# Proof of ${(a-b)}^2$ formula in Geometrical Method

## Formula

${(a-b)}^2 \,=\, a^2+b^2-2ab$

### Proof

The $a$ minus $b$ whole square formula can be derived by the geometrical approach on the basis of areas of squares and rectangles.

01

#### Basic steps for construction

Look at the animation to split the whole square as a small square and two rectangles.

1. Consider a square and length of the each side is $a$. Therefore, the area of the square is $a^2$.
2. Divide the square as two rectangles by a perpendicular line to opposite sides of the square. If length of one side of one rectangle is $b$, then the length of one side of second rectangle is $a-b$.
3. Consider the rectangle whose sides are $a-b$ and $b$. Split this rectangle by a perpendicular line but it should divide the rectangle as a small square and a small rectangle. Therefore, the length of side of the small square is equal to $a-b$. The lengths of the small rectangle are $a-b$ and $b$ geometrically.
02

#### Mathematical Analysis

Thus, the square whose area is $a^2$, is divided as one small square and two rectangles. Now, evaluate the areas of all three figures geometrically.

1. The length of each side of the square is $a-b$. So, the area of the small square is $(a-b) \times (a-b) = (a-b)^2$.
2. The lengths of sides of one rectangle are $a$ and $b$ geometrically. Hence, the area of the rectangle is $a \times b = ab$
3. The lengths of sides of second rectangle are $b$ and $a-b$. Therefore, the area of the second rectangle is $b \times (a-b) = b(a-b)$
03

#### Expressing Geometrical Analysis in Mathematics

Observe the geometrical analysis carefully in this picture to understand how to express the expansion of square of the binomial $a-b$ in mathematical form.

The area of small square ${(a-b)}^2$ can be obtained by subtracting the sum of the areas of rectangles from the area of the actual square.

${(a-b)}^2 = a^2 -[ab+b(a-b)]$

Now, simply the equation to obtain expansion of the $a-b$ whole square geometrically in mathematical form.

$\implies {(a-b)}^2 = a^2 -(ab+ba-b^2)$

$\implies {(a-b)}^2 = a^2 -(2ab-b^2)$

$\implies {(a-b)}^2 = a^2 -2ab + b^2$

$\,\,\, \therefore \,\,\,\,\,\, {(a-b)}^2 = a^2 + b^2 -2ab$

Thus, the $a-b$ whole square algebraic identity is verified in geometrical approach.

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