A window of a house is h metres above the ground. From the window, the angles of elevation and depression of the top and the bottom of another house situated on the opposite side of the lane are found to be α and β, respectively. Prove that the height of

Name: A window of a house is h metres above the ground. From the window, the angles of elevation and depression of the top and the bottom of another house situated on the opposite side of the lane are found to be α and β, respectively. Prove that the height of
Uploaded: Mon, 2021-02-01 19:18
Description: A window of a house is h metres above the ground. From the window, the angles of elevation and depression of the top and the bottom of another house situated on the opposite side of the lane are found to be α and β, respectively. Prove that the height of

A window of a house is h metres above the ground. From the window, the angles of elevation and depression of the top and the bottom of another house situated on the opposite side of the lane are found to be α and β, respectively. Prove that the height of the other house is h (1 + tan α cot β) metres.

Answers (1)

Solution. According to the question :

Let the height of the other house is X.
In $\bigtriangleup$ DCF.
$\tan \alpha = \frac{P}{B}= \frac{EC}{DC}$
$\tan \alpha =\frac{x-h}{DC}$
$DC= \frac{x-h}{\tan \alpha }\cdots \left ( 1 \right )$
In $\bigtriangleup$ DAB
. $\tan \beta = \frac{P}{B}= \frac{DA}{AB}$
$\tan \beta = \frac{h}{AB}= \frac{h}{DC}$ $\left ( \because AB= DC \right )$
$\tan \beta = \frac{h}{DC}$
$DC= \frac{h}{\tan \beta }\cdots \left ( 2 \right )$
from equation (1) and (2)
$\frac{X-h}{\tan \alpha}= \frac{h}{\tan \beta}$
$X-h= \frac{\tan \alpha h}{\tan \beta }$
$X= \frac{\tan \alpha h+\tan \beta h}{\tan \beta}$
$X= \frac{h\left ( \tan \alpha +\tan \beta \right )}{\tan \beta}$
separately divide
$X= h\left ( \frac{\tan \alpha }{\tan \beta }+1 \right )$
$X= h\left ( 1+\tan \alpha \cot \beta \right )$ $\left ( \because \frac{1}{\tan \theta }= \cot \theta \right )$
$X= h\left ( 1+\tan \alpha \cot \beta \right )$
Hence proved