The coordinates of three of the vertices of parallelogram ABCD are A(2, 0), B(4, 3), C(3, 1). How many possibilities are there for the position of the fourth vertex? What are coordinates of the fourth vertex?

If you are given 3 vertices of a parallelogram, as a Geometry student, you will be asked to find the exact coordinates of the 4th vertex.  Just so that you can put your mind at ease, there can be different answers depending on how you decide to "draw" or "shape" your parallelogram.  So if your answer is different than another student, it does not mean that you have done it incorrectly.

Suppose I am given 3 points as such, and I am looking for the coordinates of D:

A (2, 3)

B (8, 1)

C (11, 5)

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 - 5)/(x - 11) = (1 - 3)/(8 - 2)

(y - 5)/(x - 11) = -2/6

6(y - 5) = -2(x - 11)

6y - 30 = -2x + 22

equation #1:  6y + 2x = 52

Slope of AD = Slope of BC

(y - 3)/(x - 2) = (5 - 1)/(11 - 8)

(y - 3)/(x - 2) = 4/3

3(y - 3) = 4(x - 2)

3y -9 = 4x - 8

equation #2:  3y - 4x = 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:  6y + 2x = 52

equation #2:  3y - 4x = 1

Multiply equation by -2

Equation 2: -2(3y - 4x = 1)

Equation 2:  -6y + 8x = -2

Add equation 1 to equation 2

equation 1:  6y + 2x = 52

equation 2:  -6y + 8x = -2

eq 1 + eq 2:  10x = 50

x = 5

Use any of the original equation to solve for y:

6y + 2x = 52

6y + 2(5) = 52

6y = 42

y = 7

So the 4th vertex for my first answer will be (5, 7)

If we draw the parallelogram in a different way using the given vertices, we will be looking for a different 4th vertex.  So in our original question:

A (2, 3)

B (8, 1)

C (11, 5)

D (x, y)

Our line segments in this case will be draw as such:

AC is parallel to BD and CD is parallel to AB.

Slope of AC = slope of BD

(5 - 3)/(11 - 2) = (y - 1)/(x - 8)

2/9 = (y - 1)/(x - 8)

2(x - 8) = 9(y - 1)

2x - 16 = 9y - 9

equation 1:  2x - 9y = 7

Slope of CD = slope of AB

(5 - y)/(11 - x) = (3 - 1)/(2 - 8)

(5 - y)/(11 - x) = 2/-6

-6(5 - y) = 2(11 - x)

-30 + 6y = 22 - 2x

equation 2:  2x + 6y = 52

Solve the following linear system:

equation 1:  2x - 9y = 7

equation 2:  2x + 6y = 52

eq 1 - eq2:  -15y = -45

-15y = -45

y = 3

Substitute y = 3 into any one of the original equation:

equation 1:  2x - 9y = 7

2x - 9(3) = 7

2x - 27 = 7

2x = 34

x = 17

So by drawing the parallelogram in a different way, the 4th vertex of this parallelogram is (17, 3).

Step-by-step explanation:

See Answer