We have the four points
(-5, 4) (-1, 2) ( 4, 4) and (0, 6)
Let's call them A, B, C and D. So
A = (-5, 4)
B = (-1, 2)
C = ( 4, 4)
D = (0, 6)
If we can show that AB || CD and BC || AD, then we'll prove that ABCD is a parallelogram. The property I'm using is the idea that the opposite sides of a parallelogram are parallel. This is what makes a parallelogram a parallelogram. It's simply based on the definition.
Note: saying AB || CD is shorthand for saying "segment AB is parallel to segment CD"
We'll need to find the slope of segment AB
Use the slope formula
m = (y2 - y1)/(x2 - x1)
m = (2 - 4)/(-1 - (-5))
m = (2 - 4)/(-1 + 5)
m = (-2)/(4)
m = -1/2
The slope of AB is -1/2
Now find the slope of CD
m = (y2 - y1)/(x2 - x1)
m = (6 - 4)/(0 - 4)
m = (2)/(-4)
m = -1/2
The slope of CD is -1/2
Since the slope of AB is also -1/2, this means the two slopes are equal. Therefore AB is parallel to CD.
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Now we must show that BC || AD
Slope of BC
m = (y2 - y1)/(x2 - x1)
m = (4 - 2)/(4 - (-1))
m = (4 - 2)/(4 + 1)
m = 2/5
Slope of AD
m = (y2 - y1)/(x2 - x1)
m = (6 - 4)/(0 - (-5))
m = (6 - 4)/(0 + 5)
m = 2/5
Like before, we get two slopes that are the same
So BC || AD
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Since we've shown that
AB || CD
and
BC || AD
this means that ABCD is a parallelogram
