Multiple choice

The angles of elevation of the top of a tower from two points on the ground at distances a metres and b metres from the base of the tower and in the same straight line are complementary. The height of the tower is $\sqrt{ab}$ metres.

  1. True

  2. False
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A Correct answer
Explanation

If the angles are complementary (theta and 90-theta), the height h satisfies tan(theta) = h/a and tan(90-theta) = h/b. Since tan(90-theta) = cot(theta) = 1/tan(theta), we have h/b = a/h, which leads to h^2 = ab, or h = sqrt(ab).

AI explanation

Let the height of the tower be h, and let the angles of elevation from distances a and b be alpha and 90 degrees minus alpha since they are complementary. Using the tangent function, we have tan alpha equals h divided by a, and tan of 90 degrees minus alpha equals h divided by b. Since tan of 90 degrees minus alpha is cot alpha, we get cot alpha equals h divided by b. Multiplying tan alpha and cot alpha gives 1, which equals h divided by a multiplied by h divided by b, resulting in h squared equals ab. Therefore, the height of the tower is the square root of ab, making the statement true.

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