In a height and distance problem, the distance between two vertical pillars is 120 metres. The height of one pillar is three times the height of the other pillar. From the midpoint of the line segment joining their feet, the angles of elevation of the tops of the two pillars are complementary to each other. Using this information, what is the height (in metres) of the taller pillar, taking √3 ≈ 1.732?

Introduction / Context:
This question belongs to the standard height and distance category in quantitative aptitude. It combines geometry with trigonometry and uses the idea of complementary angles of elevation to relate the heights of two vertical pillars to the horizontal distance between them. Understanding how tangent behaves for complementary angles is the key idea tested here.

Given Data / Assumptions:

Distance between the feet of the two pillars = 120 metres.
Heights of the pillars are in the ratio 1 : 3, so let the shorter pillar have height h metres and the taller pillar have height 3h metres.
Observation point is the midpoint of the line joining the feet of the pillars, so it is 60 metres away from each pillar.
Angles of elevation of the tops of the two pillars from the midpoint are complementary, so their sum is 90 degrees.
Take √3 ≈ 1.732 where needed for numerical evaluation.

Concept / Approach:
The main concepts used are tangent of an angle in a right triangle and the property of complementary angles. At the midpoint, vertical height and horizontal distance form right triangles with the line of sight as the hypotenuse. For an angle θ, tan θ = opposite side / adjacent side. If two angles are complementary, say θ and 90 − θ, then tan θ × tan(90 − θ) = 1, because tan(90 − θ) = cot θ and tan θ × cot θ = 1. We use this property to form an equation in h, the unknown height of the shorter pillar, and then calculate the taller height as 3h.

Step-by-Step Solution:
Step 1: Let the shorter pillar have height h metres and the taller pillar have height 3h metres. The midpoint is 60 metres from each pillar. Step 2: Let the angle of elevation to the top of the shorter pillar be β. Then tan β = h / 60. Step 3: Let the angle of elevation to the top of the taller pillar be α. Then tan α = 3h / 60 = h / 20. Step 4: Given that the two angles are complementary, α + β = 90 degrees. Therefore, tan α × tan β = 1. Step 5: Substitute the values: (h / 20) * (h / 60) = 1, so 3h^2 / 3600 = 1, which simplifies to h^2 = 1200. Step 6: Hence h = √1200 = √(400 * 3) = 20√3 ≈ 20 * 1.732 = 34.64 metres. Step 7: The taller pillar has height 3h = 3 * 34.64 = 103.92 metres.

Verification / Alternative check:
We can quickly check the relation tan α × tan β = 1. Using h = 20√3, tan β = h / 60 = 20√3 / 60 = √3 / 3. Tan α = 3h / 60 = 60√3 / 60 = √3. Their product is (√3 / 3) * √3 = 3 / 3 = 1, which confirms that the corresponding angles are complementary as given in the question. The calculated height therefore satisfies both the ratio of heights and the trigonometric condition.

Why Other Options Are Wrong:
34.64 metres gives the height of the shorter pillar, not the taller pillar required in the question. 51.96 metres and 69.28 metres correspond to 1.5h and 2h respectively and do not satisfy the given trigonometric condition when used as the taller height. The only value consistent with both the ratio and the complementary angle condition is 103.92 metres.

Common Pitfalls:
A common mistake is to forget that the midpoint is equidistant from both pillars and to use 120 metres instead of 60 metres as the horizontal distance in the tangent expressions. Another mistake is to assume that complementary angles directly force tan α = 1 / tan β without explicitly forming the product tan α × tan β = 1. Finally, some learners mistakenly use the shorter height instead of the taller height in the final answer. Careful reading of the question and correct formation of the trigonometric equation avoids these errors.

Final Answer:
The height of the taller pillar is 103.92 metres.