Question
If an equilateral triangle has a perimeter of 48, what is the length of the perpendicular bisector of nay of its sides?
16square root symbol3
8
8square root symbol3
16
16square root symbol3
8
8square root symbol3
16
Answers
GPT-4o mini
In an equilateral triangle, all sides are equal. Given that the perimeter is 48, we can find the length of each side (\(s\)) by dividing the perimeter by 3:
\[
s = \frac{48}{3} = 16
\]
Next, to find the length of the perpendicular bisector of one of its sides, we can use the properties of an equilateral triangle. The perpendicular bisector runs from the vertex opposite the side to the midpoint of that side.
In an equilateral triangle, the height (\(h\)), which is also the perpendicular bisector from a vertex to the base, can be calculated using the formula for the height of an equilateral triangle:
\[
h = \frac{\sqrt{3}}{2} s
\]
Substituting the length of the side (\(s = 16\)) into the formula gives:
\[
h = \frac{\sqrt{3}}{2} \times 16 = 8\sqrt{3}
\]
Thus, the length of the perpendicular bisector of any side of the equilateral triangle is:
\[
\boxed{8\sqrt{3}}
\]
\[
s = \frac{48}{3} = 16
\]
Next, to find the length of the perpendicular bisector of one of its sides, we can use the properties of an equilateral triangle. The perpendicular bisector runs from the vertex opposite the side to the midpoint of that side.
In an equilateral triangle, the height (\(h\)), which is also the perpendicular bisector from a vertex to the base, can be calculated using the formula for the height of an equilateral triangle:
\[
h = \frac{\sqrt{3}}{2} s
\]
Substituting the length of the side (\(s = 16\)) into the formula gives:
\[
h = \frac{\sqrt{3}}{2} \times 16 = 8\sqrt{3}
\]
Thus, the length of the perpendicular bisector of any side of the equilateral triangle is:
\[
\boxed{8\sqrt{3}}
\]