Example 3. The side length of square $ABCD$ is $a$. Fold the square so that $C$ falls on the midpoint $G$ of $AD$. Find the length of the crease $EF$.

Fold the square so that $C$ falls on the midpoint $G$ of $AD$. Find the length of the crease $EF$.
Draw $E H \perp$ $C D$, with the foot of the perpendicular being $H$. Given $E F$ $\perp C G$, then $\triangle E F H \cong$ $\triangle C G D$.
Thus, $E F=C G$
$$
=\frac{\sqrt{5}}{2} a \text {. }
$$