To solve the problem using the Side-Side-Side (SSS) similarity method, we will determine the ratio of the corresponding sides of triangles \(\triangle ABC\) and \(\triangle DEF\) and then use this ratio to find the side lengths of \(\triangle DEF\).

### Step 1: Calculate the side lengths of \(\triangle ABC\)
- \(AC = 3 \, \text{cm}\)
- \(CB = 4 \, \text{cm}\)
- \(AB = 5 \, \text{cm}\)

### Step 2: Determine the scale factor
Given that \(\triangle ABC\) and \(\triangle DEF\) are similar, the ratio of the sides of \(\triangle DEF\) to \(\triangle ABC\) is consistent for all three sides.

### Step 3: Calculate the area of \(\triangle ABC\)
As previously calculated, the area of \(\triangle ABC\) is:
\[
\text{Area of } \triangle ABC = 6 \, \text{cm}^2
\]

### Step 4: Use the area to find the scale factor
Given the area of \(\triangle DEF\) is \(26.46 \, \text{cm}^2\), we find the scale factor \(k\) as follows:

\[
\text{Area ratio} = \frac{\text{Area of } \triangle DEF}{\text{Area of } \triangle ABC} = \frac{26.46}{6} \approx 4.41
\]
\[
k^2 = 4.41 \quad \Rightarrow \quad k = \sqrt{4.41} \approx 2.1
\]

### Step 5: Use the scale factor to find the side lengths of \(\triangle DEF\)
Since the triangles are similar, the side lengths of \(\triangle DEF\) are \(k\) times the corresponding side lengths of \(\triangle ABC\).

\[
DF = k \times AC = 2.1 \times 3 = 6.3 \, \text{cm}
\]
\[
FE = k \times CB = 2.1 \times 4 = 8.4 \, \text{cm}
\]
\[
DE = k \times AB = 2.1 \times 5 = 10.5 \, \text{cm}
\]

### Answer:
The side lengths of \(\triangle DEF\) using the SSS method are:
- \(DF = 6.3 \, \text{cm}\)
- \(FE = 8.4 \, \text{cm}\)
- \(DE = 10.5 \, \text{cm}\)

Would you like more details on how to apply the SSS similarity method or have any other questions?

Here are some related questions you might find useful:
1. How does the SSS similarity theorem differ from other similarity theorems?
2. Can we determine the angles of \(\triangle DEF\) given this information?
3. How would the result change if the triangles were not similar?
4. What is the difference between congruence and similarity in triangles?
5. How can we apply the SSS method to other types of geometric figures?

**Tip:** The SSS similarity method is powerful because it only requires the side lengths of the triangles to determine if they are similar. Make sure the corresponding sides are correctly identified before applying the method.

Learn how to solve triangle similarity problems using the Side-Side-Side (SSS) method. Calculate the side lengths of triangle DEF based on the similarity ratio derived from triangle ABC's sides. Includes step-by-step instructions and a detailed example. Suitable for Grades 9-12.

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