the angles of elevation of top of a tower from two points at a distances a and b meters from - Maths - Some Applications of Trigonometry

Question

the angles of elevation of top of a tower from two points at a
distances a and b meters from the base and in the same straight
line with it are complementry prove that the height of the tower is
meters

Priyanka Kedia · Accepted Answer

$L e t h e i g h t o f t h e t o w e r A B b e h m e t r e s . I n Δ C A B, \tan θ = \frac{A B}{A C} \Rightarrow \tan θ = \frac{h}{a} ...... (1) I n Δ D A B, \tan (90^{0} - θ) = \frac{A B}{A D} \Rightarrow \tan (90^{0} - θ) = \frac{h}{b} \Rightarrow \cot θ = \frac{h}{b} ...... (2) M u l t i p l y e q u a t i o n (1) a n d (2), w e h a v e, \tan θ \times \cot θ = \frac{h}{a} \times \frac{h}{b} \Rightarrow \tan θ \times \frac{1}{\tan θ} = \frac{h^{2}}{a b} \Rightarrow 1 = \frac{h^{2}}{a b} \Rightarrow h^{2} = a b \Rightarrow h = \sqrt{a b} m e t r e s H e n c e, t h e h e i g h t o f t h e t o w e r i s \sqrt{a b} m e t r e s .$

the angles of elevation of top of a tower from two points at a distances a and b meters from the base and in the same straight line with it are complementry .prove that the height of the tower is meters.