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