Question

Euclid showed geometrically the distributed law of multiplication. Let `bar(AB) and bar(BC)` be two straight lines, tracing a rectangle and let `bar(BC)`be cut at random at the points D & E. Use this fact to show the distributed law?

Answer 1

Please see below.

Explanation:

Let `BD=p`, `DE=q` and `EC=r`

then `BC=p+q+r`. Also let `AB=a` and hence `AB=DH=EJ=a`.

Therefore area of rectangle `ABCN=ABxxBC=axx(p+q+r)` as `BC=BD+DE+EC=p+q+r`

Also area of rectangles `ABDH`, `DEJH` and `ECNJ` are

`ABDH=ABxxBD=axxp`

`DEJH=DHxxDE=ABxxDE=axxq`

`ECNJ=EJxxEC=ABxxEC=axxr`

it is evident from the image that area of rectangle `ABCN` is sum of areas of rectangles `ABDH`, `DEJH` and `ECNJ`.

Hence, `axx(p+q+r)=axxp+axxq+axxr`

which is nothing but the distributive law.