Complementary angles from two points a and b — find the tower height: From two points P and Q on the same straight line through the foot of a tower, at distances a and b from the base, the angles of elevation of the top are complementary. What is the height of the tower?

Introduction / Context:
Complementary angles (θ and 90° − θ) imply tan(θ) · tan(90° − θ) = 1. With the same tower height h and two horizontal distances a and b, we can connect the tangents and obtain a simple closed-form for h.

Given Data / Assumptions:

• Point P at distance a; angle of elevation = θ.
• Point Q at distance b; angle of elevation = 90° − θ.
• Same height h and level ground.

Concept / Approach:
Write tan θ = h/a and tan(90° − θ) = cot θ = a/h. But from Q we also have tan(90° − θ) = h/b ⇒ equate and use tan θ · tan(90° − θ) = 1 to relate a, b, and h.

Step-by-Step Solution:
From P: tan θ = h/aFrom Q: tan(90° − θ) = h/b = cot θ = a/hSo h/b = a/h ⇒ h² = ab ⇒ h = √ab (height is positive)

Verification / Alternative check:
Then tan θ = h/a = √ab / a = √(b/a) and tan(90° − θ) = h/b = √ab / b = √(a/b). Their product is 1, confirming complementarity.

Why Other Options Are Wrong:
a·b, a b (ambiguous), a²b², or a + b do not satisfy the trig identities for complementary angles and the geometry here.

Common Pitfalls:
Forgetting tan(90° − θ) = cot θ; attempting to add distances instead of using product relations.

Final Answer:
√ab