Multiple choice

The angles of elevation of the top of a tower from two points at a distance, $x$ and $y$ metres from the base and in the same straight line with it are complementary. Find the height of the tower.

  1. $h=\sqrt{xy}$
  2. $h=\dfrac{x}{y}$
  3. $h=\sqrt{y}$
  4. $h=\sqrt{x}$
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A Correct answer
Explanation

Let the height be h and the angles be theta and 90-theta. Then tan(theta) = h/x and tan(90-theta) = h/y. Since tan(90-theta) = cot(theta) = 1/tan(theta), we have h/y = x/h, which leads to h^2 = xy or h = sqrt(xy).

AI explanation

Let the height of the tower be h and the complementary angles of elevation be theta and (90 degrees minus theta). From the distances x and y, we have tan(theta) = h/x and tan(90 degrees minus theta) = h/y, where the latter simplifies to cot(theta) = h/y. Multiplying the two equations gives 1 = h^2/(xy). Solving for h gives the height as sqrt(xy).

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