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.
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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 is18cm
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:–