Let the shortest length is represented by the variable a. The other lengths are multiples of a.

If \(\displaystyle{a}={8}\sqrt{{{3}}}\)

\(\displaystyle{a}\sqrt{{{3}}}={\left({8}\sqrt{{{3}}}\right)}\sqrt{{{3}}}={24}\)

\(\displaystyle{2}{a}={2}{\left({8}\sqrt{{{3}}}\right)}={16}\sqrt{{{3}}}\)

Therefore, the height - \(\displaystyle{24}\), hypotenuse - \(\displaystyle{16}\sqrt{{{3}}}\)

If \(\displaystyle{a}={8}\sqrt{{{3}}}\)

\(\displaystyle{a}\sqrt{{{3}}}={\left({8}\sqrt{{{3}}}\right)}\sqrt{{{3}}}={24}\)

\(\displaystyle{2}{a}={2}{\left({8}\sqrt{{{3}}}\right)}={16}\sqrt{{{3}}}\)

Therefore, the height - \(\displaystyle{24}\), hypotenuse - \(\displaystyle{16}\sqrt{{{3}}}\)