Question

The angle of elevation of the top of a tower from two points distant s and t from its foot are complementary. prove that the height of the tower is √st​

Answer 1

To prove :

• Height of tower = √(st)

Proof :

Let us assume that ;

• AB be the tower with B as its foot .

• Angle of elevation of the top of the tower from the point C is ∅ .

• Angle of elevation of the top of the tower from the point D is (90° - ∅) .

• Also , CB = s and BD = t .

( For figure , please refer to the attachment )

In ∆ABC ,

=> tan∅ = AB/CB

=> tan∅ = AB/s

=> s•tan∅ = AB --------(1)

In ∆ABD ,

=> tan(90° - ∅) = AB/BD

=> cot∅ = AB/t

=> t•cot∅ = AB ----------(2)

Now ,

Multiplying eq-(1) and (2) , we get ;

=> s•tan∅ × t•cot∅ = AB × AB

=> st × tan∅ × cot∅ = AB²

=> st × tan∅ × (1/tan∅) = AB²

=> st = AB²

=> √(st) = AB

=> AB = √(st)

Hence ,

The height of the tower is √(st) .

Proved