Question

Solve the following questions. (Any two)
Sum of areas of two squares is 244 cm² and the difference between their perimeter is 8 cm.
Find the ratio of their diagonals.
i.
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Answer 1

Step-by-step explanation:

Given:-

Sum of areas of two squares is 244 cm² and the difference between their perimeter is 8 cm.

To find:-

Find the ratio of their diagonals.

Solution:-

Let the side of the first square be X cm

Let the side of the second square be Y cm

(X > Y)

Area of the first square = Side ×Side= X^2 sq.cm

Area of the second square = Y^2 sq.cm

Sum of areas of two squares = X^+Y^2 sq.cm

According to the given problem

Sum of areas of two squares = 244 cm^2

=>X^2 + Y^2 = 244 ---------(1)

Perimeter of a square = 4× length of the side

Perimeter of the first square = 4X cm

Perimeter of the second square = 4Y cm

Their difference = 4X-4Y cm

According to the given problem

The difference between Perimeters of the two squares = 8 cm

=>4X-4Y=8

=>4(X-Y)=8

=>X-Y =8/4

=>X-Y = 2 cm -------------(2)

We know that

(a-b)^2 = a^2 -2ab +b^2

=>(X-Y)^2 = X^2 -2XY +Y^2

=>2^2 = 244-2XY

=>4-244 =-2XY

=>-240 = -2XY

=>XY = -240/-2

XY = 120 -------------(3)

(X+Y)^2 = X^2+Y^2+2XY

=>(X+Y)^2 = 244+2(120)

=>(X+Y)^2 = 244+240

=>(X+Y)^2 = 484

=>X+Y = √484

X+Y= 22------------(4) (taking positive)

From (2) &(4)

X+Y = 22

X-Y = 2

(+)

________

2X+0=24

________

=>2X=24

=>x=24/2

=>X=12

Substituting The value of X in (4) then

=>12+Y = 22

=>Y=22-12

=>Y=10

The values of X = 12 and Y=10

Side of the first square = 12 cm

Diagonal of a square = √2 ×side

Diagonal of the first square = 12√2 cm

Side of the second square = 10 cm

Diagonal of the second square = 10√2 cm

Ratio of their diagonals = 12√2:10√2

=>12:10

=>6:5

Therefore,The ratio = 6:5

Answer:-

The rato of the diagonals of the two squares is 6:5

Check:-

Side of the first square = 12 cm

Area of the first square = 12^2 = 144 sq.cm

Side of the second square = 10 cm

Area of the second square = 10^2

=100 sq.cm

Their sum = 144+100 = 244 sq.cm

Perimeter of first square = 4×12 =48 cm

Perimeter of second square = 4×10 = 40 cm

Their difference = 48-40 = 8 cm

Verified the given relations

Used formulae:-

• Diagonal of a square = √2 ×side
• Area of the first square = Side ×Side
• Perimeter of a square = 4× length of the side
• (a-b)^2 = a^2 -2ab +b^2
• (a+b)^2 = a^2 +2ab+b^2