Question

Verify it
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Answer 1

Answer:

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

Area of a square = (side)²

Area of a rectangle = lb

Materials:  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 paper

Cut a square of side b (say 2 units) on a paper

Cut two rectangles of length a (4 units) and breadth b (2 units) on paper  

(↓ Figure 1)

Draw a square PQRS of (a+ b) = (4 + 2), 6 units on white paper sheet as shown in fig. 1

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. 2

Observation: Area of the square PQRS on the white sheet paper

(a+b)² = (4+2)² = 6 x 6 = 36 units² --- (i)

Area of two coloured squares I and II

Area of 1st square = a² = 4² = 16 units²

Area of 2nd square = b² = 2² = 4 units²

Area of 2 coloured rectangles III and IV = 2(a x b) = 2(4 x 2) = 16 units²

Total area of 4 quadrilaterals = a² + b² + 2(ab)

                                                 = 16+4+16

                                                 = 36 units² --- (ii)

Area of square ABCD = Total area of four quadrilaterals = 36 units²

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.

You can do the others in the same way.

Answer 2

Explanation:

We have,

(a+b)²

It can be written as, (a+b)(a+b)

By using distributive property,

(a+b)(a+b)= a²+ab+ab+b²

Which is equal to a²+2ab+b²