A tree standing on a horizontal plane is leaning towards east. At two points situated at distances a and b exactly due west on it, the angles of elevation of the top are respectively $α a n d β$ . Prove that the height of the top from the ground is $\frac{(b - a) t a n α t a n β}{t a n α - t a n β}$ .

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Solution

Let height of the tree = AB = h
Distance between first observe point to foot of tower BD = a
Distance between secondt observe point to foot of tower BC = b
α, β are the angle of elevation to the top of the tower.
∠ACB = α and ∠ADB = β.
In ΔADB

$t a n α = \frac{h}{a}$

a = $\frac{h}{t a n α}$ ---- (1)

$I n △ A C B$ ,

$t a n β = \frac{h}{b}$

b = $\frac{h}{t a n β}$ ----(2)

Substract (1) from (2)

$b - a = \frac{h}{t a n β} - \frac{h}{t a n α}$

$\frac{(b - a) t a n α t a n β}{t a n α - t a n β}$

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