Multiple choice

PQ is a vertical tower. P is the foot, Q the top of the tower, A, B, C are three points in the horizontal plane through P. The angles of elevation of Q from A, B, C are equal and each is equal to $\theta$. The sides of the triangle ABC are a, b, c and the area of the triangle ABC is $\Delta$. The height of the tower is

  1. (abc) tan $\theta/ \, 4\,\Delta$
  2. (abc) sin $\theta/ \, 4\,\Delta$
  3. (abc) cot $\theta/ \, 4\,\Delta$

  4. None
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A Correct answer
Explanation

In triangle ABC, the circumradius R = abc / 4 * Delta. The height h = R * tan(theta). Thus h = (abc * tan(theta)) / 4 * Delta.

AI explanation

Since the angles of elevation from points A, B, and C are all equal to theta, the foot of the tower P must be equidistant from all three vertices, making P the circumcentre of triangle ABC. Let R be the circumradius, so the height of the tower h equals R tan theta. Using the extended law of sines, R equals abc divided by 4 delta, so substituting this into the height equation gives h equals (abc tan theta) divided by 4 delta.

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