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Home » If the perimeter of a rectangular plot is     m and the length of its diagonal is     m, find its area.
If the perimeter of a rectangular plot is

    \[68\]

m and the length of its diagonal is

    \[26\]

m, find its area.
Class 10 | Exercise 1 | ICSE | Maths | ML Aggarwal | Quadratic Equations in One Variable
If the perimeter of a rectangular plot is

    \[68\]

m and the length of its diagonal is

    \[26\]

m, find its area.

Given,

Perimeter =

    \[68\]

m and diagonal =

    \[26\]

m

So, Length + breadth = Perimeter/

    \[2\]

=

    \[68/2\]

=

    \[34\]

m

Let’s consider the length of the rectangular plot to be ‘x’ m

Then, breadth =

    \[(34-x)\]

m

Now, the diagonal of the rectangular plot is given by

length2 + breadth2 = diagonal2 [By Pythagoras Theorem]

    \[\begin{array}{*{35}{l}} {{x}^{2}}~+\text{ }{{\left( 34\text{ }\text{ }x \right)}^{2}}~=\text{ }{{26}^{2}} \\ {{x}^{2}}~+\text{ }1156\text{ }+\text{ }{{x}^{2}}~\text{ }68x\text{ }=\text{ }676 \\ 2{{x}^{2}}~\text{ }68x\text{ }+\text{ }1156\text{ }\text{ }676\text{ }=\text{ }0 \\ 2{{x}^{2}}~\text{ }68x\text{ }+\text{ }480\text{ }=\text{ }0 \\ {{x}^{2}}~\text{ }34x\text{ }+\text{ }240\text{ }=\text{ }0 \\ \end{array}\]

[Dividing by

    \[2\]

]

By factorization method, we have

    \[\begin{array}{*{35}{l}} {{x}^{2}}~\text{ }24x\text{ }\text{ }10x\text{ }+\text{ }240\text{ }=\text{ }0 \\ x\left( x\text{ }\text{ }24 \right)\text{ }\text{ }10\left( x\text{ }\text{ }24 \right)\text{ }=\text{ }0 \\ \left( x\text{ }\text{ }10 \right)\text{ }\left( x\text{ }\text{ }24 \right)\text{ }=\text{ }0 \\ \end{array}\]

So,

    \[\begin{array}{*{35}{l}} x\text{ }\text{ }10\text{ }=\text{ }0\text{ }or\text{ }x\text{ }\text{ }24\text{ }=\text{ }0 \\ x\text{ }=\text{ }10\text{ }or\text{ }x\text{ }=\text{ }24 \\ \end{array}\]

As length is greater than breadth,

Thus, length =

    \[24\]

m and breadth =

    \[\left( 34\text{ }\text{ }24 \right)\text{ }m\text{ }=\text{ }10\]

m

And, area of the rectangular plot =

    \[24\text{ }\times \text{ }10\text{ }=\text{ }240\text{ }{{m}^{2}}\]

i

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