Question

(a - b)2 = a2 + b2 - 2ab verify this identity by cutting and pasting paper

Answer 1

Answer:

$(x - y)2 = (x + y)2 - 4xy \\ = x {2} + y2 + 2xy - 4xy \\ x2 + y2 - 2xy$

Answer 2

Objective =

To verify the identity ( a + b ) 2 = a2 + b2 - 2ab by paper cutting .

Prerequisite knowledge =

1 . Area of square = ( side )2 .

2 . Area of rectangle = l × b .

Material required =

A sheet of white paper , three sheet of glazed paper ( different colour ) , a pair of scissor glue stick and geometry box .

Producer =

Take distinct value of a and b , say a 4 unit and b 2 unit .

1 . Cut a square of side a ( say 4 unit ) glazed paper.

2 . Cut a square of side b ( says 2 unit ) on glazed paper.

3 . Now cut two rectangles of length a ( 4 unit ) breadth b ( 2 unit ) for third glazed paper.

4 . Draw a square of PQRS of ( a+ b ) =(4 + 2) , 6 unit on white paper .

5 . Paste the square 1 and 2 and 2 rectangles 3 and 4 on the same white square paper arrange all the pieces on a white square sheet.

Observation =

1 . Area of the square PQRS on the white sheet paper .

( a + b ) 2 = (4 + 2 ) = 6×6 = 36 sq .unit .

2 . Area of two coloured square 1 and 2 area of 1st square = a2 = 42 =16 sq. unit .

Area of 2 nd square = b2 =22 = 8 sq. unit .

3 . Area of two coloured rectangles 3 and 4 = 2 ( a × b) = 2 (4×2) =16 sq. unit .

Now , total area of four quadrilateral ( calculated )

=a2 + b2 + 2(ab)

=16+ 4 + 16

=36 sq unit.

Area of square ABCD = Total area of four quadrilateral = 36 sq. unit .

Equating = (1) and (2) .

Area of square PQRS = Area of square ABCD

i. e., ( a + b ) 2 = a2 + b2 = 2ab.

Result =

Algebraic identity ( a + b ) 2 = a2 + 2 ab + b2 is verified .