Understanding Bearings
Bearings are measured clockwise from the north direction. In this case, the bearing of line AB is 137° 30'. To convert this to standard coordinates:
- North is 0°.
- East is 90°.
- South is 180°.
- West is 270°.
Calculating the Bearing Angles
To find the angle in the Cartesian coordinate system:
- From the north (0°), we can find the angle from the positive x-axis (east):
90° - 137° 30' = -47° 30' (This translates to an angle of 312° 30' in standard position).
Breaking Down the Components
The components of line AB can be calculated using trigonometric functions:
- X-component (Δx) = Length × cos(angle)
- Y-component (Δy) = Length × sin(angle)
Here, the length of AB is 125.8 m and the angle is 312° 30'.
Finding Coordinates of Point B
Using the calculated components:
- Δx = 125.8 × cos(312° 30')
- Δy = 125.8 × sin(312° 30')
Now, we can find the coordinates of point B:
- X-coordinate of B = X-coordinate of A + Δx
- Y-coordinate of B = Y-coordinate of A + Δy
With A's coordinates as (0, 123.45):
- B's X = 0 + Δx
- B's Y = 123.45 + Δy
Conclusion
By calculating Δx and Δy using the appropriate trigonometric functions, you can determine the exact coordinates of point B, including its X-coordinate.