Question 979976: show that the vertices represented by the complex numbers 6-i, 7+3i, 8+2i & 7-2i form a parallelogram
Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website!
Let us introduce complex numbers
x = 6 - i;
y = 7 + 3i;
u = 8 + 2i;
v = 7 - 2i.
They are vertices of a quadrilateral in the complex plane.
Now, calculate these differences of complex numbers:
y - z = 1 + 4i, and
u - v = 1 + 4i.
The difference y-z represents the side zy of the quadrilateral (the segment from the vertex z to the vertex y).
The difference u-v represents the side vu of the quadrilateral (the segment from the vertex v to the vertex u).
As you see, the complex numbers y-z and u-v are equal.
It means that the corresponding segments are parallel and have the same length.
Hence, our quadrilateral is a parallelogram.
See my lessons on complex numbers in this site:
REVIEW of lessons on complex numbers.
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