Question

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

a. 18√3
b. 9√3
c. 18
d. 9

Answer 1

To find the length of the perpendicular bisector of one side of an equilateral triangle given its perimeter, we will follow these steps:

1. **Find the length of one side of the triangle**:
The perimeter of an equilateral triangle is the sum of the lengths of all three sides. Since the perimeter is 54, the length of one side (s) can be calculated as follows:
\[
s = \frac{\text{Perimeter}}{3} = \frac{54}{3} = 18.
\]

2. **Locate the crucial points**:
In an equilateral triangle, the perpendicular bisector of any side also serves as the height of the triangle from the opposite vertex to that side.

3. **Calculate the height (perpendicular bisector)**:
We can use the Pythagorean theorem to find the height. An equilateral triangle height can be calculated from the formula:
\[
\text{Height} = \frac{\sqrt{3}}{2} s.
\]
Plugging in our value of \(s = 18\):
\[
\text{Height} = \frac{\sqrt{3}}{2} \cdot 18 = 9\sqrt{3}.
\]

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

The correct answer is \( \text{b. } 9\sqrt{3} \).