# The perimeter of a triangular plot of land is 2400ft. The longest side is 200 ft less than twice the shortest. The middle side is is 200 ft ,ess than the longest side. Finde the lengths of the three sides of the triangular plot

Question
Modeling data distributions
The perimeter of a triangular plot of land is 2400ft. The longest side is 200 ft less than twice the shortest. The middle side is is 200 ft ,ess than the longest side. Finde the lengths of the three sides of the triangular plot

2021-01-01
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

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