Question

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

Answer 1

To find the length of the perpendicular bisector of a side of an equilateral triangle with a perimeter of 48, we can follow these steps:

1. **Find the length of a side of the triangle:**
Since the triangle is equilateral, all three sides are equal. The perimeter \( P \) is given by:

\[
P = 3s
\]

where \( s \) is the length of each side. Given \( P = 48 \), we can solve for \( s \):

\[
3s = 48 \quad \Rightarrow \quad s = \frac{48}{3} = 16
\]

2. **Determine the vertex height (altitude) of the triangle:**
The perpendicular bisector of a side of an equilateral triangle also serves as its altitude. To find the height \( h \) of the equilateral triangle with side length \( s = 16 \), we use the formula for the height of an equilateral triangle:

\[
h = \frac{\sqrt{3}}{2} s
\]

Substituting \( s = 16 \):

\[
h = \frac{\sqrt{3}}{2} \times 16 = 8\sqrt{3}
\]

3. **Length of the perpendicular bisector:**
The length of the perpendicular bisector of one side (which is the altitude) is:

\[
h = 8\sqrt{3}
\]

Thus, the length of the perpendicular bisector of any side of the equilateral triangle is \(\boxed{8\sqrt{3}}\).