Final answer:
To find the length of a side of an equilateral triangle with an altitude of 12 feet, we identify that the altitude forms two 30-60-90 right triangles within the equilateral triangle. Using the ratio 1:√3:2 for the sides of a 30-60-90 triangle, the side length is found to be 8√3 feet.
Step-by-step explanation:
To find the length of the side of an equilateral triangle given the length of the altitude, we can use the properties of special triangles. Since an equilateral triangle has all sides equal and all angles equal, drawing an altitude creates two 30-60-90 right triangles. For these triangles, the ratio of the lengths of the sides opposite the 30°, 60°, and 90° angles is 1:√3:2.
The altitude is the side opposite the 60° angle, so using the altitude length of 12 feet, we can set up a proportion to find the side of the equilateral triangle using the ratio above:
1:√3 = x:12
To solve for x, which is the length of the side opposite the 30° angle (half of the equilateral triangle's side), multiply both sides by 12:
12(1) = x(√3)
The full length of the equilateral triangle's side is 2 times x:
2x = 2(12/√3)
Therefore, after rationalizing the denominator by multiplying the numerator and the denominator by √3 we get:
2x = 2(12√3/3)
2x = 8√3
The side length of the equilateral triangle is 8√3 feet.