Let's denote the lengths of the legs as \(a\) and \(b\), where \(b\) is the leg with a known length of 6 feet.
Given that the hypotenuse (\(c\)) is twice the length of one leg, we can express this relationship as:
\[ c = 2a \]
We also know the Pythagorean theorem for a right-angled triangle:
\[ c^2 = a^2 + b^2 \]
Substitute the expression for \(c\) from the first equation into the Pythagorean theorem:
\[ (2a)^2 = a^2 + 6^2 \]
\[ (2a)^2 = a^2 + 6^2 \]
Expand and simplify:
\[ 4a^2 = a^2 + 36 \]
Subtract \(a^2\) from both sides:
\[ 3a^2 = 36 \]
Divide by 3:
\[ a^2 = 12 \]
Take the square root of both sides. Since we're dealing with lengths, consider both the positive and negative square roots:
\[ a = \pm \sqrt{12} \]
Further simplify:
\[ a = \pm 2\sqrt{3} \]
So, the possible values for \(a\) are \(2\sqrt{3}\) and \(-2\sqrt{3}\).
Now, we know the lengths of the legs (\(a\) and \(b\)) and the hypotenuse (\(c\)):
\[ a = 2\sqrt{3} \, \text{feet} \]
\[ b = 6 \, \text{feet} \]
\[ c = 2a = 4\sqrt{3} \, \text{feet} \]
The lengths of the three sides of the triangle are \(2\sqrt{3}\) feet, \(6\) feet, and \(4\sqrt{3}\) feet.