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# The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.

Given:

The diagonal of a rectangular field is 60 meters more than the shorter side.

The longer side is 30 meters more than the shorter side.

To do:

We have to find the sides of the field.

Solution:

Let the length of the shorter side be $x$ m.

This implies, the length of the longer side$=x+30$ m.

The length of the diagonal$=x+60$ m.

We know that,

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. (Pythagoras theorem)

Therefore,

$(x)^2+(x+30)^2=(x+60)^2$

$x^2+x^2+60x+900=x^2+120x+3600$

$2x^2-x^2+60x-120x+900-3600=0$

$x^2-60x-2700=0$

Solving for $x$ by factorization method, we get,

$x^2-90x+30x-2700=0$

$x(x-90)+30(x-90)=0$

$(x-90)(x+30)=0$

$x+30=0$ or $x-90=0$

$x=-30$ or $x=90$

Length cannot be negative. Therefore, the value of $x$ is $90$.

$x+30=90+30=120$

The lengths of the sides of the field are $90$ m and $120$ m.

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