Question

Three vertices of a parallelogram taken in order are (1,2),(2,4),(3,7) find the 4th vertex

Answer 1

We are given 3 points, and have to look for the coordinates of D:

Let A, B, C, D be the vertices
A (1,2) ; B (2,4) ; C (3,7) ; 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 (1,2)
B (2,4)
C (3,7)
D (x,y)

Slope of CD = Slope Slope of AB
(y - 7)/(x - 3) = (4-2)/(2-1)
(y - 7)/(x - 3) = 2/1
(y - 7) = 2(x - 3)
y-7 = 2x-6
equation #1:  2x-y = -1

Slope of AD = Slope of BC
(y - 2)/(x - 1) = (7 - 4)/(3 - 2)
(y - 2)/(x - 1) = 3/1
(y - 2) = 3(x - 1)
y -2 = 3x - 3
equation #2: 3x-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:  2x-y=-1
equation #2: 3x-y = 1

Subtract equation 1 to equation 2

equation 1:  2x-y = -1
equation 2:  3x-y = 1
eq 1 - eq 2:   - x = -2
x = 2

Use any of the equation 1 to solve for y:

2x-y= - 1
2*2 - y = - 1
4-y=-1
y=4+1
y = 5

So the 4th vertex will be (2,5)