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To solve for the length \( x \) in the triangle using the information provided, we can apply the **Law of Cosines**. The Law of Cosines is suitable because we are given two angles and the included side, which is a classic case for this law.

The formula for the Law of Cosines is:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

Where:
- \( c \) is the side we are solving for (in this case, \( x \)),
- \( a \) and \( b \) are the other two sides (8 cm and the unknown side opposite the 19° angle),
- \( C \) is the angle opposite side \( c \) (123°).

### Steps to solve the problem:

1. We can apply the **sine rule** first to find the length of the third side, then apply the Law of Cosines.

Given:
- Angle A = 19°
- Angle B = 123°
- Side \( c = 8 \) cm.

Using the **sine rule** to find side \( b \), the side opposite the 19° angle:
\[
\frac{b}{\sin(B)} = \frac{c}{\sin(A)}
\]

Let me calculate this now.Using the sine rule, we find that the length of side \( b \) (opposite the 19° angle) is approximately \( 3.11 \) cm.

Now, we can use the **Law of Cosines** to find the length \( x \) (opposite the 123° angle):

\[
x^2 = b^2 + c^2 - 2bc \cdot \cos(123^\circ)
\]

Let me compute this next.The length of side \( x \) is approximately \( 10.04 \) cm when rounded to two significant figures.

Would you like more details on any step or have any questions about this solution?

Here are 5 related questions for further exploration:
1. How would the solution change if the angle opposite side \( x \) was altered?
2. Can the Law of Cosines be applied in a right triangle? If not, why?
3. How does the Law of Sines differ from the Law of Cosines?
4. How would you solve for the area of this triangle?
5. How can trigonometric identities simplify solving non-right triangles?

**Tip**: In any triangle, if you know two angles, you can always find the third angle because the sum of the angles in a triangle is always \( 180^\circ \).

Work out the length x, given a triangle with angles 123°, 19°, and a side of 8 cm between them.

Learn how to find the length of side x in a triangle using the Law of Sines and Law of Cosines. The problem includes angles of 123° and 19°, with a side of 8 cm between them. Suitable for students in Grades 10-12.

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