If  `Delta ABC ~ Delta PQR `  , perimeter of `Delta ABC = 32 cm, `perimeter of `Delta `PQR = 48 cm and PR = 6 cm, then find the length of AC.

Answer

To solve the problem, we will use the properties of similar triangles. Since triangles ABC and PQR are similar, their corresponding sides are in proportion.

### Step-by-step Solution:

1. **Identify the given information:**
- Perimeter of triangle ABC = 32 cm
- Perimeter of triangle PQR = 48 cm
- Length of side PR = 6 cm

2. **Set up the ratio of the perimeters:**
Since the triangles are similar, the ratio of their perimeters is equal to the ratio of their corresponding sides. Therefore, we can set up the following proportion:
\[ \frac{\text{Perimeter of } ABC}{\text{Perimeter of } PQR} = \frac{32}{48} \]

3. **Simplify the ratio:**
\[ \frac{32}{48} = \frac{2}{3} \]
This means that the ratio of any corresponding sides of triangles ABC and PQR is also \( \frac{2}{3} \).

4. **Set up the proportion for the sides:**
Let AC be the corresponding side to PR in triangle ABC. We can write:
\[ \frac{AC}{PR} = \frac{2}{3} \]
Substituting the known value of PR:
\[ \frac{AC}{6} = \frac{2}{3} \]

5. **Cross-multiply to solve for AC:**
\[ 3 \cdot AC = 2 \cdot 6 \]
\[ 3 \cdot AC = 12 \]

6. **Divide both sides by 3:**
\[ AC = \frac{12}{3} = 4 \text{ cm} \]

### Final Answer:
The length of AC is **4 cm**.