Question

the three vertices of a parallelogram taken in the in order are (-1,0),(3,1) and (2,2) respectively. find the coordinates of the fourth vertex

Answer 1

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) = 1/4
4(y - 2) = (x-2)
4y - 8 = x-2
equation #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+1
equation #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 = 6
equation #2:  y + x = - 1


Add equation 1 to equation 2

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

Using the equation#2 to solve for x:

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

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

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Answer 2

midpoint of AC = midpoint of BD
=> -1+2/2 , 0+2/2 = 3+x/2 , 1+y/2
=> 1/2, 2/2 = 3+x/2 , 1+y/2
=> 1/2 = 3+x/2 1+y/2=1
=> 6+2x=2 1+y=2
=> 2x=4 y=2-1
=> x = -4/2 y=1
therefore x = -2 & y = 1
coordinate of D is (-2,1)