5. As shown in the figure, quadrilaterals $A B C D$ and $C E F G$ are two squares. Given that $\frac{S_{\triangle A B I}}{S_{\triangle E F I}}=27$, find $\frac{A B}{E F}$.
【Analysis】Let $A B=x, E F=y$,
$$
\begin{array}{l}
\frac{A I}{I E}=\frac{S_{\triangle A B F}}{S_{\triangle B E F}}=\frac{\frac{1}{2} \times x \times(x+y)}{\frac{1}{2} \times y \times y}=\frac{x(x+y)}{y^{2}} \\
\frac{I F}{I B}=\frac{S_{\triangle A E F}}{S_{\triangle A B E}}=\frac{\frac{1}{2} \times y \times(x+y)}{\frac{1}{2} \times x \times x}=\frac{x(x+y)}{x^{2}}
\end{array}
$$
$S_{\triangle A B I}$ and $S_{\triangle E F I}$ share an angle (bird head model)
$$
\frac{S_{\triangle A B I}}{S_{\triangle E F I}}=\frac{A I \times B I}{I E \times I F}=\frac{x(x+y)}{y^{2}} \times \frac{x^{2}}{y(x+y)}=\frac{x^{3}}{y^{3}}=27
$$

So $\frac{A B}{E F}=\frac{x}{y}=3$.
【Answer】3