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To solve this problem, let's go through the steps to find the perimeter of the new square after dilation.

### Problem Analysis:

1. **Given:**
   - The side of the original square is 8 inches.
   - The square is dilated by a scale factor of \( \frac{1}{4} \).

2. **Find:**
   - The perimeter of the new square after dilation.

### Step-by-Step Solution:

1. **Calculate the side length of the new square:**

   The dilation of a shape by a scale factor affects all dimensions. The side length of the original square is multiplied by the scale factor to get the side length of the new square.

   \[
   \text{New side length} = \text{Original side length} \times \text{Scale factor}
   \]

   Substituting the values:

   \[
   \text{New side length} = 8 \times \frac{1}{4} = 2 \, \text{inches}
   \]

2. **Calculate the perimeter of the new square:**

   The perimeter of a square is given by:

   \[
   \text{Perimeter} = 4 \times \text{side length}
   \]

   Substituting the new side length:

   \[
   \text{Perimeter} = 4 \times 2 = 8 \, \text{inches}
   \]

### Conclusion:

The perimeter of the new square after dilation is **8 inches**. Therefore, the correct answer is:

**D) 8 inches.**

Would you like further details or have any questions about this solution?

### Related Questions:

1. How would the area of the square change with the same dilation factor?
2. If the scale factor was \( \frac{1}{2} \) instead of \( \frac{1}{4} \), what would the new perimeter be?
3. How do dilation and scale factors affect different geometric shapes?
4. What would be the side length if the original square had a perimeter of 32 inches instead?
5. How do you calculate the perimeter of other polygons after dilation?

### Tip:
When dealing with dilation problems, remember that the scale factor directly affects all linear dimensions, such as perimeter, but the area is affected by the square of the scale factor.

The side of a square measures 8 inches. If the square is dilated by a scale factor of 1/4 to create a new square, what is the perimeter of the new square?

Learn how to calculate the perimeter of a square dilated by a scale factor of 1/4. The original side length is 8 inches, and the new perimeter is 8 inches after dilation. Perfect for middle school geometry students.

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