To solve for the length of the perpendicular bisector of any side of the equilateral triangle, we first need to determine the length of one side of the triangle.
Given that the perimeter of the triangle is 48, we can calculate the length of a side (s) as follows:
\[ \text{Perimeter} = 3s \] \[ 48 = 3s \] \[ s = \frac{48}{3} = 16 \]
Now that we know the length of each side is 16, we can find the length of the perpendicular bisector of one of the sides. In an equilateral triangle, the perpendicular bisector of a side also serves as the altitude and median.
The formula for the altitude \( h \) of an equilateral triangle with side length \( s \) is given by:
\[ h = \frac{\sqrt{3}}{2} s \]
Substituting the value of \( s \) into the formula:
\[ h = \frac{\sqrt{3}}{2} \times 16 = 8\sqrt{3} \]
Therefore, the length of the perpendicular bisector of any side of the equilateral triangle is \( 8\sqrt{3} \).
Thus, the correct response is: \[ 8\sqrt{3} \]
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