To find the length of the perpendicular bisector of a side in an equilateral triangle, we can use the properties of a 30-60-90 triangle.
- The side of the equilateral triangle is 36 inches.
- When you draw the perpendicular bisector of one side, it divides the equilateral triangle into two 30-60-90 triangles.
- In a 30-60-90 triangle, the sides are in the ratio of \(1 : \sqrt{3} : 2\).
Let's denote:
- The length of the side (hypotenuse) of the equilateral triangle as 36 inches.
- Each half of the base of the triangle is \( \frac{36}{2} = 18 \) inches.
In the 30-60-90 triangle formed:
- The shortest side (opposite the 30-degree angle) is \(18\) inches (half of the base).
- The longer leg (which is the length of the perpendicular bisector we need to find) corresponds to the opposite side of the 60-degree angle.
From the property of 30-60-90 triangles, the formula for the longer leg is: \[ \text{Longer leg} = \text{short leg} \times \sqrt{3} = 18 \times \sqrt{3} \]
Thus, the length of the perpendicular bisector is: \[ 18\sqrt{3} \text{ inches} \]
Therefore, the closest response available in your options that corresponds to this length is: \(\text{18}\sqrt{3}\) inches.
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