Question

The angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m from the base of the tower and in the same straight line with it are complementary. The height of the tower is ___ m.

Answer 1

a + b = ${90}^{\circ }$ (complementary angles)

DB = 4 m

AB = 9 m

Let BC = x (height)

In $\Delta$BCD,

$\tan$ b =$\frac{x}{4}$

In $\Delta$ABC,

$\cot a=\frac{9}{x}$

But $\cot a=\cot$(90 - b) = $\tan$ b

Hence, $\frac{9}{x}$ = $\frac{x}{4}$

$\Rightarrow$ $x^2$ = 36

$\Rightarrow x=\mp$ 6

$\Rightarrow$ x = 6, since distance is positive.

So, the height of the building is 6 m.