The image depicts a triangle with a smaller triangle inside it. The larger triangle is labeled as $\triangle DCE$, and the smaller triangle as $\triangle ABC$. The task is to solve for $x$, which appears to be the length of the side $AC$ of the smaller triangle.

Given Information:

1. The length of side $DE$ (base of the larger triangle) is $8 \, \text{cm}$.
2. The height $DC$ of the larger triangle is $5 \, \text{cm}$.
3. The height $AB$ of the smaller triangle is $2 \, \text{cm}$.
4. The base $BC$ of the smaller triangle is $3 \, \text{cm}$.

Approach:

Since the smaller triangle $\triangle ABC$ and the larger triangle $\triangle DCE$ appear to be similar triangles (they share the same apex and have proportional sides), we can use the concept of similar triangles to find $x$. The corresponding sides of similar triangles are proportional.

We will set up a proportion between the corresponding sides of the triangles to solve for $x$.

Let me calculate this for you.The length of side $AC$ of the smaller triangle is $7.5 \, \text{cm}$.

Let me know if you have further questions or if you'd like a deeper explanation. Here are some additional related questions to explore:

1. How can you determine if two triangles are similar?
2. What are the properties of similar triangles in geometry?
3. How can the proportionality of sides be used in real-world applications?
4. How do you prove that two triangles are congruent?
5. What is the difference between similar triangles and congruent triangles?

Tip: Always check for common properties (e.g., angle and side relations) when working with similar triangles.