Question

The area of a square is twice that of a rectangle. The length of the rectangle is greater than the length of each side of the square by 5 cm and the breadth is less by 12 cm. Find the perimeter of the rectangle.​

Answer 1

Given :

• Area of square = Twice of area of Rectangle.

• Length of the Rectangle = Side of the square + 5 cm.

• Breadth of the Rectangle = Side of the square - 12 cm.

To find :

The Perimeter of the Rectangle.

Solution :

Let the side of the square be x cm.

Hence ,

• Length of the Rectangle will be (x + 5) cm.

• Breadth of Rectangle will be (x - 12) cm.

Now ,

According to the Question ,

Two times the area of square is equal to the Area of Rectangle. i.e,

$\boxed{\bf{A_{(Square)} = 2A_{(Rectangle)}}}$⠀⠀⠀⠀⠀⠀Eq.(i)

So , first we have to find the area of the Rectangle and area of square in terms of x and then by substituting the area in Eq.(i) , we can find the required value.

Area of the square :

We know the formula for area of a square i.e,

$\underline{\bf{A = a^{2}}}$

Where :

• A = Area of the Square

• a = Side of the square

By using this formula and substituting the value of side (in terms of x) , we get :

$:\implies \bf{A = a^{2}}$

$:\implies \bf{A = x^{2}}$

$\boxed{\therefore \bf{A = x^{2}}}$

Hence the area of the square in terms of x is x².

Area of the Rectangle :

We know the formula for area of a Rectangle i.e,

$\underline{\bf{A = L \times B}}$

Where :

• A = Area of the Rectangle

• L = Length of the Rectangle

• B = Breadth of the Rectangle

By using this formula and substituting the value of Length and Breadth (in terms of x) , we get :

$:\implies \bf{A = L \times B}$

$:\implies \bf{A = (x + 5) \times (x - 12)}$

$:\implies \bf{A = x(x - 12) + 5(x - 12)}$

$:\implies \bf{A = x^{2} - 12x + 5x - 60}$

$:\implies \bf{A = x^{2} - 7x - 60}$

$\boxed{\therefore \bf{A = x^{2} - 7x - 60}}$

Hence the area of the Rectangle in terms of x is x² - 7x - 60.

Now by substituting the values of area of Square and Rectangle (in terms of x) and by solving it , we get :

$:\implies \bf{A_{(Square)} = 2A_{(Rectangle)}}$

$:\implies \bf{x^{2} = 2(x^{2} - 7x - 60)}$

$:\implies \bf{x^{2} = 2x^{2} - 14x - 120}$

$:\implies \bf{0 = -x^{2} + 2x^{2} - 14x - 120}$

$:\implies \bf{0 = x^{2} - 14x - 120}$

Since, the Equation formed is a Quadratic equation , we can solve it by using the formula for Quadratic equation.

Formula for Quadratic equation :-

$\boxed{\bf{\alpha = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}}}$

Here ,

• a = 1
• b = (-14)
• c = (-120)

Using the formula for Quadratic equation and substituting the values in it, we get :

$:\implies \bf{\alpha = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}}$

$:\implies \bf{x = \dfrac{-(-14) \pm \sqrt{(-14)^{2} - 4 \times 1 \times (-120)}}{2 \times 1}}$

$:\implies \bf{x = \dfrac{14 \pm \sqrt{196 - (-480)}}{2}}$

$:\implies \bf{x = \dfrac{14 \pm \sqrt{196 - (-480)}}{2}}$

$:\implies \bf{x = \dfrac{14 \pm \sqrt{196 + 480}}{2}}$

$:\implies \bf{x = \dfrac{14 \pm \sqrt{676}}{2}}$

$:\implies \bf{x = \dfrac{14 \pm 26}{2}}$

$:\implies \bf{x = \dfrac{14 + 26}{2}\:;\;x = \dfrac{14 - 26}{2}}$

$:\implies \bf{x = \dfrac{40}{2}\:;\;x = \dfrac{(-12)}{2}}$

$:\implies \bf{x = 20\:;\;x = (-6)}$

$\boxed{\therefore \bf{x = 20;(-6)}}$

Hence the value of x is 20 and (-6).

Since , we have taken the side of the square is x , the side of the square is 20 (Not (-6) since the side of a square can't be negative).

Now let us find the length and breadth of the rectangle.

• Length of Rectangle :

$:\implies \bf{l = (x + 5)}$

Now putting the value x in the above equation, we get :

$:\implies \bf{l = (20 + 5)}$

$:\implies \bf{l = 25}$

$\boxed{\therefore \bf{l = 25\:cm}}$

Hence the length of the Rectangle is 25 cm.

• Breadth of Rectangle :

$:\implies \bf{b = (x - 12)}$

Now putting the value x in the above equation, we get :

$:\implies \bf{b = (20 - 12)}$

$:\implies \bf{b = 8}$

$\boxed{\therefore \bf{b = 8\:cm}}$

Hence the breadth of the Rectangle is 8 cm.

Perimeter of the Rectangle :

We know the formula for Perimeter of a Rectangle i.e,

$\underline{\bf{P = 2(L + B)}}$

Where :

• P = Perimeter
• L = Length
• B = Breadth

Using the formula for perimeter of a Rectangle and substituting the values in it, we get :

$:\implies \bf{P = 2(L + B)}$

$:\implies \bf{P = 2(25 + 8)}$

$:\implies \bf{P = 2 \times 33}$

$:\implies \bf{P = 66}$

$\boxed{\therefore \bf{P = 66\:cm}}$

Hence the perimeter of the Rectangle is 66 cm.