To find the perimeter of the dilated trapezoid, we first need to find the perimeter of the original trapezoid before dilation.
The sides of the trapezoid are as follows:
- Base 1: 10 inches
- Base 2 (top): 4 inches
- Side 1: 8 inches
- Side 2: 8 inches
Now we calculate the perimeter of the original trapezoid:
\[ \text{Perimeter} = \text{Base 1} + \text{Base 2} + \text{Side 1} + \text{Side 2} \]
Substituting in the values:
\[ \text{Perimeter} = 10 + 4 + 8 + 8 = 30 \text{ inches} \]
Next, we apply the dilation using a scale factor of \( \frac{1}{2} \). Dilation affects all linear dimensions, including lengths. Thus, the new lengths after dilation will be:
- New Base 1 = \( 10 \times \frac{1}{2} = 5 \) inches
- New Base 2 = \( 4 \times \frac{1}{2} = 2 \) inches
- New Side 1 = \( 8 \times \frac{1}{2} = 4 \) inches
- New Side 2 = \( 8 \times \frac{1}{2} = 4 \) inches
Now, we calculate the perimeter of the dilated trapezoid:
\[ \text{New Perimeter} = \text{New Base 1} + \text{New Base 2} + \text{New Side 1} + \text{New Side 2} \]
Substituting in the new lengths:
\[ \text{New Perimeter} = 5 + 2 + 4 + 4 = 15 \text{ inches} \]
Thus, the perimeter of the dilated trapezoid is \( \boxed{15} \) inches.
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