Conditions for a parallelogram

Question

If 4 points taken in clockwise order A(1,-1);B(-2,0);C(1,2); and D(x,y) Form the vertices of a parallelogram, then what is the coordinate of D?

Answer 1

Hints:

1. The diagonals of a parallelogram bisect each other.
2. The mid point of the line segment joining $(x_1,y_1)$ and $(x_2,y_2)$ is $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.

I am sure you can do the rest.

Answer 2

Let $A(1,-1), B(-2,0), C(1,2), D(h,k)$, $m_{AB}=m_{CD} \implies \frac{-1}{3}=\frac{2-k}{1-h} \implies 6-3k=h-1~~~(1)$. Similarly, $m_{BC}=m_{AD} \implies \frac{2}{3}=\frac{k+1}{h-1} \implies 2(h-1)=3(k+1)~~~(2)$ Solving (1) and (2), we get $D(4,1)$.