To solve the problem, we first need to find the length of one side of the equilateral triangle. Since the perimeter of the triangle is 48, we can divide this by 3 to find the length of one side (\(s\)):
\[ s = \frac{48}{3} = 16 \]
Next, we want to find the length of the perpendicular bisector of any side of the triangle. In an equilateral triangle, the perpendicular bisector also serves as the altitude.
The formula for the length of the altitude \(h\) of an equilateral triangle with side length \(s\) is given by:
\[ h = \frac{\sqrt{3}}{2} s \]
Substituting \(s = 16\) into the formula, we get:
\[ h = \frac{\sqrt{3}}{2} \times 16 = 8\sqrt{3} \]
Thus, the length of the perpendicular bisector (or the altitude) of any side of the equilateral triangle is:
\[ \boxed{8\sqrt{3}} \]