Multiple choice

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

  1. $\sqrt{x}$
  2. $\sqrt{y}$
  3. $\sqrt{xy}$
  4. $\sqrt{\displaystyle{\frac{x}{y}}}$
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C Correct answer
Explanation

Let the height be h. The angles are theta and 90-theta. Then tan(theta) = h/x and tan(90-theta) = cot(theta) = h/y. Multiplying these gives tan(theta) * cot(theta) = (h/x) * (h/y), so 1 = h^2 / (xy), which means h = sqrt(xy).

AI explanation

Let the height of the tower be h. If the angle of elevation from the first point is theta, the equation is tan theta equals h divided by x. The angle from the second point is complementary, so it is 90 degrees minus theta, giving tan(90 minus theta) equals h divided by y. Using the cofunction identity, cot theta equals h divided by y. Since tan theta and cot theta are reciprocals, multiplying the two equations gives tan theta times cot theta equals (h over x) times (h over y). This simplifies to 1 equals h squared over xy, so the height h is the square root of xy.

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