This problem involves finding the perimeter of one triangle ($\triangle$ ABC) given information about a similar triangle ($\triangle$ PQR) and the relationship between their sides.

Understanding Similar Triangles Perimeter

When two triangles are similar, like $\triangle$ ABC and $\triangle$ PQR ($\triangle$ ABC ~ $\triangle$ PQR), their corresponding sides are in the same ratio. A key property of similar triangles is that the ratio of their perimeters is equal to the ratio of their corresponding sides.

Let $P_{ABC}$ be the perimeter of $\triangle$ ABC and $P_{PQR}$ be the perimeter of $\triangle$ PQR.

The property states:

$$ \frac{P_{ABC}}{P_{PQR}} = \frac{\text{Side of } \triangle \text{ ABC}}{\text{Corresponding Side of } \triangle \text{ PQR}} $$

Identifying Corresponding Sides

Since $\triangle$ ABC ~ $\triangle$ PQR, the side BC in $\triangle$ ABC corresponds to the side QR in $\triangle$ PQR.

Calculating Perimeter Ratio

We are given:

• Side BC = 4 cm
• Corresponding side QR = 8 cm
• Perimeter of $\triangle$ PQR ($P_{PQR}$) = 32 cm

First, find the ratio of the corresponding sides:

$$ \text{Ratio of Sides} = \frac{\text{BC}}{\text{QR}} = \frac{4 \text{ cm}}{8 \text{ cm}} = \frac{1}{2} $$

Now, use the property that the ratio of the perimeters is equal to the ratio of the corresponding sides:

$$ \frac{P_{ABC}}{P_{PQR}} = \frac{\text{BC}}{\text{QR}} $$

Step-by-Step Perimeter Calculation

Substitute the known values into the formula:

$$ \frac{P_{ABC}}{32 \text{ cm}} = \frac{1}{2} $$

To find the perimeter of $\triangle$ ABC ($P_{ABC}$), multiply both sides of the equation by 32 cm:

$$ P_{ABC} = \frac{1}{2} \times 32 \text{ cm} $$

$$ P_{ABC} = 16 \text{ cm} $$

Therefore, the perimeter of $\triangle$ ABC is 16 cm.