A window of a house is h meters 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) meters

Solution:

Given, a window of a house is h m above the ground.

From the window, the angle of elevation and depression of the top and bottom of another house situated on the opposite side of the lane are found to be α and β.

We have to prove that the height of the other house is h (1 + tan α cot β) meters.

From the figure,

W be the window of the house

Height of the window from the ground, BW = h m

Let OQ be another house.

Height of the another house, QO = Hm

Angle of elevation, ∠QWM = 𝛼

Angle of depression, ∠WOB = β

Consider, BO = xm

In triangle QWM,

tan𝛼 = QM/WM

Since, QM = QO - MO,

tan𝛼 = H - h/x

x = H - h/tan𝛼--------------------(1)

Considering triangle WOB,

tanβ = WB/BO

tanβ = h/x

x = h/tanβ--------------------------(2)

By comparing (1) and (2), we get,

(H - h)/tan α = h/tan β

(H - h) tan β = h tan𝛼

H tanβ - h tanβ = h tan𝛼

H tanβ = h tan𝛼 + h tanβ

H tanβ = h (tan𝛼 + tanβ )

H = (h (tan𝛼 + tanβ))/tanβ

H = h (tan𝛼 /tanβ + 1)

H = h (1 + tan𝛼 /tanβ)

H = h (1 + tan𝛼 cotβ)

Therefore, the height of the other house is h (1 + tan𝛼 cotβ).

✦ Try This: To a man standing outside his house, the angles of elevation of the top and bottom of a window are 60° and 45° respectively. If the height of the man is 180 cm and if he is 5 m away from the wall, what is the height of the window? (√3 = 1.732)

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 8

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NCERT Exemplar Class 10 Maths Exercise 8.4 Problem 17

A window of a house is h meters 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) meters

Summary:

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. It is proven that the height of the other house is h ( 1 + tan α cot β ) metres

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