OAP=\alpha$. Then in the alternate segment,
$\angle AQP=\alpha$ so that
$2\alpha +\theta ={90}^o$. $\therefore \theta +\alpha ={90}^o-\alpha$ ...(1)
As angle between chord AP and tangent at A is the same as the angle subtended by segment AP at any point Q on the circumference. Now,
$PQ=OQ-OP=d\left(\tan \right(\alpha +\theta )-\tan \alpha )$
$=d(\cot \alpha -\tan \alpha )$ by (1)
$=2d\frac{({\cos }^2\alpha -{\sin }^2\alpha )}{2\sin \alpha \cos \alpha }$
$=2d\cot 2\alpha =2d\cot ({90}^o-\theta )$ by (1)
$=2d\tan \theta$
and $OP=d\tan \alpha =d\tan ({45}^o-\theta /2)$ by (1)