Class 09 To verify the algebraic identity :(a+b+c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

 

Activity 6 

OBJECTIVE	MATERIAL REQUIRED
To verify the algebraic identity :	Hardboard, adhesive, coloured
(a+b+c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca	papers, white paper.
METHOD OF CONSTRUCTION

 1.   Take a hardboard of a convenient size and paste a white paper on it.

 2.   Cut out a square of side a units from a coloured paper [see Fig. 1].

 3.   Cut out a square of side b units from a coloured paper [see Fig. 2].

 4.   Cut out a square of side c units from a coloured paper [see Fig. 3].

 5.   Cut out two rectangles of dimensions a× b, two rectangles of dimensions b × c and two rectangles of dimensions c × a square units from a coloured paper [see Fig. 4].

6.   Arrange the squares and rectangles on the hardboard as shown in Fig. 5.

DEMONSTRATION

From the arrangement of squares and

 rectangles in Fig. 5, a square ABCD is

 obtained whose side is (a+b+c) units.

 Area of square ABCD = (a+b+c)² . 

Therefore, (a+b+c)2  = sum of all the
squares and rectangles shown in Fig. 1 to
Fig. 4.	Fig. 5

 =   a² + ab + ac + ab + b² + bc + ac + bc + c²

 =  a² + b² + c² + 2ab + 2bc + 2ca

 Here, area is in square units.

 OBSERVATION

 On actual measurement:

a = ..............	,	b = ..............	, c = ..............	,
So, a2 = ..............	,	b2 = ..............	, c2= ..............	, ab=	..............	,
bc= ..............	,	ca = ..............	,2ab = ..............	,	2bc =	..............,
2ca= ..............	,	a+b+c = ..............	,	(a+b+c)2 =	..............	,

 Therefore, (a+b+c)² = a² + b² +c² +2ab + 2bc + 2ca

 APPLICATION

 The identity may be used for

 1.   simiplification/factorisation of algebraic expressions

 calculating the square of a number expressed as a sum of three convenient numbers.