Question

A house of height $ 100m $ subtends a right angle at the window of an opposite house. If the height of the window is $ 64m $ , then the distance between the two houses is:
(A) $ 48m $
(B) $ 36m $
(C) $ 54m $
(D) $ 72m $

Answer 1

Hint: The given question belongs to the height and distances concept of trigonometry domain. In the problem, we are given the angle subtended by a house on the window of another house along with the height of the house and window. So, we first draw a figure to represent the situation and understand the problem more clearly. We introduce some variables for the distance between the two houses and the angles. Then, we make use of the trigonometric ratios in the triangle.

Complete step-by-step answer:
So, the height of the house is $ 100m $ and the height of the window is $ 64m $ . Also, the angle subtended by the house on the window is a right angle.
So, we first draw a figure representing the problem given to us.

Let us assume the height of the house is AB and the window is DC.
Then, let us make a horizontal line from the window to the house intersecting AB at F and let the distance between the two houses be k.
So, a rectangle CBFD is formed. Now, opposite sides are of equal lengths in a rectangle.
So, we get, $ BF = CD = 64m $ .
Let us assume the angle FDB to be X and angle ADF be Y. Now, we know that the angle subtended by the house at the window is a right angle.
So, $ X + Y = {90^ \circ } $ . Hence, angles X and Y are complementary angles.
Now, in triangle ADF, we have,
 $ \tan \angle ADF = \dfrac{{{\text{Perpendicular}}}}{{{\text{Base}}}} = \dfrac{{AF}}{{DF}} $
 $ \Rightarrow \tan \angle ADF = \dfrac{{AB - BF}}{{DF}} $
Substituting the values of AB, BF and CD, we get,
 $ \Rightarrow \tan \angle ADF = \dfrac{{100m - 64m}}{k} $
Substituting the value of angle ADF and simplifying the equation, we get,
 $ \Rightarrow \tan Y = \dfrac{{36}}{k} $
Now, in triangle BDF, we have,
 $ \tan \angle FDB = \dfrac{{BF}}{{DF}} $
Substituting the values of BF, DF and angle FDB, we get,
 $ \Rightarrow \tan X = \dfrac{{64}}{k} $
Now, we know that angles X and Y are complementary angles of each other. So, we have,
 $ X + Y = {90^ \circ } $
Taking Y to the right side of equation,
 $ \Rightarrow X = {90^ \circ } - Y $
Taking tangent on both sides of the equation, we get,
 $ \Rightarrow \tan X = \tan \left( {{{90}^ \circ } - Y} \right) $
Now, we know that tangent and cotangent are complementary ratios. So, we have, $ \tan \left( {{{90}^ \circ } - x} \right) = \cot x $ .
 $ \Rightarrow \tan X = \cot Y $
Also, tangent and cotangent are reciprocal functions of each other. So, we get,
 $ \Rightarrow \tan X = \dfrac{1}{{\tan Y}} $
Substituting the values of tangent of angles X and Y,
 $ \Rightarrow \dfrac{{64}}{k} = \dfrac{1}{{\left( {\dfrac{{36}}{k}} \right)}} $
Simplifying the equation,
 $ \Rightarrow \dfrac{{64}}{k} = \dfrac{k}{{36}} $
Cross multiplying the terms,
 $ \Rightarrow {k^2} = 64 \times 36 $
Taking square roots on both the sides, we get,
 $ \Rightarrow k = \sqrt {64 \times 36} m $
Simplifying the calculations, we get,
 $ \Rightarrow k = 8 \times 6m $
 $ \Rightarrow k = 48m $
Therefore, the distance between the houses is $ 48 $ meters.
So, the correct answer is “48 Meters”.

Note: We should have a strong grip over the concepts of trigonometry and height and distances in order to deal with such kinds of problems. We should have accuracy in calculations, derivatives and arithmetic in order to be sure of the final answer. Definitions of trigonometric ratios such as tangent and cotangent should be remembered to solve such questions.