Question

The tops of two towers of height x and y, standing on level ground, subtend angles of ${30}^{\circ }$ and ${60}^{\circ }$ respectively at the centre of the line joining their feet, then find the ratio of x, y.

Answer 1

Let AB and CD be the two towers of heights x and y, respectively.

Suppose E is the centre of the line joining the feet of the two towers i.e. BD.
Now, in $\Delta$ ABE,
$\frac{AB}{BE}=\tan {30}^{\circ }$
$\Rightarrow \frac{x}{BE}=\frac{1}{\sqrt{3}}$
$\Rightarrow BE=\sqrt{3}x.......\left(1\right)$
Also,
In $\Delta$ CDE,
$\frac{CD}{DE}=\tan {60}^{\circ }$
$\Rightarrow DE=\frac{y}{\sqrt{3}}......\left(2\right)$
Now $BE=DE$ ....(3) (E is mid-point of BD)
So, from (1), (2) and (3) we get
$\sqrt{3}x=\frac{y}{\sqrt{3}}$
$\Rightarrow \frac{x}{y}=\frac{1}{3}$
Hence, the ratio of x and y is 1 : 3.