Find the length of a side of an equilateral triangle with altitude of length 6.

let each side of the triangle be 2x units

then x^2 + 6^2 = (2x^2)
x^2 + 36 = 4x^2
3x^2 = 36
x^2 = 12
x = 2√3
2x = 4√3

or
knowing the ratio of sides of a 30-60-90 triangle
is 1:√3:2
h/2 = 6/√3
√3h=12
h = 12/√3
= 12/√3(√3/√3) = 12√3/3 = 4√3

To find the length of a side of an equilateral triangle with an altitude of length 6, we can use the Pythagorean theorem.

Step 1: Draw the equilateral triangle and label the altitude as 6 units.

Step 2: Since the given triangle is equilateral, all sides are congruent. Let's call the length of each side "s".

Step 3: Draw the altitude from one of the vertices to the midpoint of the opposite side. This will divide the equilateral triangle into two congruent right triangles.

Step 4: Use the Pythagorean theorem to find the length of one side. In one of the right triangles, we have the hypotenuse as the side "s" and one leg as 6 units.

According to the Pythagorean theorem, the sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse.

So, we have the equation: 6^2 + (s/2)^2 = s^2.

Simplifying this equation: 36 + s^2/4 = s^2.

Multiply both sides of the equation by 4 to eliminate the fraction: 144 + s^2 = 4s^2.

Subtract s^2 from both sides: 144 = 3s^2.

Divide both sides by 3: s^2 = 48.

Take the square root of both sides to find s: s = √48.

Simplifying the square root: s ≈ 6.93.

Therefore, the length of a side of the equilateral triangle with an altitude of length 6 is approximately 6.93 units.