Use what you know about the relationships in 30-60-90 right triangles to solve the following problem. A stained-glass window is in the shape of an equilateral triangle with sides that are 36 inches long. How long is the perpendicular bisector of any side?(1 point)

Question

Responses

18 inches
18 inches

183–√ inches
18
3

inches
18 Start Root 3 End Root inches

36 inches
36 inches

9 inches

Answer 1

In an equilateral triangle, the perpendicular bisector of any side also serves as the altitude and the median. To find the length of the altitude (perpendicular bisector) of an equilateral triangle, we can use the relationship in a 30-60-90 triangle.

For an equilateral triangle with side length \( s \):

1. The altitude divides the triangle into two 30-60-90 triangles.
2. In a 30-60-90 triangle, the ratios of the sides are \( 1 : \sqrt{3} : 2 \).
3. The side opposite the 30-degree angle is \( \frac{s}{2} \), which is half of the side of the equilateral triangle: \[ \text{Short side} = \frac{36 , \text{inches}}{2} = 18 , \text{inches} \]
4. The side opposite the 60-degree angle (the altitude) is given by: \[ \text{Altitude} = \text{Short side} \times \sqrt{3} = 18 \sqrt{3} , \text{inches} \]

Therefore, the length of the perpendicular bisector of any side is:

18√3 inches.