How Many Different Parallograms Given 3 Points

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Given points A=(-1,2), B=(6,4), and C=(1,-20), the problem asks for the number of different points D that can form a parallelogram with A, B, and C. One solution identified is D=(-6,-22), but the answer sheet states there are three possible points. The discussion highlights the need to consider different configurations of the points as vertices of the parallelogram. By sketching the points and analyzing the relationships between the sides, it becomes clearer how multiple solutions can arise. Understanding the properties of parallelograms is key to solving this problem effectively.
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Homework Statement


This is a gre math subject test practice question: Consider the points A=(-1,2), B=(6,4), and C=(1,-20) in the plane. For how many different points D in the plane are A, B, C, and C the vertices of a parallelogram?


Homework Equations



none given

The Attempt at a Solution


I calculated one solution, D=(-6, -22), by figuring out what the coordinates had to be so that DC || AB and AD || BC. The answer sheet says that there are THREE different points. If someone could explain how this is even possible, I'd be much obliged.
 
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there's also DA||BC
 
darkchild said:

The Attempt at a Solution


I calculated one solution, D=(-6, -22), by figuring out what the coordinates had to be so that DC || AB and AD || BC. The answer sheet says that there are THREE different points. If someone could explain how this is even possible, I'd be much obliged.

Seems calculation is long way round if question is just how many points?

Sketch and it turn take two sides as sides of parallelogram.
 

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