Hi,

If 

 and  

then WXZY is a parallelogram.

(2,2)

(5,2)  m = 0 for WX and for ZY, m = 0

(6,-2)

(2,2)  m = -1 for ZX and for YW m = -1



 

Answer by thejackal(72)   (Show Source): You can put this solution on YOUR website!
 this should not be too difficult.



First let's recall what a parallelogram is. A parallelogram has opposite sides parallel and equal in length. Also, opposite angles are equal (angles WXZ and WYZ are the same, and angles YWX and YZX are the same).



next lets set out to prove that the opposite sides are equal: (distance formula)



if the opposite sides are equal then WX = YZ and WY = XZ



Since X = (5,2) and W = (2,2) then WX = |5-2| = 3

Since Z = (6,-2) and Y = (3,-2) then YZ = |3-6| = 3



(one down one to go)



Since W = (2,2) and Y = (3,-2) the WY = |2--2| = 4

Since X = (5,2) and Z = (6,-2) then XZ = |2--2| = 4



by the way the symbol | means magnitude 





Now the tricky part. (Slope formula)



To finish our proof we need to know the interior angles of the quadrilateral. Unfortunately with only two sides we can't do that we need to form a triangle using the points WYZ calculate the distance WZ and using the SSS triangle formula to calculate the angle. Hope I haven't lost you there? don't worry i'll explain step by step.



Step 1. 



If you've drawn this figure out on paper, we need to extend the distance YZ a bit so that we form a right angled triangle. The point to which we extend YZ must be perpendicular to W call it point U. So that WUZ is a right angled triangle. Now we apply the slope formula.



WU^2 + UZ^2 = WZ^2



WU will be equal to 4 because we are only interested in the y -axis and point U definitely lays on the same line as YZ so |2--2| = 4



UZ = 4 as well because now we are interested in the x-axis of point W and Z

so |2-6| = 4



therefore WZ^2 = 4^2 + 4^2 = 32 hence WZ = root(32) roughly = 5.6568



We already proved that the distance are equal and opposite so the other diagonal 

 XY is exactly the same size as WZ



Last stretch (triangles)



Knowing the distances of each side, we can find the angles

inside our quadrilateral. 

the SSS formula (SSS is simply 3 sides known) is 

cos A = (b^2 + c^2 - a^2)/2bc

Angle A = inverse of Cos A



in this case we start with angle Y so that WY is our c (always left) YZ is our b (always right of the angle) and our a which is always opposite the angle you are trying to find is WZ



so Cos Y = (3^2 + 4^2 - 5.6568^2)/(2*3*4) = (9+16-31.999)/(24) = -0.29164

so Y is = Cos^-1(-0.29164) = 106.9562 



since the measurements on the opposite side WXZ are exactly the same if we extend WX by one to calculate WZ then we can stop here because we know for sure the angles will be equal. 

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