Question

A tower stands at the center of a circular park.

$A$ and $B$ are two points on the boundary of the park such that $(AB=a)$subtends an angle of $60°$ at the foot of the tower and the angle of elevation of the top of the first tower from $A$ or $B$ is $30°$.

The height of the tower is

• A. $\frac{2a}{\sqrt{3}}$
• B. $2a\sqrt{3}$
• C. $\frac{a}{\sqrt{3}}$
• D. $\sqrt{3}$

Answer 1

The correct option is C

$\frac{a}{\sqrt{3}}$

Explanation for the correct option:

STEP 1: Find the height of the tower:

Let $OP$ be the tower. Also $OAB$ forms an equilateral triangle as all sides are equal of length $a$.

That is, $\left(OA=OB=AB\right)=a$

STEP 2: Use the property of $\mathit{t}\mathit{a}\mathit{n}\mathit{\theta }$ in right angled triangle $AOP$ to find $OP$.

In right triangle $AOP$,

$\Rightarrow$$\begin{array}{rcl}\tan 30°& =& \frac{OP}{OA}\end{array}$

$\Rightarrow$ $\begin{array}{rcl}\frac{1}{\sqrt{3}}& =& \frac{OP}{a}\end{array}$

$\Rightarrow$ $\begin{array}{rcl}OP& =& \frac{a}{\sqrt{3}}\end{array}$

Thus, the height of the tower is $\frac{\mathbf{a}}{\sqrt{\mathbf{3}}}$.

Hence, option (C) is correct.