Question
Special Right Triangles Quick Check 1 of 51 of 5 Items 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 36 inches 36 inches 9 inches 9 inches 183–√ inches 18 Start Root 3 End Root inches 18 inches 18 inches Skip to navigation
Answers
GPT-4o mini
To find the length of the perpendicular bisector of any side of an equilateral triangle, we can follow these steps:
1. **Identify the Triangle Type**: In an equilateral triangle, all sides are equal, and all angles are 60 degrees.
2. **Find the Height**: The height of an equilateral triangle can be calculated using the formula from the properties of a 30-60-90 triangle. The height divides the equilateral triangle into two 30-60-90 triangles.
3. **Application of 30-60-90 Triangle**: In a 30-60-90 triangle, the ratio of the lengths of the sides is:
- The shorter leg (opposite the 30° angle) is \( \frac{1}{2} \) of the hypotenuse.
- The longer leg (opposite the 60° angle) is \( \frac{\sqrt{3}}{2} \) of the hypotenuse.
In this case, the hypotenuse of the right triangle is the side of the equilateral triangle, which is 36 inches.
4. **Calculating the Height**:
- The shorter leg (half of the base of the equilateral triangle) is \( \frac{36}{2} = 18 \) inches.
- The longer leg (the height of the triangle) is:
\[
\text{Height} = 18\sqrt{3} \text{ inches}
\]
Thus, the length of the perpendicular bisector of any side (which is also the height of the equilateral triangle) is \( 18\sqrt{3} \) inches.
**Correct Answer**: \( 18\sqrt{3} \) inches
1. **Identify the Triangle Type**: In an equilateral triangle, all sides are equal, and all angles are 60 degrees.
2. **Find the Height**: The height of an equilateral triangle can be calculated using the formula from the properties of a 30-60-90 triangle. The height divides the equilateral triangle into two 30-60-90 triangles.
3. **Application of 30-60-90 Triangle**: In a 30-60-90 triangle, the ratio of the lengths of the sides is:
- The shorter leg (opposite the 30° angle) is \( \frac{1}{2} \) of the hypotenuse.
- The longer leg (opposite the 60° angle) is \( \frac{\sqrt{3}}{2} \) of the hypotenuse.
In this case, the hypotenuse of the right triangle is the side of the equilateral triangle, which is 36 inches.
4. **Calculating the Height**:
- The shorter leg (half of the base of the equilateral triangle) is \( \frac{36}{2} = 18 \) inches.
- The longer leg (the height of the triangle) is:
\[
\text{Height} = 18\sqrt{3} \text{ inches}
\]
Thus, the length of the perpendicular bisector of any side (which is also the height of the equilateral triangle) is \( 18\sqrt{3} \) inches.
**Correct Answer**: \( 18\sqrt{3} \) inches