Geometrically verify the identity: $(a + 2 b) (a - 2 b) = a^{2} - (2 b)^{2}$

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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 $=$ $a^{2} - (2 b)^{2}$
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 $=$ $a (a - 2 b)$
Step 6: Area of blue rectangle $=$ $2 b (a - 2 b)$
Step 7: Area of full square $=$ area of pink rectangle $+$ area of blue rectangle.
i.e., $a^{2} - (b)^{2} = a (a - 2 b) + 2 b (a - 2 b)$
$a^{2} - (2 b)^{2} = (a + 2 b) (a - 2 b)$
Hence, geometrically we proved the identity $a^{2} - (2 b)^{2} = (a + 2 b) (a - 2 b)$ .

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