Possible Fourth Corners of a Parallelogram Given Three Non-Collinear Points

In summary, the possible fourth corners of the parallelogram formed by (1,1), (4,2), and (1,3) are (4,4), (4,0), and (-2,2). This is possible because any triangle can describe 3 parallelograms, and the points given form a triangle.
  • #1
zeion
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1

Homework Statement



If three corners of a parallelogram are (1,1), (4,2), and (1,3), what are all the possible fourth corners?


Homework Equations





The Attempt at a Solution



The only possible fourth corner is (4,4) if the other 3 points are set.
The solution says (4,0) and (-2,2) can also be corners. How is this possible
 
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  • #2
Take the problem description to a graph of the points given, and literally test the answer key solution visually; is it also a vertex of a parallelogram? Now, if so, how can you account for this symbolically?
 
  • #3
Note that given 3 non-collinear points A, B, & C, you can create a triangle [itex]\triangle[/tex]ABC

A parallelogram, when divided along one of its diagonals, results in two identical triangles rotated 180[itex]^\circ[/tex] with respect to each other.
So, any triangle can describe 3 parallelograms.
 

1. What is a parallelogram fourth corner?

A parallelogram fourth corner refers to the point or vertex that completes a parallelogram. It is the opposite corner to the one specified or known, and is necessary to have in order to fully define the shape.

2. How do you find the fourth corner of a parallelogram?

To find the fourth corner of a parallelogram, you can use the properties of a parallelogram. The opposite sides of a parallelogram are equal in length and parallel to each other. So, if you know the length of one of the sides and the angle between it and the known corner, you can use trigonometric functions to determine the coordinates of the fourth corner.

3. Can there be multiple fourth corners in a parallelogram?

No, there can only be one fourth corner in a parallelogram. A parallelogram is a specific type of quadrilateral with its own unique properties, and having multiple fourth corners would violate those properties.

4. Why is the fourth corner important in a parallelogram?

The fourth corner is important in a parallelogram because it helps to fully define and describe the shape. Without the fourth corner, the parallelogram would be incomplete and we would not be able to accurately measure or calculate its properties.

5. Can a parallelogram exist without a fourth corner?

No, a parallelogram cannot exist without a fourth corner. As mentioned before, the fourth corner is a necessary component in defining the shape of a parallelogram. Without it, the shape would not be a parallelogram, but rather a different type of quadrilateral.

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