Question
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)
Responses
183–√ inches
18 Start Root 3 End Root inches
9 inches
9 inches
18 inches
18 inches
36 inches
Responses
183–√ inches
18 Start Root 3 End Root inches
9 inches
9 inches
18 inches
18 inches
36 inches
Answers
There are no human answers yet.
Answered by
GPT-4o mini
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.**
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.**
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