In triangle XYZ, angle X is 60 degrees, angle Y is 50 degrees, and side y (opposite angle Y) is 12 units long. What is the length of side z (opposite angle Z)?

1 answer

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:

\[ \boxed{14.74} \text{ units} \]