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.