Question

A parallelogram has consecutive vertices A (-2,0), P (0,-2) and C (2,4). Find the fourth vertex.

Answer 1

Let the fourth vertex be D(a,b)
Now equate the midpoint of AC and PD.
After equating you will find x=0 and y=6

Answer 2

One way to do this is to find the equations of the lines parallel to AP and PC  which pass through the points C  and  A  respectively . Then solve the 2 equations for x and y.

equation of  line parallel to AP 

y - y1 = m (x - x1)     
m =  -(-2) / -2  = -1

point C is (2,4)

y - 4 = -1(x - 2)

y =  -x + 6   equation (1)

equation of the line parallel to pc:-

slope m =  6/2 = 3

y - 0 = 3(x - (-2)

y =  3x + 6    equation (2)

solving (1) and (2):-

- x + 6 = 3x +  6
x = 0

and y = 3(0) + 6 = 6

So the fourth vertex is at (0,6)