T in a Triangle

Triangle $ABC$ is equilateral. Line segments $\overline{ED}$ and $\overline{FG}$ are perpendicular and share the same length. Line segment $\overline{ED}$ is parallel to $\overline{AC}$ .

Not to scale.

What is $\frac{x}{s}$ if $x$ is the length of either $\overline{ED}$ or $\overline{FG}$ and $s$ is the side length of $\triangle ABC$ ?

NOTE: The answer must be in the nearest thousandths.