To find the length of side \( z \) (which is opposite angle \( Z \)), we can use the Law of Sines. According to this law:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
In triangle XYZ, we have:
- \( \angle X = 60^\circ \)
- \( \angle Y = 50^\circ \)
First, we can calculate \( \angle Z \):
\[ \angle Z = 180^\circ - \angle X - \angle Y = 180^\circ - 60^\circ - 50^\circ = 70^\circ \]
Now we know:
- \( \angle Y = 50^\circ \)
- Side \( y \) (opposite \( Y \)) = 12
- \( \angle Z = 70^\circ \)
- Side \( z \) (opposite \( Z \)) is what we're solving for.
Using the Law of Sines:
\[ \frac{y}{\sin Y} = \frac{z}{\sin Z} \]
Substituting in the values we have:
\[ \frac{12}{\sin(50^\circ)} = \frac{z}{\sin(70^\circ)} \]
Now calculate \( \sin(50^\circ) \) and \( \sin(70^\circ) \):
- \( \sin(50^\circ) \approx 0.7660 \)
- \( \sin(70^\circ) \approx 0.9397 \)
Now we can substitute these values back into our equation:
\[ \frac{12}{0.7660} = \frac{z}{0.9397} \]
Cross-multiply to solve for \( z \):
\[ 12 \cdot 0.9397 = z \cdot 0.7660 \]
Calculating the left side:
\[ 12 \cdot 0.9397 \approx 11.2764 \]
Now solve for \( z \):
\[ z = \frac{11.2764}{0.7660} \approx 14.74 \]
Rounding gives us approximately \( 15.0 \).
Thus, the length of side \( z \) is approximately \( 15.0 \).
So the answer is:
15.0