Question

If an equilateral triangle has a perimeter of 48, what is the length of the perpendicular bisector of any of its side?(10 points)
Responses

83–√
8 square root of 3

163–√
16 square root of 3

8
8

16
16

Answer 1

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

1. **Calculate the side length:**
Since the perimeter of the equilateral triangle is 48, and there are 3 equal sides, each side length \( s \) is:
\[
s = \frac{48}{3} = 16
\]

2. **Find the length of the perpendicular bisector:**
In an equilateral triangle, the perpendicular bisector of a side also serves as the altitude and median. We can use the formula for the height (h) of an equilateral triangle:

\[
h = \frac{\sqrt{3}}{2} s
\]
Plugging in the side length \( s = 16 \):
\[
h = \frac{\sqrt{3}}{2} \times 16 = 8\sqrt{3}
\]

Therefore, the length of the perpendicular bisector of any of its sides is \( 8\sqrt{3} \).

So the correct response is:
**8 square root of 3**.