Given that \( \angle ZYZ = 60^\circ \) and the lengths of two sides, XY and YZ, we can use the law of cosines to find the length of the third side, XZ.

The law of cosines states:

\( XZ^2 = XY^2 + YZ^2 - 2 \cdot XY \cdot YZ \cdot \cos(\angle ZYZ) \(

Plugging in the given values:

\( XZ^2 = 3^2 + 4^2 - 2 \cdot 3 \cdot 4 \cdot \cos(60^\circ) \(

Since \( \cos(60^\circ) = \frac{1}{2} \), we simplify further:

\( XZ^2 = 9 + 16 - 12 \(

\( XZ^2 = 13 \(

Taking the square root of both sides:

\( XZ = \sqrt{13} \(