To solve this problem, we need to find the area of the triangle formed by the points $A$, $B$, and $C$ using the given bearings and distances. Let's break down the problem step by step:

Step 1: Convert Bearings to Angles

Bearings are measured clockwise from the north. We need to determine the internal angles of the triangle.

1. Bearing from $A$ to $B$: $50^\circ$
2. Bearing from $B$ to $C$: $210^\circ$
3. Bearing from $C$ back to $A$: $310^\circ$

We need to find the angles $\angle ABC$, $\angle BCA$, and $\angle CAB$.

• The internal angle at $B$ is the difference between the bearing of $B$ to $C$ and $A$ to $B$: $\angle ABC = 210^\circ - 50^\circ = 160^\circ$

• The internal angle at $C$ is the difference between the bearing of $C$ to $A$ and $B$ to $C$: $\angle BCA = 310^\circ - 210^\circ = 100^\circ$

• The internal angle at $A$ can be calculated as: $\angle CAB = 360^\circ - (\angle ABC + \angle BCA) = 360^\circ - (160^\circ + 100^\circ) = 100^\circ$

Step 2: Use the Law of Cosines

We need to find the lengths of all sides. We know the side $AB = 10$ km.

Assume $BC = x$ and $AC = y$.

We can use the Law of Cosines in triangle $ABC$:

1. For side $BC$: $x^2 = AB^2 + AC^2 - 2(AB)(AC)\cos(\angle BAC)$

2. For side $AC$: $y^2 = AB^2 + BC^2 - 2(AB)(BC)\cos(\angle ABC)$

However, since we only need the area, we can use the Sine Rule for Area. The area $\text{Area}$ of the triangle can be found by: $\text{Area} = \frac{1}{2} \times AB \times BC \times \sin(\angle ABC)$ But this formula needs both sides and the included angle. Instead, let’s use the cosine method:

$\text{Area} = \frac{1}{2} \times AB \times AC \times \sin(\angle CAB)$

Since this seems more complex than necessary, let’s simplify and proceed using trigonometry:

The distance calculation or the simplified formula needed will be complex so let's skip to solving for an explicit triangle area. With the 3 angles, we can use Heron's formula. Using:

$A = \frac{1}{2} \times AB \times AC \times \sin(\angle ABC)$

But exact side lengths needed to complete it are more suited to exact side calculations rather than 3 internal angles simplification.