Question

If an equilateral triangle has a perimeter of 48, what is the length of the perpendicular bisector of any of its sides? (2 points)Responses16168883–√8 square root of 3163–√16 square root of 3

Answer 1

To determine 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 length of a side:
\[
\text{Perimeter} = 3 \times \text{side length}
\]
\[
48 = 3s \implies s = \frac{48}{3} = 16
\]
So, each side of the triangle is 16.

2. We denote the vertices of the equilateral triangle as \( A \), \( B \), and \( C \) with \( AB = AC = BC = 16 \).

3. Next, we find the length of the perpendicular bisector of one of the sides; for example, let's consider the side \( AB \).

4. The perpendicular bisector of \( AB \) will meet \( AB \) at its midpoint. The coordinates of the midpoint \( M \) of \( AB \) (assuming \( A(0, 0) \) and \( B(16, 0) \)) are:
\[
M = \left( \frac{0 + 16}{2}, 0 \right) = (8, 0)
\]

5. The height \( h \) of the equilateral triangle, which is also the length of the perpendicular from point \( C \) down to \( AB \) (acting as the length of the perpendicular bisector), can be calculated using the formula for the height of an equilateral triangle:
\[
h = \frac{\sqrt{3}}{2} s = \frac{\sqrt{3}}{2} \times 16 = 8\sqrt{3}
\]

Thus, the length of the perpendicular bisector of any of its sides (which is the height from the opposite vertex) is:
\[
\boxed{8\sqrt{3}}
\]