The angles of elevation of the top of the tower from two points at distance a and b from the base and in the

Question

same straight line with it are complementary...proove that the height of the tower is root (ab)

Answer 1

Let h be the height of the tower, A and B be the angles of elevation from two points at distance a and b, respectively. B = 90 - A.
Then h = a tanA and h = b tanB = b tan(90 - A) = b cotA = b/tan A
so h^2 = a tanA * b/tan A = ab
h = sqrt(ab)

Answer 2

Draw the diagram.

you should find that h/a = tan x
and because the other angle is complementary
b/h = tan x

so b/h = h/a h^2 = ab h = sqrt(ab)