Consider the point A(-1,2), B(6,2) and C(1,-20) in the plane. For how many different points D in the plane are A, B, C and D the vertices of Parallelogram?
- One
- Two
- Three
- Four
Explanation
For 3 non-collinear points, you can make a parallelogram in 3 different ways depending on which 2 points you take as ends of a diagonal.
The 3 possible D's:
- AB is one side → D = A + C - B = (-1+1-6, 2-20-2) = (-6, -20)
- AC is one side → D = A + B - C = (-1+6-1, 2+2+20) = (4, 24)
- BC is one side → D = B + C - A = (6+1+1, 2-20-2) = (8, -20)
Rule: In parallelogram, A + C = B + D or any permutation.
Since A, B, C are not in a straight line, all 3 D’s are different.
Last verified on 04-06-2026
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