Question

A flag staff on the top of a house subtends the same angle $\alpha$ at two points distant a and b from the house and on the same side of it. Prove that the length of flag-staff is $(a+b)\tan \alpha$.

Answer 1

Let BC represent the flag staff on the house AB such that $\angle PBC=\angle BQC=\alpha$ where AP = a, AQ = b. Let M by the mid-point of PQ and L that of BC. If O be the centre of the circle then angle at centre O is 2 $\alpha$. As in last part to circle through B and C will pass through P and Q.
$BC=2LC=2QLtan\alpha$
or $BC=2\left[AM\right]tan\alpha =2[AP+PM]tan\alpha$
$=2[a+\frac{1}{2}(b-a\left)\right]tan\alpha$
$=(a+b)tan\alpha$