Question

If(2,1),(3,4) (0,1) are the vertices taken in order of a paralleogram find the fourth vertex

Answer 1

Let the vertices be as follows,
A(2,1) ; B(3,4) ; C(0,1); 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.

A(2,1) ; B(3,4) ; C(0,1); D(x, y)
Slope of CD = Slope Slope of AB
(y - 1)/(x - 0) = (4 - 1)/(3 - 2)
(y - 1)/(x) = 3/1
(y - 1) = 3x
-1 = 3x - y
equation #1:  3x - y = -1

A(2,1) ; B(3,4) ; C(0,1); D(x, y)
Slope of AD = Slope of BC
(y - 1)/(x - 2) = (1 - 4)/(0 - 3)
(y - 1)/(x - 2) = -3/-3
(y - 1)/(x - 2) = 1/1
y-1=x-2
equation #2: x - y = 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:  3x - y = -1
equation #2:  x - y = 1


Subtract equation 2 from equation 1
equation #1:  3x - y = -1
equation #2:   x - y = 1
eq 1 - eq 2:   2x = -2
x = -1

Using the equation#2 to solve for y:
x - y = 1
-1 + y = 1
y=2

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

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