Parallelogram- Geometry Question?

Question

Prove that the points W(-6, 2), X(-1, 3), Y(2, -3), and Z(-3, -4) represent the vertices of a parallelogram.

Answer 1

W,X (-6, 2),(-1, 3)
m= y2-y1/x2-x1
m= 3-2/ -1-(-6)
m= 3-2/-1+6
m= 1/5

X,Y (-1, 3),(2, -3)
m= y2-y1/x2-x1
m= -3-3/2-(-1)
m= -3-3/2+1
m= -6/3
m=-2

Y,Z (2, -3),(-3, -4)
m= y2-y1/x2-x1
m= -4-(-3)/-3-2
m= -4+3/-3-2
m= -1/ -5
m= 1/5

W,Z (-6, 2),(-3, -4)
m=y2-y1/x2-x1
m= -4-2/-3-(-6)
m= -4-2/-3+6
m= -6/3
m= -2

yes it is a paralellogram, WX is equal to YZ and and XY is equal to WZ, and a parallelogram's two sides are equal

Answer 2

Make a line between pairs of points i.e. (-6,2) to (-1,3) has change in x value of 5 and change in y of 1; so
slope = change in y / change in x
= 1/5.

Next pair (2,-3) to (-3,-4). Change in x is -5 and change in y is -1. So
slope = change in y / change in x
= 1/5.

These slope are the same. The line segment WX is parallel to YZ.

At this point (if not earlier) we quickly sketch the vertices on a cartesian system to verify how the points are oriented.

We still need to show that WZ is parallel to XY.

Okay, to this end, consider the line segment from (-6,2) to (-3,-4). Here the change in x is 3 and the change in y is -6. The slope is
slope = change in y / change in x
= -6 / 3 = -2

Now finally for the segment XY from (-1,3) to (2,-3). The change in x is 3, and the change in Y is -6. The slope is
slope = change in y / change in x
= -6 / 3 = -2.

We are finished.

It's important to include your sketch of the vertices (labeled) in the cartesian plane. This aids your argument, and helps you understand and construct your proof.

Good Luck!

Answer 3

vector WX = (5 , 1)
vector XY = (3 , - 6)
vector ZY = (5 , 1)
vector WZ = (3 , -6)
Vectors WX and ZY are parallel and of equal magnitude.
Vectors XY and WZ are parallel and of equal magnitude.
WX + XY + YZ + ZW = 0 ie form a closed figure, WXYZ.
WXYZ is therefore a parallelogram.