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

183–√ inches
18 Start Root 3 End Root inches

9 inches
9 inches

18 inches

Answer 1

To find the length of the perpendicular bisector of any side of an equilateral triangle, we can set up a right triangle using the properties of the equilateral triangle.

1. **Side Length**: Each side of the equilateral triangle is 36 inches.

2. **Height of the Triangle**: The perpendicular bisector from the vertex to the midpoint of the opposite side also serves as the height of the triangle.

3. **Midpoint Calculation**: The midpoint of one side of the triangle divides it into two equal segments of 18 inches (since 36 inches ÷ 2 = 18 inches).

4. **Right Triangle Formation**: We can now see that we form a right triangle, where:
- One leg (half of the side) is 18 inches,
- The other leg (the height) is what we need to find,
- The hypotenuse is the side of the equilateral triangle, which is 36 inches.

5. **Using the Pythagorean Theorem**:
\[
\text{height}^2 + (18 \text{ inches})^2 = (36 \text{ inches})^2
\]
\[
\text{height}^2 + 324 = 1296
\]
\[
\text{height}^2 = 1296 - 324
\]
\[
\text{height}^2 = 972
\]
\[
\text{height} = \sqrt{972} = \sqrt{36 \times 27} = 6\sqrt{27} = 6 \times \sqrt{9 \times 3} = 18\sqrt{3}
\]

Thus, the length of the perpendicular bisector (which is essentially the height of the triangle) is \(18\sqrt{3}\) inches.

The correct response is:
**18√3 inches**.