Question

Draw and explain the identity geometrically: $(3x-2y)(3x+2y)=(3x{)}^2-(2y{)}^2$

Answer 1

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 $=$ $(3x{)}^2-(2y{)}^2$
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 $=$ $x3(3x-2y)$
Step 6: Area of blue rectangle $=$ $2y(3x-2y)$
Step 7: Area of full square $=$ area of pink rectangle $+$ area of blue rectangle.
i.e., $(3x{)}^2-(2y{)}^2=3x(3x-2y)+2y(3x-2y)$
$(3x{)}^2-(2y{)}^2=(3x+2y)(3x-2y)$
Hence, geometrically we proved the identity $(3x{)}^2-(2y{)}^2=(3x+2y)(3x-2y)$.