Question

The length and breadth of a rectangle are 10 centimetres and 4 centimetres. It is made larger, increasing the length and breadth by the same amount.

Find algebraic expressions to compute the perimeters and areas of such enlarged rectangles in terms of the increase in the lengths of sides.

Answer 1

Length of the rectangle = 10 cm

Breadth of the rectangle = 4 cm

Let the length and the breadth increase by x cm.

∴ New length = (10 + x) cm

New breadth = (4 + x) cm

Perimeter of the enlarged rectangle = 2 (Length + Breadth)

= 2(10 + x + 4 + x) cm

= 2(2x + 14) cm

= (4x + 28) cm

Area of the enlarged rectangle = Length × Breadth

= (10 + x) × (4 + x) cm²

= 10(4 + x) + x(4 + x) cm²

= (40 + 10x + 4x + x²) cm²

= {40 + (10 + 4)x + x²} cm²

= (40 + 14x + x²) cm²

Thus, the algebraic expressions for the area and perimeter of the enlarged rectangle are 40 + 14x + x² and 4x + 28 respectively.

Now, let the length and breadth decrease by x cm.

∴ New length = (10 − x) cm

New breadth = (4 − x) cm

Perimeter of the new rectangle = 2 (Length + Breadth)

= 2(10 − x + 4 − x) cm

= 2(14 − 2x) cm

= (28 − 4x) cm

Area of the new rectangle = Length × Breadth

= (10 − x) × (4 − x) cm²

= 10(4 − x) − x(4 − x) cm²

= (40 − 10x − 4x + x²) cm²

= {40 − (10 + 4)x + x²} cm²

= (40 − 14x + x²) cm²

Thus, the algebraic expressions for the area and perimeter of the new rectangle are 40 − 14x + x² and 28 − 4x respectively which are not equal to the algebraic expressions for the area and perimeter of the enlarged rectangle.

Hence, we cannot use the same algebraic expressions in both the cases.