Question

The angle of elevation of the top of a $24{\text{ m}}$ high pillar from the window of a building $6{\text{ m}}$ above the ground and the angle of depression of its base are complementary angles. The distance of building from pillar is:
(A) $2\sqrt 3 {\text{ m}}$
(B) $8\sqrt 3 {\text{ m}}$
(C) $12\sqrt 3 {\text{ m}}$
(D) $6\sqrt 3 {\text{ m}}$

Answer 1

Hint: Analyse the situation with a diagram. Consider two different triangles for angle of elevation and angle of depression and compare the results obtained from both of them. Use formula $\tan \theta = \dfrac{{{\text{Perpendicular}}}}{{{\text{Base}}}}$

Complete step-by-step answer:

Consider the above diagram. AE is representing the building while E is the position of window. BC is a pillar.
According to the question, the angle of elevation from the window and angle of depression of the base of the building are complementary angles i.e. their sum is ${90^ \circ }$. So if $\angle CAB = x$ then $\angle CED = 90 - x$.
Let the length AB, which is the distance of the building from pillar, to be $h$, $AB = h$.
Now, in triangle ABC:
$
   \Rightarrow \tan x = \dfrac{{BC}}{{AB}} \\
   \Rightarrow \tan x = \dfrac{{24}}{h}{\text{ }}.....{\text{(1)}} \\
$
Again in triangle DCE,
$ \Rightarrow \tan \left( {90 - x} \right) = \dfrac{{CD}}{{DE}}$
From the figure, we have:
$ \Rightarrow DE = AB = h$
$ \Rightarrow CD = CB - DB$ and $DB = EA = 6{\text{ m}}$ whereas $CB = 24{\text{ m}}$.
Thus, $CD = 24 - 6 = 18{\text{ m}}$
Putting these values in above expression, we’ll get:
$ \Rightarrow \tan \left( {90 - x} \right) = \dfrac{{18}}{h}$
We know that $\tan \left( {90 - x} \right) = \cot x$. So we have:
$ \Rightarrow \cot x = \dfrac{{18}}{h}{\text{ }}.....{\text{(2)}}$
Now multiplying equation (1) and (2) we’ll get:
$ \Rightarrow \tan x\cot x = \dfrac{{24}}{h} \times \dfrac{{18}}{h}$
We know that $\tan x\cot x = 1$. Using this, we’ll get:
$
   \Rightarrow 1 = \dfrac{{6 \times 4 \times 6 \times 3}}{{{h^2}}} \\
   \Rightarrow {h^2} = {6^2} \times {2^2} \times 3 \\
$
Taking square root both sides, we have:
$
   \Rightarrow h = 6 \times 2\sqrt 3 \\
   \Rightarrow h = 12\sqrt 3 \\
$
Thus the distance of the building from the pillar is \[12\sqrt 3 \].

(C) is the correct option.

Note: The formula of $\tan \theta $ is used in a triangle whenever a relation between two non-hypotenuse sides of the triangle is required. Other trigonometric ratios can also be used if hypotenuse is to be determined or a relation between hypotenuse and any other side is to be compared.