Complementary angles from two points on one line: From two points on the same straight line through the foot of a vertical tower, at distances a and b (a > b), the angles of elevation of the top are complementary. Find the height of the tower.

Introduction / Context:When two angles of elevation sum to 90°, one is θ and the other is 90° − θ. Using tan and cot relations gives a clean product form for the height.

Given Data / Assumptions:

• Distances: a and b from the tower foot on the same line.
• Angles: θ at distance a, and (90° − θ) at distance b.
• Tower height = H.

Concept / Approach:tan θ = H / a and tan(90° − θ) = cot θ = H / b. Multiply the equations and eliminate tan/cot to solve for H.

Step-by-Step Solution:

Verification / Alternative check:Choose θ = 45°: then a = b = H, giving H = √(H^2) = H, consistent. For a ≠ b, the formula still satisfies both equations.

Why Other Options Are Wrong:Forms with a^2 ± b^2 do not satisfy both tangent relations at once; √a(a − b) lacks symmetry expected from complementary angles.

Common Pitfalls:Forgetting cot θ = 1 / tan θ; taking square root without noting H > 0; assuming a specific angle instead of using identities.

Final Answer:√(ab)