Question

Sum of the areas of two squares is 400 square cm. If the difference between their perimeters is 16 cm, find the sides of the two squares.

Answer 1

Hint: We will first write the formula for perimeter and squares and assume the sides of both the squares. Then by putting all these things in the given conditions, we will form some equation which on solving will lead to a result.

Complete step-by-step answer:
Let us first write the formulas of both the perimeter and area of a square.
Perimeter of a square is given by $P = 4 \times side$ and area of a square is given by $A = {(side)^2}$.
Now, let the sides of two squares be $x$ cm and $y$ cm.
Without loss of generality, let $x > y$.
Now, the perimeters of both the squares is $4x$ cm and $4y$ cm.
We are given in the question that:- difference between their perimeters is 16 cm.
So, $4x - 4y = 16$
Taking 4 common from LHS:-
$4(x - y) = 16$
Now, taking the 4 from LHS to RHS:-
$x - y = \dfrac{{16}}{4}$
Now, simplifying the RHS, we will have:-
$x - y = 4$
$ \Rightarrow y = x - 4$ …………..(1)
Now, let us find the areas of both the squares by putting the values in the formula: $A = {(side)^2}$.
So, their areas are ${x^2}$ square cm and ${y^2}$ square cm.
As per the information in the question: the sum of the areas of two squares is 400 square cm.
So, ${x^2} + {y^2} = 400$ ……….(2)
Putting (1) in (2):
${x^2} + {(x - 4)^2} = 400$
Now, using the formula ${(a - b)^2} = {a^2} + {b^2} - 2ab$.
$ \Rightarrow {x^2} + {x^2} + 16 - 8x = 400$
\[ \Rightarrow 2{x^2} - 8x - 384 = 0\]
Taking 2 common, we will get:-
\[ \Rightarrow {x^2} - 4x - 192 = 0\]
We can rewrite it as:- \[ \Rightarrow {x^2} + 12x - 16x - 192 = 0\]
\[ \Rightarrow x(x + 12) - 16(x + 12) = 0\]
\[ \Rightarrow (x + 12)(x - 16) = 0\]
So, either $x = - 12$ or $x = 16$.
Since the side cannot be negative. Therefore the side of one of the squares is 16 cm.
For another, putting this value in (1):-
Side of another square = 16 – 4 = 12 cm

Note: The students must notice that we have used the term “without loss of generality”, we cannot use this term when we already are given something specific about each of the squares. In this question, there was no sequence of squares. So, we used it. You may choose the other possible value as well.
The students may use the direct formula method to find the roots of quadratic equations instead of factorization method as well.