Question

If an equilateral triangle has a perimeter of 48 what is the length of the perpendicular bisector of any of its sides?

8 square root 3
16 square root 3
16
8

Answer 1

To find the length of the perpendicular bisector of one side of an equilateral triangle with a perimeter of 48, we first need to determine the length of each side of the triangle.

1. **Calculate the side length:**
The perimeter \( P \) of an equilateral triangle is given by:
\[
P = 3s
\]
where \( s \) is the length of each side. Given the perimeter is 48:
\[
3s = 48 \implies s = \frac{48}{3} = 16
\]

2. **Finding the length of the perpendicular bisector:**
The perpendicular bisector of a side in an equilateral triangle can also be considered as the height of the triangle when dropping a perpendicular from the opposite vertex to the midpoint of that side.

The height \( h \) of an equilateral triangle with side length \( s \) can be calculated using the formula:
\[
h = \frac{\sqrt{3}}{2} s
\]
Substituting \( s = 16 \):
\[
h = \frac{\sqrt{3}}{2} \times 16 = 8\sqrt{3}
\]

Hence, the length of the perpendicular bisector of any side of the equilateral triangle is

\[
\boxed{8\sqrt{3}}
\]