To find the perimeter of an equilateral triangle when given the altitude, we start by recalling a few properties of equilateral triangles.
In an equilateral triangle, all three sides are equal, and the altitude creates two 30-60-90 right triangles. The altitude can be expressed in terms of the length of a side (let’s call it ‘s’). For an equilateral triangle, the altitude is calculated as:
Altitude = (sqrt(3)/2) * side
In this case, we know the altitude is 15 m. Therefore, we can set up the equation:
15 = (sqrt(3)/2) * s
To find ‘s’, we rearrange the equation:
s = 15 * (2/sqrt(3))
Next, we simplify this:
s = 30/sqrt(3)
Now, to rationalize the denominator:
s = 30 * (sqrt(3)/3) = 10 * sqrt(3)
Now that we know the length of one side, we can find the perimeter of the triangle. The perimeter (P) of an equilateral triangle is given by:
P = 3 * s
Thus:
P = 3 * (10 * sqrt(3)) = 30 * sqrt(3)
Finally, calculating the numerical value:
Using the approximation of sqrt(3) as about 1.732, we find:
P ≈ 30 * 1.732 ≈ 51.96 m
So, the perimeter of the equilateral triangle is approximately 51.96 meters.