Question

verify the identity (a+b+c)^2-a^2+b^2+c^2+2ab+2bc+2ca

Answer 1

Answer:

Step 1: Draw a square and cut into 9 parts.

Step 2: There are 3 squares (red, yellow, green) and 6 rectangles (2 pink, 2 purple, 2 blue)

Step 3: Area of the full square = (a+b+c)

2

Step 4: Now we have to find the area of 3 inside square(red, yellow, green) = a

2

+b

2

+c

2

Step 5: Consider the area of 2 pink rectangle = length × breadth =b.a+b.a=2ab

Step 6: Area of 2 purple rectangle =a.c+a.c=2ac and Area of 2 blue rectangle =b.c+b.c=2bc

Step 7: Area of full square = area of 3 inside square + area of 2 pink rectangle + area of 2 purple rectangle + area of 2 blue rectangle.

i.e., (a+b+c)

2

=a

2

+b

2

+c

2

+2ab+2bc+2ac

Hence, geometrically we proved the identity (a+b+c)

2

=a

2

+b

2

+c

2

+2ab+2bc+2ac