Similar triangles with corresponding sides in proportion.

Problem Analysis:

The image shows two triangles. The task is to determine the value of \( x \), which represents the length of one side of the second triangle. To solve this, we need to analyze the relationship between the two triangles.

#### Key Observations:
1. The first triangle has sides of lengths 5 cm and 4 cm.
2. The second triangle has sides of lengths \( x \) and 8 cm.
3. The problem does not explicitly state that the triangles are similar, congruent, or related in any specific way. However, given the context and typical problem structures, it is reasonable to assume that the triangles are similar (since no other information suggests otherwise).

#### Assumption:
If the triangles are similar, their corresponding sides are proportional. This means the ratios of the corresponding sides of the two triangles are equal.

Step-by-Step Solution:

1. Identify Corresponding Sides:
- Let us assume the side of length 5 cm in the first triangle corresponds to the side of length 8 cm in the second triangle.
- Let the side of length 4 cm in the first triangle correspond to the side of length \( x \) in the second triangle.

2. Set Up the Proportion:
Since the triangles are similar, the ratios of corresponding sides are equal:
\[
\frac{\text{Side in Triangle 1}}{\text{Corresponding Side in Triangle 2}} = \frac{\text{Another Side in Triangle 1}}{\text{Corresponding Side in Triangle 2}}
\]
Using the identified corresponding sides:
\[
\frac{5}{8} = \frac{4}{x}
\]

3. Solve for \( x \):
Cross-multiply to solve the proportion:
\[
5 \cdot x = 8 \cdot 4
\]
\[
5x = 32
\]
Divide both sides by 5:
\[
x = \frac{32}{5}
\]
\[
x = 6.4
\]

Final Answer:

\[
\boxed{6.4}
\]

Explanation:

The value of \( x \) is determined by the similarity of the triangles and the proportional relationship between their corresponding sides. By setting up and solving the proportion, we find that \( x = 6.4 \) cm. This solution assumes the triangles are similar, which is a common assumption in such problems unless otherwise stated.