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

Hint: Let the length of the shorter side be x, so form the equation considering the above condition that longer side is 30 metre more than the shorter side and the diagonal of a rectangular field is 60 metre more than the shorter side. And as we know that in the rectangle adjacent sides and diagonal will form a right angled triangle so apply Pythagoras theorem to it and hence on solving the value of x will be obtained.

Complete step-by-step answer:
As per the given the length of diagonal of a rectangular field is 60 metre more than the shorter side and longer side is 30 metre more than the shorter side.
So, let the length of the shorter side be \[x\].
So, the diagonal will be 60 metre more than the shorter side which can be given as \[x + 60\]
And the length of longer side is 60 metre more than the shorter side so it will be \[x + 30\]

Diagram:

As we know that in rectangle adjacent sides and diagonal will form a right angled triangle,
Now, applying Pythagoras theorem in the above triangle,
We can see that
\[{\left( {x + 60} \right)^2} = {\left( x \right)^2} + {\left( {x + 30} \right)^2}\]
Now, simplifying the above term using \[{(a + b)^2} = {a^2} + {b^2} + 2ab\],
\[ \Rightarrow {x^2} + 120x + 3600 = {x^2} + {x^2} + 60x + 900\]
Hence, on further simplification we get,
\[ \Rightarrow \] \[{x^2} - 60x - 2700 = 0\]
Now, we factorize the above quadratic to solve for the value of x,
\[ \Rightarrow \] \[{x^2} - 90x + 30x - 2700 = 0\]
Now, take the terms common from the above equation
\[ \Rightarrow \] \[x\left( {x - 90} \right) + 30\left( {x - 90} \right) = 0\]
On further simplifying, we get,
\[ \Rightarrow \] \[\left( {x + 30} \right)\left( {x - 90} \right) = 0\]
Hence, the sides cannot be negative so the value of x is

\[ \Rightarrow \] \[\left( {x - 90} \right) = 0\]
Hence, value of x is
\[ \Rightarrow \] \[x = 90\]
And the longer side of the field is \[x + 30 = 90 + 30 = 120metre\]
The diagonal is \[x + 60 = 90 + 60 = 150metre\].
So, the length of the field is 120 metre and breadth is 90 metre.

Note: Use the above condition to form an equation and solve the calculation and apply Pythagoras theorem properly.
Note that the side cannot have a negative value so, we simply ignore that value of x.
A rectangle is a quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal. It can also be defined as a parallelogram containing a right angle.