a, b and c be the shortest, middle and longest sides of the triangle respectively.

Then according to given data, we get

\(\displaystyle{a}+{b}+{c}={2400}\to{\left({1}\right)}{\left[:'\text{Perimeter = sum of all sides}\right]}\)

\(\displaystyle{c}={2}{a}-{200}\to{\left({2}\right)}\) [logest side is 200 feet less than whe twice the shortest]

\(\displaystyle{b}={c}-{200}\to{\left({3}\right)}\) [middle side is 200 feet less than the logest side]

From equation (1), we have

\(\displaystyle{a}={2400}-{b}-{c}\)

\(\displaystyle\therefore\) equation (2) becomes,

\(\displaystyle{c}={2}{\left({2400}-{b}-{c}\right)}-{200}\Rightarrow{c}={4800}-{2}{b}-{2}{c}-{200}\)

\(\displaystyle\Rightarrow{3}{c}={4600}-{2}{b}\Rightarrow{3}{c}={4600}-{2}{b}\Rightarrow{3}{c}={4600}-{2}{\left({c}-{200}\right)}\) [using equation (3)]

\(\displaystyle\Rightarrow{3}{c}={4600}-{2}{c}+{400}\Rightarrow{5}{c}={5000}\Rightarrow{c}={1000}\)

Now substituting \(\displaystyle{c}={1000}\) in equation (3), we get \(\displaystyle{b}={1000}-{200}\Rightarrow{b}={800}\)

Now substituting \(\displaystyle{c}={1000}\) in equation (2), we get \(\displaystyle{1000}={2}{a}-{200}\Rightarrow{2}{a}={1200}\Rightarrow{a}={600}\)

Hence the lengths of the sides are 600 ft, 800 ft and 1000 ft

Then according to given data, we get

\(\displaystyle{a}+{b}+{c}={2400}\to{\left({1}\right)}{\left[:'\text{Perimeter = sum of all sides}\right]}\)

\(\displaystyle{c}={2}{a}-{200}\to{\left({2}\right)}\) [logest side is 200 feet less than whe twice the shortest]

\(\displaystyle{b}={c}-{200}\to{\left({3}\right)}\) [middle side is 200 feet less than the logest side]

From equation (1), we have

\(\displaystyle{a}={2400}-{b}-{c}\)

\(\displaystyle\therefore\) equation (2) becomes,

\(\displaystyle{c}={2}{\left({2400}-{b}-{c}\right)}-{200}\Rightarrow{c}={4800}-{2}{b}-{2}{c}-{200}\)

\(\displaystyle\Rightarrow{3}{c}={4600}-{2}{b}\Rightarrow{3}{c}={4600}-{2}{b}\Rightarrow{3}{c}={4600}-{2}{\left({c}-{200}\right)}\) [using equation (3)]

\(\displaystyle\Rightarrow{3}{c}={4600}-{2}{c}+{400}\Rightarrow{5}{c}={5000}\Rightarrow{c}={1000}\)

Now substituting \(\displaystyle{c}={1000}\) in equation (3), we get \(\displaystyle{b}={1000}-{200}\Rightarrow{b}={800}\)

Now substituting \(\displaystyle{c}={1000}\) in equation (2), we get \(\displaystyle{1000}={2}{a}-{200}\Rightarrow{2}{a}={1200}\Rightarrow{a}={600}\)

Hence the lengths of the sides are 600 ft, 800 ft and 1000 ft