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A parallelogram has three of its vertices at (-1,0), (2,4) and (2,-4). What is the positive difference between the greatest possible perimeter and the least possible perimeter of the parallelogram?

 Feb 27, 2020
 #1
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We  have  3  possible  points  for  the  last  vertex

 

These  are

 

[ -1 + 2 , 0 + 4 ]  - [ 2 , -4 ]  =  [  [ 1 , 4] - [2 , - 4 ] =   ( -1, 8  )

The perimeter  of this parallelogram = 

2  [  sqrt  [ (-1-2)^2 + (8-4)^2]  + 8  ]  =

2 [ sqrt [ 9 + 16 ] +  8 ]  =

2 [ 5  + 8 ] =

26

 

Another possible  point is

[ 2 + 2, 4 - 4]  - [-1,0] =  [ 4, 0]  - [ -1,0]  = (5,0)

The perimeter of this parallelogram = 

2 [ sqrt [ (5-2)^2  + (0 - 4)^2]  + sqrt [ (2- -1)^2 + (4 - 0)^2) ] ] =

2 [ sqrt (3^2 + 4*2)  + sqrt (3^2 + 4^2) ]  =

2 sqrt (25)  + sqrt (25) ] =

2 [ 5 + 5 ] = 

20

 

We don't need to calculate  the third possible point......this point will not chage  either  the greatest perimeter of 26  or the smallest perimeter  of 20

 

So.....the difference in perimeters  =   6

 

 

 

cool cool cool

 Feb 27, 2020

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