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The problem involves finding the length of side \( x \) in a triangle where the lengths of the other two sides are given, along with an adjacent angle. Based on the arrangement in the image, we can apply the **Law of Cosines**, which is useful for finding unknown side lengths in triangles when two sides and the included angle are known.

### Solution
In this triangle, we have:
- \( a = 7 \)
- \( b = 8 \)
- \( c = x \) (the side we need to find)
- \(\theta = 120^\circ\) (the angle between sides of length 7 and 8)

The Law of Cosines states:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(\theta)
\]

Substitute the values:
\[
x^2 = 7^2 + 8^2 - 2 \cdot 7 \cdot 8 \cdot \cos(120^\circ)
\]

Since \(\cos(120^\circ) = -0.5\), we get:
\[
x^2 = 49 + 64 + 2 \cdot 7 \cdot 8 \cdot 0.5
\]
\[
x^2 = 49 + 64 + 56
\]
\[
x^2 = 169
\]
\[
x = \sqrt{169} = 13
\]

### Answer
\[
x = 13
\]

Would you like further details or have any questions?

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Here are some additional questions to explore:

1. What is the Law of Cosines, and when is it used?
2. How does the angle between two sides affect the length of the opposite side?
3. Can the Law of Cosines be used in right triangles, and if so, how?
4. What is the relationship between cosine values and angles greater than 90°?
5. How do different methods, like the Law of Sines, compare for solving non-right triangles?

**Tip:** For any triangle, if you know two sides and the included angle, the Law of Cosines is the best method to find the third side.

Find the length of side x in a triangle with sides 5, 7, and an included angle of 120°, using the Law of Cosines.

Learn how to find the length of a triangle's side using the Law of Cosines. This example includes a triangle with sides 5, 7, and an angle of 120°. The solution involves using the formula c^2 = a^2 + b^2 - 2ab * cos(θ). Suitable for students in Grades 9-11.

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