We are tasked with determining the distance from point $Z$ to point $X$, given the following conditions:

• $X$ is due north of $Y$, and the distance between $X$ and $Y$ is 2 km.
• $Z$ is due east of $Y$ and has a bearing of $S 35^\circ 12' E$ (or $35^\circ 12'$ south of east) from $X$.

Step-by-Step Approach:

1. Bearings and Coordinates Setup:

   • Place $Y$ at the origin $(0,0)$.
   • Since $X$ is due north of $Y$, $X$ is at the point $(0, 2)$ km.
   • Since $Z$ is due east of $Y$, its position is along the x-axis at some point $(d, 0)$, where $d$ is the unknown distance between $Y$ and $Z$.

2. Using the Bearing:

   • The bearing $S 35^\circ 12' E$ from $X$ to $Z$ gives the angle between the line from $X$ to $Z$ and the southward direction from $X$. This angle is measured clockwise from south, and it translates into an angle of $90^\circ - 35^\circ 12' = 54^\circ 48'$ with the horizontal (x-axis) in standard position.

3. Trigonometric Relations:

   • Using the law of sines or basic trigonometry, we can establish the relationships between the distances. Since we know the angle and the positions, we can form the right triangle and solve for the unknown distance.

I'll calculate the distance now.The distance from point $Z$ to point $X$ is approximately 3,470 meters.

Would you like further details or have any questions?

Here are 5 related questions for deeper understanding:

1. How do we interpret bearings when solving geometry problems like this one?
2. Can the law of sines or cosines be used in problems involving bearings?
3. What are other methods to solve right-angled triangle problems in navigation?
4. How do we convert between degrees, minutes, and radians when calculating angles?
5. Why is trigonometry useful for solving navigation-related distance problems?

Tip: When solving problems involving bearings, always ensure you understand the reference direction (e.g., north, east) to set up the triangle correctly.