Math Doubts

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

animation for square of a+b
  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.

a+b whole square
  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

animation for 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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