Question

Sum of areas of two squares is 244 cm and theri difference between paerimeters is 8 cm.find the ratio of their diagonals

Answer 1

Let the sides of two square be x and y respectively.

So area of first sqaure = x²
And area of second square = y²

Given the sum of their squares is 244

=> x² + y² = 244...........(i)

Now perimeter of first sq. = 4x
and that of second = 4y

Given difference of their perimeter is 8

=> 4x - 4y = 8

=> 4(x - y) = 8

=> x - y = 8/4 = 2.............(ii)

Now squaring both sides in equation ii

(x - y)² = (2)²

=> x² + y² - 2xy = 4

Substituting value of x² + y² from eqn. (i)
we get

244 - 2xy = 4

=> -2xy = 4 - 244

=> -2xy = -240

=> 2xy = 240...............(iii)

Now in equation (i)

Adding 2xy on both sides,

x² + y² + 2xy = 244 + 2xy

Putting value of 2xy in R.H.S from eqn. (iii)
x² + y² + 2xy = 244 + 240

=> (x + y)² = 484

=> (x + y)² = (22)²

=> x + y = 22 .............(iv)

Now adding i and iv eqn.


x + y + x - y = 22 + 2

=> 2x = 24

=> x = 24/2 = 12

and hence, from eqn. iv
x + y = 22

=> 12 + y = 22

=> y = 22 - 12
=> y = 10


Now we know that diagonal of a square = a√2

where a = side

Hence diagonal of first square
= x√2
=> 12√2

And that of the second
= y√2
=> 10√2

Ratio of diagonal

=> 12√2 : 10√2

Simplifying,

6:5


Hope it helps dear friend ☺️✌️

Answer 2

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