https://file.aiquickdraw.com/mathbot/file/1724587451166epu6lm6b.jpg

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.

Explore how to solve for x in a problem involving similar triangles. Given dimensions include base and height measurements of triangles DCE and ABC, applying similarity theorems and proportional sides.

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