Question

Sum of the areas of two squares is $468$${m^2}$. If the difference of their perimeters is $24$m, find the sides of the two squares.

Answer 1

Hint: We are given the sum of the area of two squares and the difference of the perimeter of two squares as we know that all sides of square are equal and area means the quantity that expresses the extent of a two-dimensional figure or shape or lamina in the plane. Perimeter means the boundary so first we will assume the length of one side of each square as a variable and then find the area and perimeter using the formula
* Area of a square having side ‘a’ units \[ = {a^2}\]
* Perimeter a square having side ‘a’ units \[4a\]
Here a is the side of the square by finding the area and perimeter and equating the sum and difference to the numbers given we form two equations and we can solve them simultaneously and find the value of variables and then we can find the area and perimeter using the measurement of sides.

Complete step-by-step solution:
Step1: we are given two squares let us assume that the side of one square be $x$and the other be $x - 6$
Also given that sum of the area of square is $468$${m^2}$
Area of square 1+ area of square 2 $ = 468$${m^2}$
Area of square1$ = {x^2}$; Area of square 2$ = {(x - 6)^2}$
$ \Rightarrow {x^2} + {(x - 6)^2} = 468$
Use identity\[{\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\]
$ \Rightarrow {x^2} + {x^2} - 2 \times x \times 6 + {6^2} = 468$
$ \Rightarrow {x^2} + {x^2} - 12x + 36 = 468$
Subtracting 468 from both the sides
$ \Rightarrow 2{x^2} - 12x - 432 = 0$
Dividing both sides by 2
$ \Rightarrow \dfrac{{2{x^2} - 12x - 432}}{2} = \dfrac{0}{2}$
$ \Rightarrow {x^2} - 6x - 216 = 0$
Step2: Compare equation with $a{x^2} + bx + c = 0$
Here $a = 1;b = - 6;c = - 216$
We know that \[D = {b^2} - 4ac\]
Substitute the values of a, b and c
$ \Rightarrow D = {( - 6)^2} - 4 \times 1 \times ( - 216)$
$ \Rightarrow D = 36 + 4 \times 216$
$ \Rightarrow D = 36 + 864$
$ \Rightarrow D = 900$
So the roots to equation are given by the formula \[x = \dfrac{{ - b \pm \sqrt D }}{{2a}}\]
Put values $b = - 6$; $D = 900$; $a = 1$
$x = \dfrac{{ - ( - 6) \pm \sqrt {900} }}{{2 \times 1}}$
$ \Rightarrow x = \dfrac{{6 \pm \sqrt {900} }}{2}$
$ \Rightarrow x = \dfrac{{6 \pm \sqrt {9 \times 100} }}{2}$
On forming the squares inside the square root
$ \Rightarrow x = \dfrac{{6 \pm \sqrt {{3^2} \times {{10}^2}} }}{2}$
Taking the square root
$ \Rightarrow x = \dfrac{{6 \pm 3 \times 10}}{2}$
$ \Rightarrow x = \dfrac{{6 \pm 30}}{2}$
Solving:
$ \Rightarrow x = \dfrac{{6 + 30}}{2}$
$ \Rightarrow x = \dfrac{{36}}{2}$
$ \Rightarrow x = 18$
Or,
$ \Rightarrow x = \dfrac{{6 - 30}}{2}$
$ \Rightarrow x = \dfrac{{ - 24}}{2}$
$ \Rightarrow x = - 12$
So, $x = 18$ & $x = - 12$
Since $x$ is side of square and $x$ cannot be negative
So, $x = 18$ is the solution
Therefore, side of a square 1 is $x = 18$m
& side of square 2 $ = x - 6$
Substituting value of x
$ = 18 - 6$
$ = 12$m

Hence sides of two squares are $18$m & $12$m

Note: In these types of questions students mainly get confused in units of perimeter and area in area units \[{m^2}\] \[c{m^2}\] and perimeter units cm or m. In this students do calculation mistakes they get confused in solving equations specially quadratic equation after solving quadratic we get two values of a variable so keep in mind negative values are always neglected as area, length, volume these units cannot be negative so choose positive values always.