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------------------------------------------------ If the angles of elevation of the top of a tower from two points at distances [tex]a[/tex] and [tex]b[/tex] from the base, and in the same straight line with it, are complementary, then the height of the tower is:

A) [tex]ab[/tex] metres
B) [tex]\sqrt{ab}[/tex] metres
C) [tex]ab[/tex] metres
D) [tex](a + b)[/tex] metres

The height of the tower, when the angles of elevation from two points in the same straight line with the base are complementary, is the square root of the product of the two distances, which is B) √ab metres.

If the angles of elevation of the top of a tower from two points at distance a and b from the base and in the same straight line with it are complementary, the height of the tower can be determined using trigonometry. Specifically, we can use the concept that the sum of complementary angles is 90 degrees. Therefore, if the angles of elevation are θ and 90 - θ, the tangent of these angles can define two right triangles with the tower height h as their common opposite side.

To find the height of the tower, we can solve the following proportions:

tan(θ) = h/a
tan(90-θ) = h/b
Since tan(90-θ) is the same as the cotangent of θ, and cot(θ) = 1/tan(θ), we can say that a/h =h/b or h² =a*b. Therefore, h = √(a*b).

So, the correct answer is the height of the tower is √(ab) metres, which corresponds to option B) √ab metres.