Question 669172: Find the slope of two diagonals of quadrilateral whose vertices are (3,1), (-1,2), (-3,-1) and (1,-2). identify the quadrilateral.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the first diagonal goes from (-1,2) to (1,-2)
the slope of this diagonal would be change in y divided by change in x which becomes (-2-2) / (1-(-1)) which becomes -4 / 2 which becomes -2.
the second diagonal goes from (-3,-1) to (3,1)
the slope of this diagonal would be change in y divided by change in x which becomes (1 - (-1)) / (3 - (-3)) which becomes 2 / 6 which becomes 1/3.
it looks like we have a parallelogram.
the diagonals appear to bisect each other.
the intersection point appears to be (0,0).
the point (1,-2) is a reflection of the point (-1,2) about the origin because (x,y) becomes (-x,-y).
the point (3,1) is a reflection of the point (-3,-1) about the origin because (x,y) becomes (-x,-y).
being reflections about the origin means that the the distance from the origin of both pairs of points is the same for each pair.
this means the length of (0,0) to (1,-2) is the same length as (0,0) to (-1,2).
this also means the length of (0,0) to (3,1) is the same length as (0,0) to (-3,-1).
since a quadrilateral is a parallelogram if the diagonals bisect each other, this should be sufficient proof.
bottom line is you found the slope as requested.
you determined from the points themselves that the ends of each diagonal are reflections about the origin.
you deduced from this that the diagonals crossed at the origin (0,0).
you deduced from this that the distance from the points to the origin had to be the same for each diagonal point pair.
since the diagonals bisect each other, you deduced that the quadrilateral had to be a parallelogram based on a geometric theorem previously proven that states that.
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