# A vertical pole is divided by any point in the ratio 9:1. If both the segments subtend equal angles to each other at a point 20m away from the foot of the pole , find the height of the pole?

Aug 5, 2018

Let the lengths of two parts be $9 x \mathmr{and} x$ m.

So total length will be $10 x$ m

So length of hypotenuse $= \sqrt{{20}^{2} + 100 {x}^{2}}$ m

As the angles subtended by two parts at 20 m distance from the foot are same, we can write

$\frac{\sqrt{{20}^{2} + 100 {x}^{2}}}{20} = \frac{9}{1}$

$\implies \frac{{20}^{2} + 100 {x}^{2}}{400} = 81$

$\implies 100 {x}^{2} = 81 \times 400 - 400 = 80 \times 400$

$\implies x = \sqrt{320} = 8 \sqrt{5}$

So length of the pole $= 80 \sqrt{5}$ m