A triangle ABC with `AB = 3 cm, BC = 6 cm and AC = 4 cm` is enlarged to A DEF such that the longest side of A DEF = 9 cm. Find the scale factor and hence, the lengths of the other sides of `DeltaDEF`.

Answer

To solve the problem, we need to find the scale factor of the enlargement from triangle ABC to triangle DEF and then use that scale factor to determine the lengths of the other sides of triangle DEF.

### Step-by-Step Solution:

1. **Identify the longest side of triangle ABC**:
- In triangle ABC, the sides are given as:
- AB = 3 cm
- BC = 6 cm (longest side)
- AC = 4 cm

2. **Identify the longest side of triangle DEF**:
- The longest side of triangle DEF is given as 9 cm.

3. **Calculate the scale factor**:
- The scale factor can be calculated using the ratio of the longest sides of the two triangles:
\[ \text{Scale Factor} = \frac{\text{Longest Side of DEF}}{\text{Longest Side of ABC}} = \frac{9 \text{ cm}}{6 \text{ cm}} = \frac{3}{2} = 1.5 \]

4. **Use the scale factor to find the other sides of triangle DEF**:
- Now we will use the scale factor to find the lengths of the other sides of triangle DEF.
- For side AB (3 cm):
\[ DE = \text{Scale Factor} \times AB = \frac{3}{2} \times 3 \text{ cm} = \frac{9}{2} \text{ cm} = 4.5 \text{ cm} \]
- For side AC (4 cm):
\[ DF = \text{Scale Factor} \times AC = \frac{3}{2} \times 4 \text{ cm} = \frac{12}{2} \text{ cm} = 6 \text{ cm} \]

5. **Summarize the lengths of the sides of triangle DEF**:
- The sides of triangle DEF are:
- DE = 4.5 cm
- EF = 9 cm (given)
- DF = 6 cm

### Final Answer:
- The scale factor is \( \frac{3}{2} \) or 1.5.
- The lengths of the sides of triangle DEF are:
- DE = 4.5 cm
- EF = 9 cm
- DF = 6 cm