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# Analytical Geometry

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Hi there,

If I have a parallelogram, ABCD, with the following coordinates:

A(-2;3), B(4;2), C(2;-1), D(x;y) and M(0;1), which is a point in the middle of AC, determine x and y.

I know how to calculate when there is only one unknown, but not 2.

I have tried some approaches on paper, like calculating the gradient of AB, then tried to use that in the distance formula using DC coordinates...but that does not seem to do it....would someone kindly please help me with this?. Thank you all very much!

Sep 15, 2017
edited by juriemagic  Sep 15, 2017

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This isn't as difficult as it seems

"D"   can be figured as one of three possible points

First possibility......  add A and B  and subtract C....so we have

(-2 + 4 - 2, 3 + 2 - -1)  =  ( 0, 6)  = (x,y) =  D1

Second possibility....add B and C  and subtract A.....so we have

(4 + 2 - -2 , 2 - 1 - 3) =  ( 8, -2)  = (x,y) =  D2

Third possibility.....[ you might have already guessed it !! ]....add A and C  and subtract B

(-2 + 2 - 4, 3 - 1 - 2)  = ( -4, 0)  = (x,y) =  D3

If the coordinates have to follow a clock-wise order, then D3  will be the "correct" point  for  "D"

Here's a graph showing the possible parallelograms :    Sep 15, 2017