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?
+128406 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
