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

Answer

To solve the problem step by step, we will use the properties of similar triangles and the relationship between their perimeters.

### Step 1: Understand the relationship between the triangles
Since triangles ABC and PQR are similar, the ratios of their corresponding sides are equal to the ratios of their perimeters.

### Step 2: Write the ratio of the perimeters
The ratio of the perimeters of triangles ABC and PQR can be expressed as:
\[ \frac{\text{Perimeter of } \triangle ABC}{\text{Perimeter of } \triangle PQR} = \frac{32 \text{ cm}}{48 \text{ cm}} \]

### Step 3: Simplify the ratio
We can simplify the ratio:
\[ \frac{32}{48} = \frac{2}{3} \]

### Step 4: Set up the proportion for the sides
Let AC be the side corresponding to PR in triangle PQR. Since PR = 6 cm, we can set up the proportion:
\[ \frac{AC}{PR} = \frac{2}{3} \]
Substituting PR into the equation gives:
\[ \frac{AC}{6} = \frac{2}{3} \]

### Step 5: Solve for AC
To find AC, we can cross-multiply:
\[ 3 \cdot AC = 2 \cdot 6 \]
This simplifies to:
\[ 3 \cdot AC = 12 \]
Now, divide both sides by 3:
\[ AC = \frac{12}{3} = 4 \text{ cm} \]

### Conclusion
The length of AC is 4 cm.