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

183–√ inches
18 Start Root 3 End Root inches

9 inches
9 inches

18 inches
18 inches

36 inches

Answer 1

To solve for the length of the perpendicular bisector of a side in an equilateral triangle, we can follow these steps:

1. Understand the Geometry: In an equilateral triangle, when we draw a perpendicular from a vertex to the opposite side, it serves as the height of the triangle. This line also bisects the side it meets (the base).

2. Identify Key Values: The side length of the equilateral triangle is given as 36 inches.

3. Relationship of Right Triangles in Equilateral Triangle: When the height (which also serves as the perpendicular bisector of the base) is drawn, it creates two 30-60-90 right triangles. In a 30-60-90 triangle:

   • The sides opposite the 30°, 60°, and 90° angles have ratios of 1:√3:2 respectively.
   • If the length of the side opposite the 60° angle (the height we want to find) is \( h \), and the hypotenuse is the side of the equilateral triangle (36 inches), then the side opposite the 30° angle (which is half the base, or \( 36/2 = 18 \) inches) corresponds to the shorter leg.

4. Set Up the Equation: In our 30-60-90 triangle:

   • If we take the shorter leg (half of the base) as 18 inches:
   • The longer leg (the height \( h \)) is given by \( h = 18 \sqrt{3} \) inches (since it is opposite the 60° angle).

Therefore, the length of the perpendicular bisector (height) of the triangle is:

18√3 inches.

Thus, the correct answer is: 18√3 inches.