The question asks for the perimeter of the smaller rectangle, given that the rectangles are similar and the dimensions of the larger rectangle and its perimeter are provided.

Step-by-Step Solution:

1. Understand the Relationship Between Similar Figures:

   • Similar rectangles have proportional sides. The sides of the smaller rectangle are scaled down proportionally from the larger rectangle.

2. Determine the Scale Factor:

   • The width of the larger rectangle is $8 \, \text{mm}$, and the width of the smaller rectangle is $4 \, \text{mm}$.
   • The scale factor is: $\text{Scale Factor} = \frac{\text{Width of Smaller Rectangle}}{\text{Width of Larger Rectangle}} = \frac{4}{8} = \frac{1}{2}.$

3. Apply the Scale Factor to the Perimeter:

   • The perimeter of similar figures is scaled by the same factor as their dimensions.
   • The perimeter of the larger rectangle is $P_1 = 24 \, \text{mm}$.
   • The perimeter of the smaller rectangle $P_2$ is: $P_2 = P_1 \times \text{Scale Factor} = 24 \times \frac{1}{2} = 12 \, \text{mm}.$

Final Answer:

The perimeter of the smaller rectangle is: $P_2 = 12 \, \text{mm}.$

Would you like a further breakdown or have additional questions?

Related Questions:

1. How do we find the scale factor for similar figures with different dimensions?
2. Can this technique be applied to other shapes like triangles or circles?
3. If the length of the smaller rectangle were given instead, how would the solution change?
4. Why does the perimeter scale proportionally for similar shapes?
5. What is the formula for the area ratio of two similar figures?

Tip:

For similar figures, always check if the scale factor applies to all dimensions consistently to ensure the figures are truly similar!