To find the length of the perpendicular bisector of any side of an equilateral triangle with a given perimeter, we first start by calculating the length of each side.
Given that the perimeter of the equilateral triangle is 48, we can find the length of one side as follows:
\[ \text{Length of one side} = \frac{\text{Perimeter}}{3} = \frac{48}{3} = 16 \]
Next, we will denote the vertices of the equilateral triangle as \( A \), \( B \), and \( C \), with each side measuring 16 units. We can calculate the length of the perpendicular bisector of side \( AB \).
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Determine the height of the triangle: The perpendicular bisector of a side in an equilateral triangle is also the altitude (height) from the opposite vertex to the midpoint of that side.
The height \( h \) of an equilateral triangle can be calculated using the formula:
\[ h = \frac{\sqrt{3}}{2} s \]
where \( s \) is the side length. Substituting \( s = 16 \):
\[ h = \frac{\sqrt{3}}{2} \times 16 = 8\sqrt{3} \]
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Finding the length of the perpendicular bisector: The length of the perpendicular bisector of side \( AB \) (which is the height from vertex \( C \) to the midpoint of \( AB \)) is equal to the height \( h \):
Therefore, the length of the perpendicular bisector is:
\[ \boxed{8\sqrt{3}} \]