From the figure we have

\(\displaystyle\overline{{{F}{G}}}=\overline{{{E}{F}}}\)

and \(\displaystyle\angle{G}{H}{F}={90}°\)

\(\displaystyle\Rightarrow\angle{E}{H}{F}={90}°\)

also \(\displaystyle\overline{{{G}{H}}}={93}\) units

Since \(\triangle GHF\) is right angle triangle, hence

\(\displaystyle\overline{{{G}{H}}}^{{2}}+\overline{{{H}{F}}}^{{2}}=\overline{{{G}{F}}}^{{2}}\)

\(\displaystyle{93}^{{2}}+\overline{{{H}{F}}}^{{2}}=\overline{{{G}{F}}}^{{2}}\)

\(\displaystyle\overline{{{G}{F}}}^{{2}}-\overline{{{H}{F}}}^{{2}}={93}^{{2}}\)-(1)

Also since \(\triangle FHE\) is right angle triangle, hence

\(\displaystyle\overline{{{F}{H}}}^{{2}}+\overline{{{H}{E}}}^{{2}}=\overline{{{E}{F}}}^{{2}}\)

\(\displaystyle\overline{{{H}{E}}}^{{2}}=\overline{{{E}{F}}}^{{2}}-\overline{{{F}{H}}}^{{2}}\)

\(\displaystyle=\overline{{{G}{F}}}^{{2}}-\overline{{{H}{F}}}^{{2}}\)

\(\displaystyle={93}^{{2}}\), using (1)

\(\displaystyle\therefore\overline{{{H}{F}}}={93}\)

Therefore \(\displaystyle\overline{{{G}{E}}}=\overline{{{G}{H}}}+\overline{{{H}{E}}}\)

\(\displaystyle={93}+{93}\)

\(\displaystyle={186}\)

\(\displaystyle\therefore\overline{{{G}{E}}}={186}\)