SOLUTION: Points P,Q, R are 3 vertices of a parallelogram. Find the coordinates of point D, the 4th vertex. Find 3 solutions. What are the 3 solutions ?

You forgot to give the coordinates or P, Q and R, so I'll
just make some up and you can model your solution after this
one using the correct coordinates:

I'll suppose that P is the point P(2,3), Q is the point Q(8,1),
and Q is the point R is R(11,5), 

Those three points form a triangle. 



To make a parallelogram out of a triangle, you make a triangle
congruent to it, reverse it, then paste it on one of the three
sides of the triangle.  Since a triangle has three sides,
there are three sides you can paste it on.

You can make one parallelogram like this by pasting it on the 
right side:



In that case we let the 4th point be S(x,y),  Then for
RS and PQ to be parallel,

1.  the slope of the line RS must be equal to the slope 
of PQ, so we use the slope formula to set those slopes 
equal:





Reducing the fraction on the right:



Cross-multiplying,







and also for SQ to be parallel to RP,

1.  the slope of the line SQ must be equal to the slope 
of RP, so we use the slope formula to set those slopes 
equal:









So we solve the system:



and we get S(17,3)

==================================

Or you can make a parallelogram by pasting it on the
top side, like this:



In that case we let the 4th point be T(x,y),  Then for
RT and QP to be parallel,

1.  the slope of the line RT must be equal to the slope 
of QP, so we use the slope formula to set those slopes 
equal:





Reducing the fraction on the right:



Cross-multiplying,















So we solve the system:



and we get T(5,7)

==================================

Or you can make a parallelogram by pasting on the
bottom side like this:



In that case we let the 4th point be U(x,y),  Then for
PU and RQ to be parallel,

1.  the slope of the line PU must be equal to the slope 
of RQ, so we use the slope formula to set those slopes 
equal:





Simplifying the fraction on the right:



Cross-multiplying,









and also for QU to be parallel to RP,

2.  the slope of the line QU must be equal to the slope 
of RP, so we use the slope formula to set those slopes 
equal:





Simplifying the fraction on the right:









So we solve the system:



and we get U(-1,-1)

Now so you can see how they all look togethsr, you
have one big triangle, similar to the original triangle,
and of course to all the parts, and with the sides twice
as long as the original's sides.



Edwin