prove that the opposite sides of parallelogram are equal

Question

All about parallelogram is here

Answer 1

This can be proved graphically.

In the x-y plane, plot y=a, where a≠0  (parallel to the x axis), y=bx (b≠0) and y=bx+c (c≠0).

There are four intersection points: O(0,0), A(a/b,a), B((a-c)/b,a), C(-c/b,0), forming parallelogram OABC. Every parallelogram shape can be defined this way by suitable choice of a, b and c.

OA=√((a/b)^2+a^2), AB=|(a-c)/b-a/b|=|-c/b|, BC=√(-c/b-(a-c)/b)^2+a^2)=√((a/b)^2+a^2), OC=|-c/b|.

We can see that OA=BC and AB=OC, so the opposite sides of a parallogram are equal.