SOLUTION: If A(2 3) B(p -2) and C(-1 1) and D(3 r) are tthe vertices of a parallelogram ABCD calculated the value of p and of r

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Question 1067636: If A(2 3) B(p -2) and C(-1 1) and D(3 r) are tthe vertices of a parallelogram ABCD calculated the value of p and of r
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
For ABCD to be a parallelogram,
opposite sides must have equal length and equal slope.
The easiest way to do that is to consider vectors.
We need to have
A-B=D-C or B-C=A-D .
If one of those pairs of opposite sides complies,
the other pair is forced to do so too.
The only problem would be if we ended with all 4 points being co-linear.
A-B=%28matrix%281%2C2%2C2-p%2C3-%28-2%29%29%29=%28matrix%281%2C2%2C2-p%2C5%29%29
D-C=%28matrix%281%2C2%2C3-%28-1%29%2Cr-1%29%29=%28matrix%281%2C2%2C4%2Cr-1%29%29
So, A-B=D-C translates into the system of equations
system%282-p=4%2C5=r-1%29 ---> highlight%28system%28p=-2%2Cr=6%29%29 .
The solution for B-C=A-D is the same.
So, B=%28matrix%281%2C2%2C-2%2C-2%29%29 and D=%28matrix%281%2C2%2C3%2C6%29%29 ,
and the parallelogram is shown below