Math Doubts

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.

a minus b whole square formula
  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.

a minus b whole square identity
  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.

(a-b)^2 formula

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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