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To solve this problem, we need to use the fact that triangle \(ABC\) is similar to triangle \(DEF\). This means the corresponding sides of the triangles are proportional.

From the diagram:
- In triangle \(ABC\), the sides are: 
  - \(AB = 4 \, \text{cm}\), 
  - \(BC = 3 \, \text{cm}\), 
  - \(AC = 5 \, \text{cm}\).
  
- In triangle \(DEF\), the sides we know are: 
  - \(EF = 9 \, \text{cm}\), 
  - \(DF = 15 \, \text{cm}\), 
  - and we are asked to find \(DE\).

Since the triangles are similar, the ratios of corresponding sides are equal:
- \(AB\) corresponds to \(DE\),
- \(BC\) corresponds to \(EF\),
- \(AC\) corresponds to \(DF\).

We are given \(AB = 4 \, \text{cm}\) and \(EF = 9 \, \text{cm}\), so the ratio of these sides is:

\[
\frac{AB}{EF} = \frac{4}{9}
\]

Now, this ratio should be the same as the ratio of \(DE\) to \(EF\), giving the equation:

\[
\frac{4}{3} = \frac{DE}{9}
\]

This matches option **A**. Therefore, the correct proportion is:

\[
\boxed{A}
\]

Would you like more details or have any other questions?

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Here are 5 related questions to further explore the topic:
1. How can you find the length of \(DE\) based on the proportion from option A?
2. Why is it important that triangles are similar when setting up proportions?
3. How would you solve for \(DE\) if different side lengths were given?
4. Can you apply this method to non-right triangles? Why or why not?
5. What is the scale factor between the two triangles?

**Tip:** Always match corresponding sides carefully when setting up proportions between similar triangles.

Which proportion can be used to find the length of DE in centimeters, given that triangle ABC is similar to triangle DEF?

Learn how to solve a problem involving the similarity of triangles and proportions. This problem helps students find the length of DE in triangle DEF using the given proportionality from triangle ABC. Suitable for Grades 7-9.

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