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.

Class 09 To verify the algebraic identity :a2 – b2 = (a + b)(a – b)

 Activity 5

 OBJECTIVE                                                                    

To verify the algebraic identity :a² – b² = (a + b)(a – b)

METHOD OF CONSTRUCTION

 MATERIAL REQUIRED

Drawing sheets, cardboard, coloured papers, scissors, sketch pen, ruler, transparent sheet and adhesive.

1.   Take a cardboard of a convenient size and paste a coloured paper on it.

2.    Cut out one square ABCD of side a units from a drawing sheet [see Fig. 1].

3.3.Cut out one square AEFG of side b units (b < a) from another drawing sheet [see Fig. 2].

4.   Arrange these squares as shown in Fig. 3.

 5.   Join F to C using sketch pen. Cut out trapeziums congruent to EBCF and GFCD using a transparent sheet and name them as EBCF and GFCD, respectively [see Fig. 4 and Fig. 5].

6. Arrange these trapeziums as shown in

Fig. 6.

 DEMONSTRATION

 Area of square ABCD = a²

 Area of square AEFG = b²

 In Fig. 3,

 Area of square ABCD – Area of square

 AEFG

 = Area of trapezium EBCF + Area of

 trapezium GFCD

 =  Area of rectangle EBGD [Fig. 6].

 =  ED×DG

 Thus, a² – b² = (a+b) (a–b)Fig. 6

Here, area is in square units.

OBSERVATION

On actual measurement:

 a = ..............,         b = .............., (a+b) = ..............,

 So, a2 = ..............,       b2 = .............., (a–b) = ..............,

 a2–b2 = .............., (a+b) (a–b) = ..............,

 Therefore, a²–b² = (a+b) (a–b)

 APPLICATION

 The identity may be used for

 1.   difference of two squares

 2.   some products involving two numbers

 3.   simplification and factorisation of algebraic expressions.

Class 09 To verify the algebraic identity :(a – b)2 = a2 – 2ab + b2

 

Activity 4 

OBJECTIVE                                                                    

To verify the algebraic identity :(a – b)² = a² – 2ab + b²

 MATERIAL REQUIRED

Drawing sheets, cardboard, coloured papers, scissors, ruler and adhesive.

METHOD OF CONSTRUCTION

1.    Cut out a square ABCD of side a units from a drawing sheet/cardboard [see Fig. 1].

 2.   Cut out a square EBHI of side b units (b < a) from a drawing sheet/cardboard [see Fig. 2].

 3.   Cut out a rectangle GDCJ of length a units and breadth b units from a drawing sheet/cardboard [see Fig. 3].

 4.   Cut out a rectangle IFJH of length a units and breadth b units from a drawing sheet/cardboard [see Fig. 4].

5. Arrange these cut outs as shown in Fig. 5.

 DEMONSTRATION

 According to figure 1, 2, 3, and 4, Area of square ABCD = a², Area of square EBHI = b²

 Area of rectangle GDCJ = ab, Area of rectangle IFJH = ab

 From Fig. 5, area of square AGFE = AG × GF = (a – b) (a – b) = (a – b)²

 Now, area of square AGFE = Area of square ABCD + Area of square EBHI 

– Area of rectangle IFJH – Area of rectangle		Fig. 5
GDCJ
= a2	+ b2 – ab – ab
= a2	– 2ab + b2

 Here, area is in square units.

 OBSERVATION

 On actual measurement: 

a = ..............	,	b = ..............		, (a – b) = ..............	,
So, a2 = ..............	,	b2 =	..............	, (a – b)2 = ..............	,
ab = ..............	,	2ab	= ..............

 Therefore, (a – b)² = a² – 2ab + b²

 APPLICATION

 The identity may be used for

 1.   calculating the square of a number expressed as a difference of two convenient numbers.

 2.   simplifying/factorisation of some algebraic expressions.

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

 Activity 3 

OBJECTIVE               

          

                                            

To verify the algebraic identity : (a + b)² = a² + 2ab + b²

 MATERIAL REQUIRED

 Drawing sheet, cardboard, cello-tape, coloured papers, cutter and ruler.

 METHOD OF CONSTRUCTION

 1.   Cut out a square of side length a units from a drawing sheet/cardboard and name it as square ABCD [see Fig. 1].

 2.   Cut out another square of length b units from a drawing sheet/cardboard and name it as square CHGF [see Fig. 2].

 Fig. 1                                                                             Fig. 2

 3.   Cut out a rectangle of length a units and breadth b units from a drawing sheet/cardbaord and name it as a rectangle DCFE [see Fig. 3].

 Cut out another rectangle of length b units and breadth a units from a drawing sheet/cardboard and name it as a rectangle BIHC [see Fig. 4].

5.   Total area of these four cut-out figures

 =  Area of square ABCD + Area of square CHGF + Area of rectangle DCFE

 +  Area of rectangle BIHC

 =  a² + b² + ab + ba = a² + b² + 2ab.

 Join the four quadrilaterals using cello-tape as shown in Fig. 5.

Clearly, AIGE is a square of side (a + b). Therefore, its area is (a + b)². The combined area of the constituent units = a² + b² + ab + ab = a² + b² + 2ab.

 Hence, the algebraic identity (a + b)² = a² + 2ab + b² Here, area is in square units.

 OBSERVATION

 On actual measurement:

 a = ..............,     b = .............. (a+b) = ..............,

 So, a² = ..............        b² = .............., ab = ..............

 (a+b)² = ..............,             2ab = ..............

 Therefore, (a+b)² = a² + 2ab + b² .

 The identity may be verified by taking different values of a and b.

 APPLICATION

 

The identity may be used for

 

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

 

2.   simplifications/factorisation of some algebraic expressions.