use the definition of a parallelogram to show that the quadrilateral with vertices A(-4,4), B(-2,0), C(6,4), and D(4,8) is a parallelogram.

Question

how do you solve this as a proof?

Answer 1

 

show line segment AB and line segment DC have the same slope which means that they are parallel

show line segment AD and line segment BC have the same slope which means that they are parallel

this should suffice because a parallelogram is a quadrilateral whose opposite  sides are parallel

you don't have to show that the opposite sides are equal because if one pair of opposite sides were not equal, the other pair of sides would not be parallel, plus the fact that you are given four distinct points with their coordinates

AB

(-4,4) and (-2,0)

(4-0)/(-4+2)=4/-2=-2

DC

(4,8) and (6,4)

(8-4)/(4-6)=4/-2=-2

AD

(-4,4) and (4,8)

(4-8)/(-4-4)=-4/-8=1/2

BC

(-2,0) and (6,4)

(0-4)/(-2-6)=-4/-8=1/2

the opposite sides have the same slope so therefore the opposite sides are parallel

therefore you have a parallelogram

Answer 2

Drew,

For a parallelogram the following should hold true

 

Length AB = length CD

length BC = length AD

Slope AB = Slope CD

Slope BC = Slope AD

 

Slope AB = (4-0)/(-4-(-2) = -2

Slope CD = (4-8)/(6-4) = -2

These are same

Slope BC = (0-4)/(-2-6) =0.5

Slope AD = (4-8)/(-4-4) = 0.5

These slopes are same too

 

Now length AB = sqrt((4-0)^2 + (-4+2)^2) = sqrt (20)

Length CD = sqrt((6-4)^2 + (4-8)^2) = sqrt 20

These are also same.

 

Similarly you can find length of BC and AD.