The altitude of an equilateral triangle is 9 meters long. Find the perimeter of the triangle. Put in radical form.
if the altitude is h, then each side is 2h/√3
so the perimeter is 3*2h/√3 = 2h√3
To find the perimeter of an equilateral triangle, we need to know the length of one side. However, we are given the altitude of the triangle, which is a vertical line segment from one vertex to the opposite side.
In an equilateral triangle, the altitude divides the triangle into two congruent 30-60-90 right triangles. Let's denote the length of the altitude as "h," and the length of the side of the equilateral triangle as "s."
In a 30-60-90 right triangle, the sides are related by the following ratios:
- The length of the side opposite the 30-degree angle is `h/2`.
- The length of the side opposite the 60-degree angle is `(sqrt(3)/2) * h`.
- The length of the hypotenuse (which is the side of the equilateral triangle) is `s`.
Given that the altitude is 9 meters, we can set up the following equation:
`(sqrt(3)/2) * h = 9`
To solve for h, we divide both sides of the equation by `(sqrt(3)/2)`:
`h = 9 / (sqrt(3)/2)`
`= 9 * (2/sqrt(3))`
`= 18 / sqrt(3)`
`= (18 * sqrt(3)) / (sqrt(3) * sqrt(3))`
`= (18 * sqrt(3)) / 3`
`= 6 * sqrt(3)`
Now that we have the length of the altitude, we can find the length of one side of the equilateral triangle:
`s = 2 * h`
`= 2 * (6 * sqrt(3))`
`= 12 * sqrt(3)`
Finally, we can find the perimeter of the equilateral triangle by multiplying the length of one side by 3:
Perimeter = 3 * s
= 3 * (12 * sqrt(3))
= 36 * sqrt(3) meters
Therefore, the perimeter of the equilateral triangle is 36 * sqrt(3) meters.
To find the perimeter of an equilateral triangle, we need to use the length of one side. We can use the altitude of the triangle to find this length.
In an equilateral triangle, the altitude splits the triangle into two congruent right triangles. Let's call the length of one side of the triangle "s".
Using the Pythagorean theorem, we can find the length of one side:
s^2 = (altitude)^2 + (base of right triangle)^2
s^2 = 9^2 + (s/2)^2
s^2 = 81 + s^2/4
Multiply both sides by 4 to eliminate the fraction:
4s^2 = 324 + s^2
3s^2 = 324
s^2 = 324/3
s^2 = 108
Now, we can find the perimeter by multiplying the length of one side by 3:
perimeter = 3s
perimeter = 3 * √108 (we want the answer in radical form)
To simplify the radical, we can factor out the largest perfect square from 108:
108 = 36 * 3
Now the equation becomes:
perimeter = 3 * √(36 * 3)
perimeter = 3 * 6 * √3
perimeter = 18√3
Therefore, the perimeter of the equilateral triangle is 18√3 meters.