Multiple choice

The angle of elevation of the top of the tower observed from each of three points A, B, C on the horizontal ground, forming a triangle ABC, is observed to be $'\theta '$. If 'R' be the circumradius of the triangle, then height of tower is :

  1. $R\tan { \theta } $
  2. $R\cot { \theta } $
  3. $R\sin { \theta } $
  4. $R\cos { \theta } $
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A Correct answer
Explanation

If the tower's foot is O, equal angles of elevation imply OA = OB = OC, so O is the circumcenter and the horizontal distance is R. In the right triangle, tan(theta) = height/R, giving height = R tan(theta).

AI explanation

Since the angle of elevation to the top of the tower is theta from three points forming triangle ABC on the ground, the base of the tower must be equidistant from A, B, and C. This means the base of the tower is the circumcenter of the triangle, making the horizontal distance to any point equal to the circumradius R. Using basic trigonometric ratios, the height of the tower is found by multiplying this distance R by tan(theta). Thus, the height of the tower is R tan(theta).

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