Verify $(x y + z)^{2} = (x y)^{2} + z^{2} + 2 x y z$ geometrically.

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Solution

Step 1: Draw a line with a point which divides $x y, z$
Step 2: Total distance of this line $= x y + z$
Step 3: Now we have to find out the square of $x y + z$ i.e., Area of big square, $A B C D =$ $(x y + z)^{2}$
Step 4: From the diagram, inside square red and yellow square, be written as $x y^{2}, z^{2}$
Step 5: The remaining corner side will be calculated as rectangular side $=$ length $\times$ breadth $=$ $x y \times z$
Therefore, Area of the big square, $A B C D =$ Sum of the inside square $+$ $2$ times the corner rectangular side.
$(x y + z)^{2} = (x y)^{2} + z^{2} + 2 x y z$
Hence, geometrically we proved the identity $(x y + z)^{2} = (x y)^{2} + z^{2} + 2 x y z$

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