At what coordinates should the fourth vertex be located?

Question

A parallelogram is drawn on a coordinate grid so that three vertices are located at (3,4), (-2,4), and (-4,1). At what coordinates should the fourth vertex be located?

Answer 1

Let fourth point be $D=(x,y)$.

$A=(3,4),B=(-2,4),C=(-4,1)$

As, Diagonals of $||_{gm}$ bisect each other $\implies $ mid-point of $AC$ is same as the mid-point of $BD$

Now, coordinates of mid-point of $AC=(\frac{3+(-4)}{2},\frac{4+1}{2})$ and coordinates of mid-point of $BD=(\frac{x+(-2)}{2},\frac{y+4}{2})$

Thus, equating $x$-coordinate and $y$-coordinate gives $x=1$ and $y=1$.

Thus, point $D$ is $(1,1)$

Answer 2

Hint: Have you drawn a picture? There are three answers, one seeming more obvious to me than the others. You can connect any two of the points to make two sides of the parallelogram. Then through the two points that have only one side drawn so far, draw a line parallel to the other side. Where the two new lines intersect is the point you want.

Added: another way to think of it is that a parallelogram can be divided into two congruent triangles by either diagonal. One of the triangles is upside down relative to the other. Your three points form a triangle, which you can reflect in any of the three sides.

Answer 3

Keep the points as A,B,C. Find mid point(A1)of B and C, Find A2 as A.A2:A2.A1 = -2:1 (external). A2 will be the 4th point opposite to A. IIIy find B2, C2.