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.
-
Side Length: Each side of the equilateral triangle is 36 inches.
-
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.
-
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).
-
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.
-
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.
1 answer