To find the perimeter of the dilated triangle, we first calculate the perimeter of the original triangle.
The lengths of the legs are given as 7 and 24, and the hypotenuse is given as 25. Let's calculate the perimeter of the triangle:
\[ \text{Perimeter} = \text{Leg 1} + \text{Leg 2} + \text{Hypotenuse} = 7 + 24 + 25 \]
Calculating this gives us:
\[ \text{Perimeter} = 7 + 24 + 25 = 56 \]
Next, since the triangle is dilated by a scale factor of 14, we multiply the perimeter of the original triangle by the scale factor:
\[ \text{Perimeter of dilated triangle} = \text{Original Perimeter} \times \text{Scale Factor} = 56 \times 14 \]
Calculating this gives:
\[ \text{Perimeter of dilated triangle} = 56 \times 14 = 784 \]
Thus, the perimeter of the dilated triangle is \( \boxed{784} \).