Quadrilateral BCDE is similar to quadrilateral FGHI. Find the measure of side HI. Figures are not drawn to scale.

Since quadrilateral $BCDE$ is similar to quadrilateral $FGHI$, we know that the corresponding side lengths are proportional. Specifically,

\[\frac{BC}{FG}=\frac{CD}{HI}=\frac{DE}{IH}\]
We are asked to find the measure of side $HI$. Notice that $HI$ appears in the denominator of the fraction $\frac{CD}{HI}$. So, we use the fact that $CD$ corresponds to $FG$ to equate those two lengths:
\[\frac{BC}{FG}=\frac{CD}{HI}\]
\[\frac{BC}{7}=\frac{7}{HI}\]
\[BC\cdot HI=49\]
\[4.2\cdot HI=49\]
Solving the above equation, we find that $HI=\boxed{11.67}$