Question

A man lying on a 6 m tall building finds that the angle of elevation of a 24 m high pillar and the angle of depression of its base are complementary angles. Find the distance of the man from the pillar.

• A. $6\sqrt{3}m$
• B. $6m$
• C. $2\sqrt{3}m$
• D. $3\sqrt{3}m$

Answer 1

The correct option is A $6\sqrt{3}m$


Given:
Man is at a height of 6 m.
Height of the building = 24 m.
Angle of elevation and angle of depression are complementary angles.

To find: Distance between man and the pillar $EC$.

Since the height of the building is 6m, AE = BC = 6 m.
Remaining height of the pillar = 24 m - 6 m = 18 m
So, BC = 18 m.

If the angle of depression is $\angle CEB=\theta$, then the angle of elevation will be $\angle CED={90}^o-\theta$.

The trigonometric ratio connecting the oposite side and adjacent side is $tan\theta$.
Hence, in triangle ECB,
$tan\theta =\frac{BC}{EC}=\frac{6}{EC}$-----(i)
Similarly, in triangle ECD,
$tan(90-\theta )=cot\theta =\frac{EC}{DC}=\frac{18}{EC}$------(ii)
We know that,
$tan\theta =\frac{1}{cot\theta }\Rightarrow \frac{6}{EC}=\frac{1}{\frac{18}{EC}}\Rightarrow \frac{6}{EC}=\frac{EC}{18}\Rightarrow E{C}^2=108$
Taking square root on both the sides, we get
$EC=6\sqrt{3}m$
The distance of man from the pillar is $6\sqrt{3}m$.