Question

If an equilateral triangle has a perimeter of 54, what is the length of the perpendicular bisector of any of its sides?(1 point)
Responses

183–√
18
3
18 square root of 3

93–√
9
3
9 square root of 3

9
9
9

18

Answer 1

To find the length of the perpendicular bisector of any side of an equilateral triangle with a perimeter of 54, we can first determine the length of each side.

1. **Calculate the side length**:
Since the triangle is equilateral, all three sides are equal. The perimeter \( P \) of the triangle is given by:
\[ P = 3s \]
where \( s \) is the length of one side. Given that the perimeter is 54:
\[ 3s = 54 \]
\[ s = \frac{54}{3} = 18 \]

2. **Determine the length of the perpendicular bisector**:
The length of the perpendicular bisector from a vertex to the midpoint of the opposite side in an equilateral triangle can be found using the formula:
\[ h = \frac{\sqrt{3}}{2} s \]
where \( h \) is the length of the height (which is also the length of the perpendicular bisector in this case).

Substituting the value of \( s = 18 \):
\[ h = \frac{\sqrt{3}}{2} \times 18 = 9\sqrt{3} \]

Thus, the length of the perpendicular bisector of any side of the equilateral triangle is \( 9\sqrt{3} \).

**Final answer**: \( 9\sqrt{3} \)