To find the perimeter of an equilateral triangle when given the altitude, we need to follow a few mathematical steps. An equilateral triangle has all three sides of equal length, and the altitude creates two 30-60-90 right triangles. This property of the triangle can be used to find the side length.
In a 30-60-90 triangle, the ratios of the lengths of the sides are as follows:
- Shorter leg (opposite the 30° angle): x
- Longer leg (opposite the 60° angle): x√3
- Hypotenuse (which is also a side of the equilateral triangle): 2x
In our case, the altitude of the equilateral triangle is 15 meters, which corresponds to the longer leg of the 30-60-90 triangle. Therefore, we can set up the equation:
x√3 = 15
To solve for x, we rearrange the equation:
x = 15 / √3
Now, to simplify, we can multiply the numerator and the denominator by √3:
x = (15√3) / 3 = 5√3
Now that we have the length of the shorter leg (x), we can find the length of the hypotenuse, which gives us the side length of the equilateral triangle:
Side length = 2x = 2 * (5√3) = 10√3
Finally, the perimeter (P) of the equilateral triangle, which is three times the length of one side, can be calculated as:
P = 3 * (10√3) = 30√3
To create a numerical approximation, we can substitute the value of √3 (approximately 1.732):
P ≈ 30 * 1.732 ≈ 51.96 m
Thus, the perimeter of the equilateral triangle with an altitude of 15 meters is approximately 51.96 meters.