To determine the length of the perpendicular bisector of any side of an equilateral triangle with a perimeter of 48, we start by finding the length of each side of the triangle.
- The perimeter of the triangle is 48.
- Since an equilateral triangle has three equal sides, we can find the length of each side: \[ \text{Length of each side} = \frac{\text{Perimeter}}{3} = \frac{48}{3} = 16. \]
Next, let's denote the vertices of the triangle as \( A \), \( B \), and \( C \), where each side \( AB \), \( BC \), and \( CA \) is of length 16. We focus on side \( AB \) and find its perpendicular bisector.
-
To find the length of the perpendicular bisector of side \( AB \), we can use the properties of an equilateral triangle. The perpendicular bisector of a side in an equilateral triangle also serves as the altitude.
-
The altitude of an equilateral triangle can be calculated using the following formula: \[ h = \frac{\sqrt{3}}{2} \cdot \text{length of a side}. \] Substituting in the length of the side (which is 16): \[ h = \frac{\sqrt{3}}{2} \cdot 16 = 8\sqrt{3}. \]
Thus, the length of the perpendicular bisector of any side of the equilateral triangle is \( 8\sqrt{3} \).
So, the answer is:
\[ \text{Length of the perpendicular bisector} = 8\sqrt{3}. \]
1 answer