A window of a house is h metres above the ground From the window, the angles of elevation and - Maths - Arithmetic Progressions

Question

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 alpha and beta, respectively Prove that the height
of the other house is h(1 + tan alpha * cot beta)

Vimala · Accepted Answer

Let us observe the following figure

The point of observation is at A The height of the window is h meters Thus, AB = CD = h meters The top and bottom of another house in the opposite lane is E and C Therefore the angles of elevation and the depression are $∠ E A D = α a n d ∠ D A C = β$ Consider the triangle ADE Thus, $\tan α = \frac{E D}{D A} E D = A D \tan α (1)$ ; Now consider the triangle ABC $\tan β = \frac{A B}{B C} \tan β = \frac{h}{B C} B C = \frac{h}{\tan β} B C = h \cot β (2)$ From the figure, it is clear that AD = BC Thus, equation (2) becomes, $A D = h \cot β (3)$ Substitute the value of AD from equation (3) in equation (1), we have $E D = A D \tan α = (h \cot β) \tan α (4)$ The height of the house is CE That is, $C E = C D + D E = h + D E (5)$ Substitute the value of ED from equation (4) in equation (5), we hve $C E = h + D E = h + h \tan α \cot β = h (1 + \tan α \cot β)$ Thus, the toltal height of the the housse is, $h (1 + \tan α \cot β)$