3. In Rt $\triangle A B C$, it is known that $\angle B A C=90^{\circ}, A B=$ $6, A C=8$, points $E$ and $F$ are on side $A B$ such that $A E=2, B F=3$. Draw a line through point $E$ parallel to $A C$, intersecting side $B C$ at point $D$, connect $F D$ and extend it, intersecting the extension of side $A C$ at point $G$. Then the length of segment $F G$ is $\qquad$
3. $\sqrt{265}$.

As shown in Figure 3.
From the problem, we know
$$
\begin{array}{l}
\triangle B E D \backsim \triangle B A C \\
\Rightarrow \frac{E D}{8}=\frac{4}{6} \\
\Rightarrow E D=\frac{16}{3} .
\end{array}
$$

In the right triangle $\triangle D E F$,
$$
\begin{array}{l}
F D=\sqrt{1^{2}+\left(\frac{16}{3}\right)^{2}}=\frac{\sqrt{265}}{3} . \\
\text { Also, } \triangle F E D \backsim \triangle F A G \Rightarrow \frac{F G}{F D}=\frac{F A}{F E}=3 \\
\Rightarrow F G=3 F D=\sqrt{265} .
\end{array}
$$