Multiple choice

What is the height of a tower if the angles of elevation of its top from two points x and y at distance of a and b respectively from the base and on the same straight line with the tower, are complementary?

  1. $\sqrt b/a$
  2. $\sqrt a/b$
  3. $\sqrt {ab}$

  4. None of these
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C Correct answer
Explanation

Let height be h. tan(theta) = h/a and tan(90-theta) = h/b. Since tan(90-theta) = cot(theta), we have cot(theta) = h/b. Thus, tan(theta) * cot(theta) = (h/a) * (h/b) = 1. So h^2 = ab, h = sqrt(ab).

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 two points, this creates the equations tan(theta) equals h divided by a, and tan(90 minus theta) equals h divided by b. Using the co-function identity tan(90 minus theta) equals cot(theta), the second equation becomes 1 divided by tan(theta) equals h divided by b. Substituting h divided by a for tan(theta) into the modified second equation and solving for h gives the square root of ab.

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