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?

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Answer 1 · 2016-09-13T08:48:57

Please see below.

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.

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