Calculate Length of Line

The length ($l$) of a line is calculated using the Pythagorean theorem with given Latitude ($L = -50\text{ m}$) and Departure ($D = -50\text{ m}$).

$l = \sqrt{L² + D²} = \sqrt{(-50)^2 + (-50)^2} = \sqrt{5000} \approx 70.7107 \text{ m} \approx 70.72 \text{ m}$

Calculate Reduced Bearing (RB / Quadrantal Bearing)

Find the acute angle $\theta$ with the meridian: $\tan \theta = \left|\frac{D}{L}\right|$.

$\tan \theta = \left|\frac{-50}{-50}\right| = 1 \implies \theta = 45°$

Determine Quadrant and Reduced Bearing

Since both Latitude ($L$) and Departure ($D$) are negative, the line lies in the 3rd quadrant (South-West quadrant). Therefore, the Reduced Bearing is $\text{S}45°\text{W}$.

$\text{RB} = \text{S}45°\text{W}$

Calculate Whole Circle Bearing (WCB)

In the 3rd quadrant, Whole Circle Bearing is calculated as $\text{WCB} = 180° + \theta$.

$\text{WCB} = 180° + 45° = 225°$

Compare Options

Option B gives the length as $70.72$ and bearing in RB as $\text{S}45°\text{W}$. Option C gives length as $70.72$ and bearing in WCB as $225°$. Since both expressions for bearing are valid, Option D ('Both B and C') is the correct choice.

$\text{Bearing} = \text{S}45°\text{W} = 225°$