Question

the length of a rectangle is twice its breadth. If the ratio of the perimeter of the rectangle and its area is 1:2 find the dimension of the rectangle.

Answer 1

$\bf{\huge{\underline{\boxed{\rm{\blue{Answer:}}}}}}$

Given:-

• the length of a rectangle is twice its breadth
• the ratio of the perimeter of the rectangle and its area is 1:2

To find :-

• The dimensions of the rectangle i.e length and breadth of the rectangle.

Solution :-

As per the first given statement,

• the length of a rectangle is twice its breadth,

Let the breadth of the rectangle= x cm

Then, the length of the rectangle = 2x cm

Perimeter and area are in the ratio 1:2,

Perimeter :Area = 1:2

$\bf\large\frac{Perimeter}{Area}$ = $\bf\large\frac{1}{2}$

Formulaes for perimeter and area of rectangle,

• Perimeter = 2 (l+b)
• Area = L × B

$\bf\large\frac{2 (l+b)}{l × b}$ = $\bf\large\frac{1}{2}$

Plug the values,

$\bf\large\frac{2 (2x +x)}{2x × x}$ = $\bf\large\frac{1}{2}$

$\bf\large\frac{2 (3x) }{2x^2}$ = $\bf\large\frac{1}{2}$

$\bf\large\frac{6x}{2x^2}$ = $\bf\large\frac{1}{2}$

Cross multiplying the terms,

2 (6x) = 1 × 2x²

12x = 2x²

•°• 2x² = 12x

•°• 2x² - 12x = 0

•°• 2x (x - 6) = 0

2x = 0 OR x - 6 = 0

x = $\bf\large\frac{0}{2}$ OR x = 6

x = 0 OR x = 6

Since, x = breadth = 6cm

Substitute this value of x in length,

length = 2x = 2 × 6 = 12 cm

$\bf{\large{\underline{\boxed{\rm{\pink{Lengt\:of\:the\:rectangle\:=\:12\:cm}}}}}}$

$\bf{\large{\underline{\boxed{\rm{\green{Breadth\:of\:the\:rectangle\:=\:x \:=\:6cm}}}}}}$

$\bf{\huge{\underline{\boxed{\rm{\blue{Verification:}}}}}}$

As given in the question,

For first case :-

• the length of a rectangle is twice its breadth

Length = 2x = 2 × 6 = 12cm

Breadth = x = 6cm

12 = twice of 6 i.e, length of the rectangle is twice the breadth.

Hence the first condition is satisfied.

For second case :-

• the ratio of the perimeter of the rectangle and its area is 1:2

Perimeter of the rectangle = 2(l+b)

Perimeter = 2 (12 + 6)

Perimeter = 2 (18)

$\bf{\large{\underline{\boxed{\rm{\pink{<strong><u> </u></strong><strong><u>Perimeter </u></strong><strong><u>=</u></strong><strong><u> </u></strong><strong><u>36cm</u></strong>}}}}}}$

Area of the rectangle = l × b

Area = 12 × 6

$\bf{\large{\underline{\boxed{\rm{\green{<u> </u><strong><u>Area </u></strong><strong><u>=</u></strong><strong><u> </u></strong><strong><u>72cm</u></strong><strong><u>^</u></strong><strong><u>2</u></strong>}}}}}}$

Ratio of perimeter and area = 1:2

Perimeter : Area = 1:2

$\bf\large\frac{36}{72}$ = $\bf\large\frac{1}{2}$

Dividing LHS by 36,

$\bf\large\frac{1}{2}$ = $\bf\large\frac{1}{2}$

LHS = RHS.

Hence verified!

Answer 2

Answer:

Length = 12 units

Breadth = 6units

Step-by-step explanation:

Let the breadth be x

then length will be 2x

According to the question,

Area : perimeter = 2 : 1

=> (l × b) : 2(l + b) = 2 : 1

=> (2x × x) : 2(2x + x) = 2 : 1

=> 2x² : 6x = 2 : 1

=> 2x²/6x = 2/1

=> 1/3x = 2

=> x = 2*3

=> x = 6unit

Hence,

breadth = x = 6units

length is 2x = 12units