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

## Formula

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

### Proof

The a plus b whole square formula is an algebraic identity but it can be derived in geometrical approach on the basis of a concept of areas of square and rectangle.

#### Calculating Area of a Square

1. Take a square. Divide the square vertically into two parts by drawing a straight line. The lengths of them are $a$ and $b$ respectively.
2. Divide the square horizontally into two parts by drawing a straight line but the lengths of them should also be $a$ and $b$ respectively.
3. The length of whole square is $a+b$ and width of this square is also $a+b$. Therefore, the area of the square is $(a+b) \times (a+b) \,=\, {(a+b)}^2$ geometrically.

#### Calculating Areas of Internal Squares and Rectangles

The geometrical process split a square as two small different squares and two small same rectangles. Now, calculate the areas of all four geometrical figures in geometrical system.

1. The length of each side of first square is $a$. So, the area of this square is $a^2$.
2. The length and width of first rectangle are $b$ and $a$ respectively. So, the area of this rectangle is $ba$.
3. The length and width of second rectangle are $a$ and $b$ respectively. Therefore, the area of this rectangle is $ab$.
4. The length of each side of the second square is $b$. Hence, the area of this square is $b^2$.

Now, add areas of all four geometrical shapes.

$a^2+ba+ab+b^2$

Mathematically, the product of $a$ and $b$ is equal to the product of $b$ and $a$. Therefore, the term $ab$ can be written as $ba$ and vice-versa.

$\implies$ $a^2+ab+ab+b^2$

$\implies$ $a^2+2ab+b^2$

$\implies$ $a^2+b^2+2ab$

#### Obtaining the a+b whole square formula

The area of a square in this example is considered as ${(a+b)}^2$.

The same square is split as two small distinct squares and two small same rectangles. The sum of areas of two small squares and two small rectangles is $a^2+b^2+2ab$.

Actually, a square is divided as two small different squares and two small same rectangles. Hence, the area of a square should be equal to the sum of the areas of two small squares and two small rectangles.

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

Geometrically, it is proved that square of $a+b$ can be expanded as $a$ squared plus $b$ squared plus $2ab$.

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