Question

The perimeter of a rectangle is 84 cm.If the area of the square formed on the diagonal of the rectangle as its side is 1(1/12)% more than the area of the rectangle, find the shorter side of the rectangle.​

Answer 1

Step-by-step explanation:

Given :-

The perimeter of a rectangle is 84 cm.If the area of the square formed on the diagonal of the rectangle as its side is 1(1/12)% more than the area of the rectangle.

To find:-

Find the shorter side of the rectangle.?

Solution:-

Let the sides of the rectangle be l cm and b cm

Given that

The perimeter of the rectangle =84 cm

=> 2(l+b) = 84

=>l+b = 84/2

=> l + b = 42 --------(1)

l = 42-b -----------(2)

We know that

Length of the diagonal of the given rectangle

= √(l^2+b^2) cm -------(3)

Area of the rectangle = lb sq.cm

Given that

The area of the square formed on the diagonal of the rectangle as its side

Area of a square = side^2 sq.units

(Since side = Diagonal )

Area of the square formed on the diagonal of the rectangle

=[√l^2+b^2]^2

= l^2+b^2 sq.cm ----------(4)

Given that

The area of the square formed on the diagonal of the rectangle as its side is 1(1/12)% more than the area of the rectangle

=>l^2+b^2 = lb+1(1/12) of lb

=>l^2+b^2 = lb+(13/12) of lb

=>l^2+b^2= lb[1+(13/12) ]

=>l^2+b^2 = lb[ (12+13)/12]

=>l^2+b^2=25/12 lb

=> 12(l^2+b^2) = 25lb

=> 12[(l+b)^2-2lb] = 25 lb

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

=> 12(l+b)^2-24lb = 25 lb

=> 12(l+b)^2 = 25lb+24lb

=> 12(l+b)^2 = 49lb

=> 12(42)^2 = 49lb

=> 12(1764) = 49 lb

=> 21168 = 49 lb

=> lb = 21168/49

=> lb = 432 -----------(5)

=> b(42-b) = 432

=> 42b -b^2 =432

=> -b^2+42b-432 = 0

=> b^2-42b+432 = 0

=>b^2-18b-24b+432=0

=> b(b-18)-24(b-18) = 0

=> (b-18)(b-24) = 0

=> b-18=0 or b-24 = 0

=> b = 18 or b= 24

The longest side = 24 cm

Shorter side = 18 cm

Answer:-

The shorter side of the rectangle for the given problem is 18 cm

Used formulae:-

• Area of a rectangle = lb sq.units

• Diagonal of a rectangle =√(l^2+b^2) units

• Area of a square = a^2 sq.units

• a = Side of the square

• l= length of a rectangle

• b= breadth of a rectangle

• Perimeter of a rectangle = 2(l+b) units

• (a+b)^2=a^2+2ab+b^2