To solve the problem, let's start by determining the length of each side of the equilateral triangle given its perimeter.
The perimeter \( P \) of an equilateral triangle is given by:
\[ P = 3s \]
where \( s \) is the length of each side. Given that the perimeter is 48, we can set up the equation:
\[ 3s = 48 \]
Dividing both sides by 3, we find:
\[ s = \frac{48}{3} = 16 \]
Now that we have the length of each side, we need to find the length of the perpendicular bisector of one of its sides.
In an equilateral triangle, the perpendicular bisector of any side also acts as the altitude, as well as the median, because all sides are equal and the angles are congruent. The formula for the length of the altitude \( h \) of an equilateral triangle with side length \( s \) is given by:
\[ h = \frac{\sqrt{3}}{2} s \]
Plugging in the value of \( s = 16 \):
\[ h = \frac{\sqrt{3}}{2} \cdot 16 = 8\sqrt{3} \]
Therefore, the length of the perpendicular bisector of any side of the equilateral triangle is \( 8\sqrt{3} \).
So, the answer is:
8√3