# 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?

Sep 13, 2016

#### Explanation:

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

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

Therefore area of rectangle $A B C N = A B \times B C = a \times \left(p + q + r\right)$ as $B C = B D + D E + E C = p + q + r$

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

$A B D H = A B \times B D = a \times p$

$D E J H = D H \times D E = A B \times D E = a \times q$

$E C N J = E J \times E C = A B \times E C = a \times r$

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

Hence, $a \times \left(p + q + r\right) = a \times p + a \times q + a \times r$

which is nothing but the distributive law.