Formula used:

Similar triangles have sides in same ratio.

Calculation:

We know that in similar triangles HFE and FGE,

\(\displaystyle\frac{{{H}{F}}}{{{F}{G}}}=\frac{{{F}{E}}}{{{G}{E}}}=\frac{{{H}{E}}}{{{F}{E}}}\)(by law of similarity)

Hence, we substitute the value of EH and HG in the equation above , we get

\(\displaystyle{F}{E}^{{2}}={H}{E}{\left({G}{E}\right)}\)

Thus, we get

\(\displaystyle{F}{E}=\sqrt{{{77}}}\)

Similar triangles have sides in same ratio.

Calculation:

We know that in similar triangles HFE and FGE,

\(\displaystyle\frac{{{H}{F}}}{{{F}{G}}}=\frac{{{F}{E}}}{{{G}{E}}}=\frac{{{H}{E}}}{{{F}{E}}}\)(by law of similarity)

Hence, we substitute the value of EH and HG in the equation above , we get

\(\displaystyle{F}{E}^{{2}}={H}{E}{\left({G}{E}\right)}\)

Thus, we get

\(\displaystyle{F}{E}=\sqrt{{{77}}}\)