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Two points at distance x and y from the base point are on the same side of the line passing through the base of a tower. The angle of elevation from these two points to the top of the tower are complementary. Then, find the height of the tower.

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\sqrt{x}

\sqrt{y}

\sqrt{xy}

\sqrt{\frac{x}{y}}

answer is C.

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Detailed Solution

It is given that the angle of elevation from point x and y on the ground to the top of the tower on the same side of the tower are complementary.
Let the tower be AB and C and D be points on the same side of the tower such that
BD = x and BC = y.

∴ CD = (y - x)

Let

∠ ADB = θ

.
Then

∠ ACB = 9 0^{\circ} - θ

, because the sum of complementary angles is 90°.
In

△

ADB,
tan

θ

\frac{AB}{DB} = \frac{AB}{x}

.... (1)
In

△ ABC

\tan (9 0^{\circ} - θ

) =

\frac{AB}{BC} = \frac{AB}{y}

We know that, tan (90° -

θ

) = cot

θ

∴ cotθ = \frac{AB}{y}

..... (2)
Also,

tan θ \times cot θ = 1

.
From (1) and (2),

∴

\frac{AB}{x} \times \frac{AB}{y} = 1

⇒

A B^{2} = xy

⇒

AB = \sqrt{xy}

∴

The height of the tower is

\sqrt{xy}

Hence, the correct option is 3.

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