To find the perimeter of the dilated triangle, we first need to calculate the perimeter of the original triangle.
The side lengths of the triangle are 6 cm, 6 cm, and 5 cm. The perimeter \( P \) of the triangle is calculated as follows:
\[ P = 6 + 6 + 5 = 17 \text{ cm} \]
Now, when the triangle is dilated with a scale factor of 4, each side length is multiplied by 4. Therefore, the new side lengths will be:
\[ 6 \times 4 = 24 \text{ cm} \] \[ 6 \times 4 = 24 \text{ cm} \] \[ 5 \times 4 = 20 \text{ cm} \]
Now we calculate the perimeter of the dilated triangle:
\[ \text{New perimeter} = 24 + 24 + 20 = 68 \text{ cm} \]
Thus, the perimeter of the dilated triangle is:
\[ \boxed{68 \text{ cm}} \]