Coordinate geometry from bearings: Point B is 50 m due east of A. The bearing of C from A is N 30° E, and the bearing of C from B is N 60° W. What is the departure (east–west coordinate) of C from A (positive east)?

Introduction / Context:
Computing coordinates from bearings is a staple in traverse reduction and intersection problems. “Departure” denotes the east–west component (positive to the east) of a line or point relative to an origin. Here we use two bearings to intersect lines and determine the point C, then read off its departure from A.

Given Data / Assumptions:

• Let A = (0, 0) with x east and y north.
• B is 50 m east of A ⇒ B = (50, 0).
• Bearing of C from A: N 30° E (azimuth 30°).
• Bearing of C from B: N 60° W (azimuth 300°).

Concept / Approach:

For azimuth θ (measured clockwise from north), the unit direction vector is (sin θ, cos θ). Form parametric lines for AC and BC and solve for their intersection. The x-coordinate of the intersection relative to A is the required departure of C from A.

Step-by-Step Solution:

Verification / Alternative check:

Sketching confirms C lies northeast of A but west of B’s meridian, consistent with an x value less than 50 m and positive. Back-substitution into both parametric equations yields consistent coordinates.

Why Other Options Are Wrong:

25 m and 50 m overstate the easting; 25√3 m and 12.5√3 m are northing-like magnitudes, not the departure computed here.

Common Pitfalls:

Using cos for eastings and sin for northings (reversed); forgetting that azimuth is from north, not from east; mixing up departure (x) with latitude (y).

Final Answer:

12.5 m