The problem in the image asks to find the perimeter of a rectangle after it undergoes a dilation with a scale factor of $\frac{1}{4}$.

Given:

• Original dimensions of the rectangle:
  • Length = 12 inches
  • Width = 4 inches
• Scale factor = $\frac{1}{4}$

Step 1: Find the new dimensions

The dilation with a scale factor of $\frac{1}{4}$ will reduce each dimension by that factor.

• New length = $\frac{1}{4} \times 12 = 3$ inches
• New width = $\frac{1}{4} \times 4 = 1$ inch

Step 2: Calculate the perimeter

The formula for the perimeter of a rectangle is: $P = 2 \times (\text{length} + \text{width})$ Substitute the new dimensions: $P = 2 \times (3 + 1) = 2 \times 4 = 8 \text{ inches}$

Final Answer:

The perimeter of the image after the dilation is 8 inches.

Would you like further details on how dilation works or any related topics?

Related Questions:

1. How would the area of the rectangle change after the dilation?
2. What happens to the perimeter if the scale factor is $\frac{1}{2}$ instead of $\frac{1}{4}$?
3. How does the scale factor affect shapes that are not rectangles, like circles or triangles?
4. If the original rectangle had a diagonal of 13 inches, what would the diagonal be after the dilation?
5. Can a dilation increase the size of a shape? What would the perimeter be if the scale factor were 2?

Tip:

When working with dilations, always remember that the scale factor applies uniformly to all dimensions of the shape, and it directly affects both perimeter and area differently!