Question

The perimeter of a square and rectangle is the same. The width of the rectangle is 6cm and it's area is 16cmsquare less than the area of the square. Find the area of the square

Answer 1

Area of square = 100 square cm

Explanation:

Let the sides of a square be = a

Perimeter of a square = 4a

Let area of square =
`a^2`

Let the Length of rectangle be =
`l`

Given: width of the rectangle = 6 cm

Area of rectangle = length x breadth

Perimeter of rectangle and square is equal.

That is,

`2(length + width) = 4a\\\\2(l + 6) = 4a\\\\l + 6 = 2a\\\\l = 2a - 6`

Therefore ,

Area of rectangle

`= Length * width \\\\= (2a - 6) * 6\\\\=6(2a - 6)`

Given area of rectangle is 16 less than area of square.

That is ,

`( 6(2a- 6) ) = a^2 - 16\\\\12a - 36 = a^2 - 16\\\\a^2 - 12a +20= 0\\\\a^2 - 2a -10a + 20 = 0\\\\a(a - 2) - 10(a - 2) = 0\\\\(a -10) ( a-2) = 0\\\\a = 10 , \ a = 2`

Check which value of 'a ' satisfies the equation:

`\underline {when \ a = 2 }\\\\Length\ of \ rectangle \ l = 2a - 6 = 2 ( 2 ) - 6 = 4 - 6 = - 2. \\\\Length \ cannot \ be \ negative \ number. \\\\ \underline{ when \ a = 10 }\\\\Length \ of \ rectangle \ , l = 2a - 6 = 2 (10) - 6 = 20 - 6 = 14\\\\satisfies \ the \ conditions. \\\\Therefore , a = 10`

That is , side of the sqaure = 10

Therefore , area of the square = 10 x 10 = 100 square cm.