Question 10.8.1

In ancient times, many algebraic facts were found geometrically, with areas. Use pictures to show the following identities:

1. $(a+b)^2 = a^2 + 2ab + b^2$ Show hint

   A square of side length $a+b$ has an area of $(a+b)^2$.

   Show answer

   Interpreted geometrically, the lhs is the area of a square with side-length $a+b$. And the rhs is a combination of the areas of squares of side-length $a$ and $b$ and two rectangles with one side $a$ and the other side $b$.

   

2. $a^2 - b^2 = (a-b)(a+b)$ Show answer

   The lhs is the area of a square of side-length $a$, with a square of side-length $b$ cut out. The rectangle down the bottom can be moved over to the right, giving a rectangle of width $a+b$ and height $a-b$, which has an area $(a+b)(a-b)$: