Verify the identity: $a^{2} - b^{2} = (a + b) (a - b)$ geometrically.

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Solution

Step 1: Draw a square and cut into $3$ parts.
Step 2: There are $1$ hided square green and $2$ rectangles (pink, blue)
Step 3: Area of the full square $=$ $a^{2} - b^{2}$
Step 4: Now we have to find the area of rectangle as shown in the figure.
Step 5: Consider the area of pink rectangle $=$ length $\times$ breadth $=$ $a (a - b)$
Step 6: Area of blue rectangle $=$ $b (a - b)$
Step 7: Area of full square $=$ area of pink rectangle $+$ area of blue rectangle.
i.e., $a^{2} - b^{2} = a (a - b) + b (a - b)$
$a^{2} - b^{2} = (a + b) (a - b)$
Hence, geometrically we proved the identity $a^{2} - b^{2} = (a + b) (a - b)$ .

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