SOLUTION: Three vertices of a parallelogram are at the points(0,0),(2,4),and(6,0).What are the coordinates of the fourth vertex?

We plot the points:


 
There are three different solutions, because we can connect those
three points in three different ways:

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1. If we connect them this way:



We notice that (2,4) is 2 units right of (0,0).  Therefore the 
x-coordinate of the 4th point of the parallelogram will be 2 units
left of (6,0).  That will be 6-2, or 4.

We also notice that (2,4) is 4 units above (0,0) and (6,0).  Therefore 
the y-coordinate of the 4th point of the parallelogram will be 4 units
below both (0,0,) and (6,0).  That will be 0-4, or -4.

So the 4th point is (4,-4).  The parallelogram is:


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2.  If we connect them this way:



We notice that (2,4) is 2 units right of (0,0).  Therefore the 
x-coordinate of the 4th point of the parallelogram will be 2 units
right of (6,0).  That will be 6+2, or 8.

We also notice that (2,4) is 4 units above (0,0) and (6,0).  Therefore 
the y-coordinate of the 4th point of the parallelogram will be 4 units
above both (0,0,) and (6,0).  That will be 0+4, or 4.

So the 4th point in this case is (4,4).  The parallelogram is:


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3.  If we connect them this way:



We notice that (2,4) is 4 units leftt of (6,0).  Therefore the 
x-coordinate of the 4th point of the parallelogram will be 4 units
left of (0,0).  That will be 0-4, or -4.

We also notice that (2,4) is 4 units above (0,0) and (6,0).  Therefore 
the y-coordinate of the 4th point of the parallelogram will be 4 units
above both (0,0,) and (6,0).  That will be 0+4, or 4.

So the 4th point in this case is (-4,4).  The parallelogram is:


Now you may wonder what all three solutions look like
when drawn on a graph:



They all form a big triangle, with a small triangle
inscribed insides, similar to the big triangle, whose
vertices are the midpoints of the three sides of
the big triangle.

Edwin