# Answer to Question #5168 in Analytic Geometry for aljon

Question #5168

three consecutive vertices of a parallelogram (1,-3),(-3,-1), (3,5). FIND THE 4TH VERTEX?

Expert's answer

If ABCD is a parallelogram and D(x,y) then the midpoint of AC(the diagonal of ABCD) is ((1+3)/2,(-3+5)/2)=(2,1);

The midpoint of BD (the diagonal of ABCD) is ((x-3)/2, (y-1)/2)

<img src="data:image/png;base64,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" alt="">

The midpoint of BD (the diagonal of ABCD) is ((x-3)/2, (y-1)/2)

<img src="data:image/png;base64,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" alt="">

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