To find the perimeter of the dilated triangle, we first need to determine the side lengths of the dilated triangle using the scale factor of \( 13 \).
The original side lengths of the triangle are:
- 11 inches
- 9 inches
- 28 inches
We can calculate the new side lengths of the dilated triangle by multiplying each side length by the scale factor \( 13 \):
- \( 11 \times 13 = 143 \) inches
- \( 9 \times 13 = 117 \) inches
- \( 28 \times 13 = 364 \) inches
Next, we can find the perimeter of the dilated triangle by adding the new side lengths together:
\[ \text{Perimeter} = 143 + 117 + 364 = 624 \text{ inches} \]
Therefore, the perimeter of the dilated triangle is
\[ \boxed{624} \text{ inches} \]