Complementary sight lines to the top and base of a pillar A man 6 ft tall observes: the angle of elevation to the top of a 24 ft pillar and the angle of depression to its base are complementary. What is the horizontal distance between the man and the pillar?

Introduction / Context:
Complementary angles (α + β = 90°) imply tan α * tan β = 1. Here, the man’s eye is at 6 ft (his height). The elevation to the pillar top uses a 18 ft rise (24 − 6), while the depression to the base uses a 6 ft drop. Combine these with the complementary condition to get the distance.

Given Data / Assumptions:

• Pillar height = 24 ft, man's eye height = 6 ft.
• Horizontal ground; line of sight straight.
• Angles of elevation (to top) and depression (to base) are complementary.
• Let distance be d ft.

Concept / Approach:
tan α = (24 − 6)/d = 18/d. tan β = 6/d. Complementary ⇒ tan α * tan β = 1 ⇒ (18/d)*(6/d) = 1.

Step-by-Step Solution:

Verification / Alternative check:
Compute tangents: tan α = 18/(6√3) = √3; tan β = 6/(6√3) = 1/√3; product = 1, so they are complementary.

Why Other Options Are Wrong:
2√3, 4√3, 8√3 produce tan products different from 1 for the given vertical legs.

Common Pitfalls:
Using 24/d for tan α (forgetting the eye level), or treating the depression angle's opposite as 24 instead of 6.

Final Answer:
6√3 ft