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

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Euclid showed geometrically the distributed law of multiplication. Let $¯ ¯¯¯¯ ¯ A B and ¯ ¯¯¯¯ ¯ B C$ $bar(AB) and bar(BC)$ be two straight lines, tracing a rectangle and let $¯ ¯¯¯¯ ¯ B C$ $bar(BC)$ 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.

Explanation:

Let $B D = p$ $BD=p$ , $D E = q$ $DE=q$ and $E C = r$ $EC=r$

then $B C = p + q + r$ $BC=p+q+r$ . Also let $A B = a$ $AB=a$ and hence $A B = D H = E J = a$ $AB=DH=EJ=a$ .

Therefore area of rectangle $A B C N = A B \times B C = a \times (p + q + r)$ $ABCN=ABxxBC=axx(p+q+r)$ as $B C = B D + D E + E C = p + q + r$ $BC=BD+DE+EC=p+q+r$

Also area of rectangles $A B D H$ $ABDH$ , $D E J H$ $DEJH$ and $E C N J$ $ECNJ$ are

$A B D H = A B \times B D = a \times p$ $ABDH=ABxxBD=axxp$

$D E J H = D H \times D E = A B \times D E = a \times q$ $DEJH=DHxxDE=ABxxDE=axxq$

$E C N J = E J \times E C = A B \times E C = a \times r$ $ECNJ=EJxxEC=ABxxEC=axxr$

it is evident from the image that area of rectangle $A B C N$ $ABCN$ is sum of areas of rectangles $A B D H$ $ABDH$ , $D E J H$ $DEJH$ and $E C N J$ $ECNJ$ .

Hence, $a \times (p + q + r) = a \times p + a \times q + a \times r$ $axx(p+q+r)=axxp+axxq+axxr$

which is nothing but the distributive law.

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

1 Answer

Explanation:

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