Question

If an equilateral triangle has a perimeter of 48, what is the length of the perpendicular bisector of any of its sides?(1 point)
Responses

163–√
16 Start Root 3 End Root

8
8

83–√
8 Start Root 3 End Root

16

Answer 1

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

The perimeter \( P \) of an equilateral triangle is given by the formula:
\[
P = 3s
\]
where \( s \) is the length of one side. Given that the perimeter is 48, we can solve for \( s \):
\[
3s = 48 \\
s = \frac{48}{3} = 16
\]

Next, we want to 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 median and the altitude.

We can calculate the altitude \( h \) of the equilateral triangle using the formula:
\[
h = \frac{\sqrt{3}}{2} s
\]
Substituting \( s = 16 \) into the formula gives:
\[
h = \frac{\sqrt{3}}{2} \cdot 16 = 8\sqrt{3}
\]

The perpendicular bisector of a side extends from the vertex opposite that side to the midpoint of the side, and since we have found that the altitude is \( 8\sqrt{3} \), we conclude that the length of the perpendicular bisector of any side of this equilateral triangle is:

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