Multiple choice

In a city, there is a tall tower located at a point. Three observation points, X, Y and Z, are positioned at distances of 5 metres, 10 metres, and 20 metres, respectively, from the base of the tower. The angles of elevation of the top of the tower from X and Z are complementary. What is the height of the tower in metres?

  1. 5

  2. 10

  3. 20

  4. 30

  5. 25
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B Correct answer
Explanation

Let height be h. Angles of elevation from X (5m) and Z (20m) are complementary (a and 90-a). tan(a) = h/5, tan(90-a) = h/20. Since tan(90-a) = cot(a) = 1/tan(a), we have h/20 = 1/(h/5) = 5/h. h^2 = 100, so h = 10.

AI explanation

Let the height of the tower be h. From point X at 5 meters, the angle of elevation α satisfies tan(α) = h/5. From point Z at 20 meters, the angle of elevation β satisfies tan(β) = h/20. Since the angles are complementary, α + β = 90 degrees, meaning tan(α) = cot(β), which equals 1/tan(β). Equating the expressions gives h/5 = 20/h, leading to h squared = 100. Therefore, the height of the tower is 10 meters.

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