Question

$\small \bf Question : 1$
Prove the following by paper cutting activity .
$\implies \boxed{ \tt(a+b)^2= a^2+b^2+2ab }$
$\quad$
$\small \bf Question : 2$
Write some of the algebraic identities. ​

Answer 1

$\small \bf Question : 1$

Prove the following by paper cutting activity .

$\implies \boxed{ \tt(a+b)^2= a^2+b^2+2ab }$

Answer:-

Objectives:-

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

Prerequisite Knowledge :-

• Area of a square = (side)².
• Area of a rectangle = l x b.

Materials Required :-

A sheet of white paper, three sheets of glazed paper (different colours), a pair of scissors, gluestick and a geometry box.

Procedure :-

• Take distinct values of a and b, say a = 4 units, b = 2 units

• Cut a square of side a (say 4 units) on a glazed paper (blue).

• Cut a square of side b (say 2 units) on glazed paper (pink).

• Now, cut two rectangles of length a (4 units) and breadth b (2 units) from third glazed paper (red).

• Draw a square PQRS of (a+ b) = (4 + 2), 6 units on white paper sheet as shown in fig. (i).

• Paste the squares I and II and two rectangles III and IV on the same white squared paper. Arrange all the pieces on the white square sheet in such a way that they form a square ABCD fig. (ii)

Observation :-

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

• (a+b)² = (4+2)² = 6 x 6 = 36 sq. units ……….(i)

• Area of two coloured squares I and II

• Area of Ist square = a² = 4² = 16 sq.units

• Area of IInd square = b² = 2² = 4 sq.units

• Area of two coloured rectangles III and IV = 2(a x b) = 2(4 x 2) = 16 sq. units

• Now, total area of four quadrilaterals (calculated)

• = a² + b² + 2(ab)
• = 16+4+16
• = 36 sq. units ……….(ii)

• Area of square ABCD = Total area of four quadrilaterals = 36 sq. units

Equating (i) and (ii):-

• Area of square PQRS = Area of square ABCD i.e., (a+b)2 = a2 + b2 + 2ab

Result :-

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

Learning Outcome:-

• The identity (a+b)2 = a2 + 2ab + b2 is verified by cutting and pasting of paper. This identity can be verified geometrically by taking other values of a and b.

There are three attachments for question 1.

$\small \bf Question : 2$

Write some of the algebraic identities.

*Answer:-

Identity I:

(a + b)² = a² + 2ab + b²

Identity II:

• (a – b)² = a² – 2ab + b²

Identity III:

• a² – b²= (a + b)(a – b)

Identity IV:

• (x + a)(x + b) = x² + (a + b) x + ab

Identity V:

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

Identity VI:

• (a + b)³ = a³ + b³ + 3ab (a + b)

Identity VII:

• (a – b)³ = a³ – b³ – 3ab (a – b)

Identity VIII:

• a³ + b³ + c³ – 3abc = (a + b + c)(a² + b² + c2 – ab – bc – ca)

Answered by

@shwetasingh1421977

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