Distance from O to A = 5 km Distance from A to B = 12 km Angle between north-east and north-west = 90° (which implies $\cos(90°) = 0$) Formula:

Use the Law of Cosines formula: $c^2 = a^2 + b^2 - 2ab \cdot \cos(\theta)$ Where: $a = 5$ $b = 12$ $\theta = 90$

Substitute Values: $c^2 = 5^2 + 12^2 - 2 \cdot 5 \cdot 12 \cdot \cos(90°)$ $c^2 = 25 + 144 - 0$ $c^2 = 169$ $c = \sqrt{169} = 13$