Question

The angles of elevation of the top of a tower from two points on the ground at distances a metres and b metres from the base of the tower and in the same straight line are complementary. The height of the tower is $\sqrt{ab}$ metres.

• A. True
• B. False

Answer 1

The correct option is A True

Let AB be the tower and let C and D be the two points of the observer. Then,

AC = a metres and AD = b metres

Let $\angle ACB=\theta$. Then, $\angle ADB=({90}^o-\theta )$.

Let AB = h metres

In right $△CAB$, we have

$\tan \theta =\frac{AB}{AD}$

$\tan \theta =\frac{h}{a}$

$h=a\tan \theta$ ......(1)

In right $△DAB$, we have,

$\tan ({90}^o-\theta )=\frac{AB}{AD}$

$\cot \theta =\frac{h}{b}$

$h=b\cot \theta$ .......(2)

From 1 and 2, we get,

$h^2=ab$

$h=\sqrt{ab}metres$