To find the length of side \( z \) in triangle \( XYZ \), we can use the Law of Sines. The Law of Sines states that:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
In our triangle:
- Let \( A = \angle Y = 50^\circ \)
- Let \( B = \angle X = 60^\circ \)
- Let \( C = \angle Z \)
First, we need to find angle \( Z \):
\[ C = 180^\circ - A - B = 180^\circ - 50^\circ - 60^\circ = 70^\circ \]
Next, we know that side \( y \) (opposite angle \( Y \)) is 12 units long. Thus, we can denote:
- \( y = 12 \)
- \( A = 50^\circ \)
- \( B = 60^\circ \)
- \( C = 70^\circ \)
Now, we can use the Law of Sines to find side \( z \):
\[ \frac{y}{\sin A} = \frac{z}{\sin C} \]
Substituting the known values:
\[ \frac{12}{\sin(50^\circ)} = \frac{z}{\sin(70^\circ)} \]
Now, we can solve for \( z \):
\[ z = \frac{12 \cdot \sin(70^\circ)}{\sin(50^\circ)} \]
Let's calculate \( \sin(50^\circ) \) and \( \sin(70^\circ) \):
- \( \sin(50^\circ) \approx 0.7660 \)
- \( \sin(70^\circ) \approx 0.9397 \)
Now substituting these values into the equation for \( z \):
\[ z \approx \frac{12 \cdot 0.9397}{0.7660} \approx \frac{11.2764}{0.7660} \approx 14.74 \]
Thus, the length of side \( z \) is approximately:
\[ \boxed{14.74} \text{ units} \]