The angles of elevation of the top of a tower from two points on the same side of the tower and in a straight line with it, at distances 4 m and 9 m from its base, are complementary. What is the height of the tower?

Introduction / Context:
This problem uses complementary angles of elevation to the top of a tower from two points at known distances from its base. Two right triangles share the same vertical height but have different bases. You must use the property that tan θ and tan(90° − θ) are reciprocal to each other to form an equation and then solve for the height of the tower.

Given Data / Assumptions:

• Distances of the two observation points from the tower base are 4 m and 9 m.
• The angles of elevation from these two points are complementary (their sum is 90 degrees).
• The tower is vertical and stands on horizontal ground.
• Let the two angles be α and 90° − α respectively.
• In a right triangle, tan(90° − α) = 1 / tan α.

Concept / Approach:
Let the height of the tower be H. Then tan α = H / 4 for the nearer point and tan(90° − α) = H / 9 for the farther point. Because tan(90° − α) = 1 / tan α, we can construct a simple equation involving H and the two base distances. Solving this equation yields the height of the tower, which can be checked against the given options.

Step-by-Step Solution:
Let H be the height of the tower. From the point at distance 4 m, tan α = H / 4. From the point at distance 9 m, angle of elevation is 90° − α, so tan(90° − α) = H / 9. But tan(90° − α) = 1 / tan α. Thus H / 9 = 1 / (H / 4) = 4 / H. Cross multiplying: H² = 36. So H = 6 m (taking the positive root for height).

Verification / Alternative check:
If H = 6 m, then tan α = 6 / 4 = 3 / 2 = 1.5. So tan(90° − α) must be 1 / 1.5 = 2 / 3. From the second point, tan(90° − α) = H / 9 = 6 / 9 = 2 / 3. The two expressions match, confirming that the angles are indeed complementary and that H = 6 m is correct.

Why Other Options Are Wrong:

• 4 m, 5 m, 7 m, 9 m: Substituting any of these values into the relationships tan α = H / 4 and tan(90° − α) = H / 9 destroys the reciprocal property required by complementary angles.
• Only 6 m satisfies H / 9 = 4 / H while keeping H positive and giving consistent triangles.

Common Pitfalls:
A common mistake is to assume that complementary angles mean the sum of their tangents is 1, which is incorrect. The correct property is that tan(90° − α) is the reciprocal of tan α. Another error is to confuse the distances 4 m and 9 m or to place them incorrectly in the triangles. A neat diagram and careful application of the tan definitions help avoid these issues.

Final Answer:
The height of the tower is 6 m.