6. Geometrical proof of identity - IV

 

Let us first, simplify the identity $(a + b)(a - b)= a^2–b^2$.

 

Multiply the expression, as shown below.

 

 

Now, we construct a figure to understand the concept.

 

 

Then we construct a rectangle using the above information.

 

In the given figure, \(AB = AD = a\).

So, the area of square \(ABCD = a^2\).

 

 

Also, \(SB = DP = b\). Then the area of the rectangle \(SBCT = ab\).

 

Similarly, the area of the rectangle \(DPRC = ab\). And the area of the square \(TQRC = b^2\).

 

Area of the rectangle \(DPQT = ab − b^2\).

 

Hence, \(\text{the area of the rectangle APQS = The area of square ABCD}\) \(\text{– area of rectangle STCB}\) \(\text{+ area of rectangle DPQT}\).

 

 

Therefore, $(a + b)(a - b)= a^2–b^2$.