To find the straight-line distance from pillar $X$ to pillar $Z$, we can use the Pythagorean theorem.

1. Determine the distances:

- Pillar $Y$ is 4 km east of pillar $X$.
- Pillar $Z$ is 4 km south of pillar $Y$.

So, the horizontal distance from $X$ to $Y$ is 4 km, and the vertical distance from $Y$ to $Z$ is 4 km.

2. Apply the Pythagorean theorem:

The distance $XZ$ is the hypotenuse of a right-angled triangle with both legs of 4 km.

$$XZ=\sqrt{(XY{)}^2+(YZ{)}^2}$$

Substituting the values:

$$XZ=\sqrt{4^2+4^2}=\sqrt{16+16}=\sqrt{32}=4\sqrt{2}\text{ km}$$

So the straight-line distance from $X$ to $Z$ is:

A. $4\sqrt{2}$ km