From the bank of a river, the angle subtended by a tree on the opposite bank is 60°. After moving 40 m directly away from the bank, the angle becomes 30°. Find the breadth of the river (in metres).

Introduction / Context:
This is the same geometry pattern as earlier: two angle observations at known horizontal separations solve for the unknown width.

Given Data / Assumptions:

• Initial angle = 60° at distance x (breadth).
• After moving 40 m away, angle = 30° at distance x + 40.

Concept / Approach:
tan 60° = h/x and tan 30° = h/(x + 40) with the same height h; eliminate h to solve for x.

Step-by-Step Solution:

Verification / Alternative check:
Check with ratios: (x + 40)/x = √3 * √3 = 3 ⇒ (x + 40) = 3x ⇒ x = 20 m ✔.

Why Other Options Are Wrong:
30, 40, 60, 80 m do not keep both tangent equations true simultaneously.

Common Pitfalls:
Interpreting “angle subtended” as something other than the elevation to the top; mixing up nearer vs farther position.

Final Answer:
20 m