Question

The angles of elevation of the top of a tower from two points at a distance, $x$ and $y$ meters from the base and in the same straight line with it are complementary. Find the height of the tower.

• A. $h=\sqrt{xy}$
• B. $h=\frac{x}{y}$
• C. $h=\sqrt{y}$
• D. $h=\sqrt{x}$

Answer 1

Answer: (A)

$h=\sqrt{xy}$

Let the height of the tower be $h.$

In right $△ABC$

$\tan \theta =\frac{h}{x}\dots \left(i\right)$

In right $△ABD$

$\tan (90-\theta )=\frac{h}{y}\dots \left(ii\right)$

From $\left(i\right)$ & $\left(ii\right)$ we get,

$\tan \theta \times \tan (90-\theta )=\frac{h}{x}\times \frac{h}{y}$

$\Rightarrow \tan \theta \cdot \cot \theta =\frac{h^2}{xy}$

$\Rightarrow h^2=xy$

$\Rightarrow h=\sqrt{xy}$

Hence, option $A$ is the correct answer.