Let the vertices be as follows,A(-1,0) ; B(3,1) ; C(2,2); D(x, y)

The line segments that make up our parallelogram is AD is parallel to BC, and CD is parallel to AB. Since the definition of a parallelogram is a quadrilateral with 2 pairs of parallel sides, we can use this fact to find the coordinate to the 4 vertex. A pair of parallel side also means that these 2 lines have the same slope. So if we do this on both pairs of parallel lines, we should have 2 equations with 2 variables, which should be simple to solve.

Slope of CD =Slope Slope of AB(y - 2)/(x - 2) = (1 - 0)/(3 - - 1)(y - 2)/(x - 2) = 1/44(y - 2) = (x-2)4y - 8 = x-2equation #1: 4y - x = 6

A(-1,0) ; B(3,1) ; C(2,2); D(x, y)Slope of AD = Slope of BC(y - 0)/(x - -1) = (2 - 1)/(2 - 3)(y)/(x + 1) = 1/-1-1(y) = (x + 1)-y= x+1equation #2: y+x = - 1

We now we have 2 equations with 2 variables, and we can use any method (substitution, linear combination, matrix)to solve for the x and y coordinates.

equation #1: 4y - x = 6equation #2: y + x = - 1

Add equation 1 to equation 2

equation 1: 4y - x = 6equation 2: y + x = -1eq 1 + eq 2: 5y = 5y = 1

Using the equation#2 to solve for x:

y + x = -11 + x = - 1x=-2

So the 4th vertex will be (-2,1)