Question

The diagonal of a rectangular field is 40 meters more than the shorter side. If the longer side is 20 meters more than the shorter side, find the sides of the field.

Answer 1

Hint: Firstly, we will assume the value of the shorter side with respect to that we assume the other values as \[x + 40\] and \[x + 20\] . Then apply the Pythagoras theorem to calculate the value of \[x\] .

Complete answer:
It is given that diagonal of a rectangular field is 40 metres more than the shorter side
Let us assume that the side of the field which is the shorter side of the rectangle is equal to \[x\] meters.
So, diagonal will be \[x + 40\] meters.
Given that the longer side is 20 metres more than the shorter side, the longer side is equal to \[x + 20\] meters.
As we all know that all the angles in a rectangle are always equal to 90 degrees.
So, Now we will use Pythagoras theorem to calculate the value of \[x\].
So, we can say that \[{\left( {Diagonal} \right)^2} = {\left( {shorterside} \right)^2} + {\left( {longerside} \right)^2}\].
By, substituting the values of variables we get,
\[{\left( {x + 40} \right)^2} = {\left( x \right)^2} + {\left( {x + 20} \right)^2}\]
By using the identity \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\] on both the left and right side of the equation.
We get, \[ \Rightarrow {x^2} + 1600 + 80x = {x^2} + {x^2} + 400 + 40x\]
By cancelling \[{x^2}\] on both sides,
\[ \Rightarrow 1600 + 80x = {x^2} + 400 + 40x\]
Let’s take on the variables and constants on the right side of the equation.
\[ \Rightarrow 0 = {x^2} + 400 + 40x - 80x - 1600\]
On simplifying further we get,
\[ \Rightarrow 0 = {x^2} - 40x - 1200\]
Now we will use splitting the middle term method to calculate the value of \[x\] .
In this method we will find out two numbers whose sum is \[ - 40\] and product is \[ - 1200\].
So, we are getting the two numbers which are \[ - 60\] and \[ + 20\] .
 \[ \Rightarrow 0 = {x^2} - 40x - 1200\]
\[ \Rightarrow 0 = {x^2} - 60x + 20x - 1200\]
Taking out common in the pairs of 2 we get,
\[ \Rightarrow 0 = x\left( {x - 60} \right) + 20\left( {x - 60} \right)\]
Taking \[\left( {x - 60} \right)\] common on right side,
\[ \Rightarrow 0 = \left( {x + 20} \right)\left( {x - 60} \right)\]
Here \[x\] has two factors which means it will have two values which can be found by equating pairs to 0.
\[\therefore\] x = - 20 and x = 60
As it is basics of geometry that any rectangle cannot have a negative side
\[\therefore x = 60\]

Hence, shorter side is \[ = x = 60m\] , longer side is \[ = x + 20 = 80m\] and the diagonal is \[ = x + 40 = 100m\]

Note: To solve these types of questions, we must make the equations according to the given statements. After conversion of the statement we will use the equations to provide us with the value which is required to get the desired results.