Question

A rectangle has an area of 12 square centimeters and a perimeter of 16 centimeters. Which of the following could be its dimensions?

Answer 1

`6 × 2`

Explanation:

For the rectangle

• Length`= ℓ`
• Breadth`= b`

Area is `12\ "cm"^2`

`ℓb = 12`

Perimeter is `16\ "cm"`

`2(ℓ + b)= 16`

`ℓ + b = 8`

Substitute `b = 12/ℓ` from first equation

`ℓ + 12/ℓ = 8`

`ℓ^2 + 12 = 8 ℓ`

`ℓ^2 - 8 ℓ + 12 = 0`

Use quadratic formula (`x = (-b +- sqrt(b^2 - 4ac))/(2a)`) to find `ℓ`

`ℓ = (-(-8) +- sqrt((-8)^2 - (4 × 1 × 12)))/(2 × 1)`

`ℓ = (8 +- sqrt(16))/2`

`ℓ = (8 +- 4)/2`

`ℓ_1 = (8 + 4)/2 = 6`

`ℓ_2 = (8 - 4)/2 = 2`

If `ℓ_1` is taken as length then `ℓ_2` is the breadth of the rectangle.