Two points P and Q lie on a straight line at distances x and y (with y > x) from the base of a vertical building. If the angles of elevation of the top of the building from P and Q are complementary, what is the height of the building in terms of x and y?

Introduction / Context:
This is a height and distance problem in trigonometry. Two observation points P and Q are on a straight line from the base of a vertical building. The angles of elevation of the top from these points are complementary, meaning they add up to 90°. The task is to express the height of the building in terms of the given horizontal distances x and y.

Given Data / Assumptions:
- P is at distance x from the base of the building.
- Q is at distance y from the base, with y > x.
- The angles of elevation from P and Q are complementary, so one is θ and the other is 90° − θ.
- The building is vertical and the ground is horizontal, forming right triangles.

Concept / Approach:
Let the height of the building be h. From right triangle trigonometry, tan of the angle of elevation equals opposite side over adjacent side, that is h divided by the horizontal distance. Using complementary angles, tan(90° − θ) is equal to cot θ. Equating these expressions allows us to relate h, x and y and solve for h in terms of x and y only.

Step-by-Step Solution:
Step 1: Let the angle of elevation from P be θ, so from Q it is 90° − θ.Step 2: From point P, tan θ = h / x, so h = x tan θ.Step 3: From point Q, tan(90° − θ) = h / y.Step 4: Use the identity tan(90° − θ) = cot θ = 1 / tan θ.Step 5: Therefore, h / y = 1 / tan θ, which gives h = y / tan θ.Step 6: Set the two expressions for h equal: x tan θ = y / tan θ.Step 7: Multiply both sides by tan θ to get x tan²θ = y.Step 8: Thus tan²θ = y / x and tan θ = √(y / x) (taking the positive root for a realistic angle of elevation).Step 9: Substitute back into h = x tan θ to get h = x * √(y / x) = √(x y).

Verification / Alternative check:
If x and y are given specific values, for instance x = 4 and y = 9, then h = √(4 * 9) = √36 = 6. Computing the angles from tan θ = 6 / 4 and tan(90° − θ) = 6 / 9 shows that they are complementary, confirming the derivation.

Why Other Options Are Wrong:
The expression xy has the wrong units (it would be distance squared). The expressions √(y/x) and √(x/y) are dimensionless and therefore cannot directly represent a physical height. Only √(xy) has the correct form and units for a length quantity derived from two distances x and y.

Common Pitfalls:
Students sometimes confuse tan θ with cot θ or forget that complementary angles satisfy tan(90° − θ) = cot θ. Another issue is not checking units, which can help quickly reject options like xy that do not have the correct dimension for a length.

Final Answer:
The height of the building is √(xy).