### Instruction: 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?### Response: To find the perimeter of a parallelogram, I need to know the lengths of its two pairs of parallel sides.
One pair of parallel sides is formed by the segments joining $(-1,0)$ and $(2,4)$ and their opposite vertices, which I don't know yet.
The length of this segment is $\sqrt{(2-(-1))^2+(4-0)^2}=\sqrt{9+16}=\sqrt{25}=5$ by the distance formula.
The other pair of parallel sides is formed by the segments joining $(2,4)$ and $(2,-4)$ and their opposite vertices, which I also don't know yet.
The length of this segment is $\sqrt{(2-2)^2+(-4-4)^2}=\sqrt{0+64}=\sqrt{64}=8$ by the distance formula.
So the perimeter of the parallelogram is $2(5+8)=26$ units, regardless of where the fourth vertex is.
This means that the greatest possible perimeter and the least possible perimeter are both 26 units.
Therefore, the positive difference between them is $26-26=0$ units.