$$
\begin{align}
EH
&=ED\sec\left(15^{\large\circ}\right)\\
&=\frac12\left(\sqrt6-\sqrt2\right)
\end{align}
$$

$$
\begin{align}
GC
&=AC-AK-KG\\
&=\sqrt2-\frac{\sqrt2}4-\frac{\sqrt6}4\\
&=\frac14\left(3\sqrt2-\sqrt6\right)
\end{align}
$$

$$
\begin{align}
\frac{x}{24}
&=\frac{EH\sin\left(60^{\large\circ}\right)}{GC\sin\left(150^{\large\circ}\right)}\\
&=\frac{\frac12\left(\sqrt6-\sqrt2\right)\frac{\sqrt3}2}{\frac14\left(3\sqrt2-\sqrt6\right)\frac12}\\
&=\frac{\frac14\left(\sqrt6-\sqrt2\right)}{\frac18\left(\sqrt6-\sqrt2\right)}\\
&=2
\end{align}
$$