Question

Three vertices of a quadrilateral are (-1 -1), (1,2) and (5, -1). What are the coordinates of two vertices that will form two different parallelograms?

Answer 1

Choose any two vertices from {(-5, 2), (7, 2), (3, -4)}

Explanation:

Any pair of the given vertices can be one diagonal of the parallelogram. Then the fourth vertex is found by adding the two points of the chosen pair and subtracting the remaining point. Then coordinates of the 4th vertex could be ...

... (-1, -1) +(1, 2) -(5, -1) = (-1+1-5, -1+2+1) = (-5, 2)

or

... (-1, -1) +(5, -1) -(1, 2) = (-1+5-1, -1-1-2) = (3, -4)

or

... (1, 2) +(5, -1) -(-1, -1) = (1+5+1, 2-1+1) = (7, 2)

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The method we have described for finding the remaining point comes from the fact that the two diagonals of a parallelogram intersect at their midpoints. If M is the midpoint of diagonals AC and of BD of parallelogram ABCD, then ...

... M = (A+C)/2 = (B+D)/2

... A+C = B+D . . . . multiply by 2

... A+C-B = D . . . . . . the fourth point is the sum of opposite points less the third one.