To solve for the length of the perpendicular bisector of any side of an equilateral triangle with a given perimeter, we first need to find the side length of the triangle.
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Find the side length: The perimeter \( P \) of an equilateral triangle is given by: \[ P = 3s \] where \( s \) is the side length. Given that the perimeter is 48: \[ 3s = 48 \] Dividing both sides by 3: \[ s = \frac{48}{3} = 16 \]
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Calculate the height (and therefore the length of the perpendicular bisector): In an equilateral triangle, the height \( h \) can be calculated using the formula: \[ h = \frac{\sqrt{3}}{2} s \] Substituting \( s = 16 \): \[ h = \frac{\sqrt{3}}{2} \times 16 = 8\sqrt{3} \]
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Understanding the perpendicular bisector: The perpendicular bisector of any side of the triangle passes through the midpoint of that side and the opposite vertex, which aligns with the height of the triangle. Therefore, the length of the perpendicular bisector is equal to the height of the triangle.
Thus, the length of the perpendicular bisector of any of its sides is: \[ \boxed{8\sqrt{3}} \]