Question 550000
If I choose pairs of points, and the parallelogram
is a rhombus, the slopes of the lines through
these pairs of points will either be parallel
or they will be negative reciprocals of
each other ( perpendicular )
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If will choose the pairs
(1) A(-5,-1) and C(-1,5) 
and
(2) B(-9,6) and D(3,-2)
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The slope of pair (1) is
( change in y ) / ( change in x ) = {{{ (5 -(-1)) / (-1 -(-5)) = 6/4 }}}
{{{ 6/4 = 3/2 }}}
The slope of pair (2) is
( change in y ) / ( change in x ) = {{{ ( -2 - 6 ) / ( 3 -(-9) ) = -8/12 }}}
{{{ -8/12 = -2/3 }}}
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These lines have slopes that are negative reciprocals of each other,
so they are perpendicular, and the parallelogram is a rhombus
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You can also show ( not necessary ) that the pairs of
points AB and CD form lines which are parallel