How to work out the unknown vertex of a parallelogram given the other three

Question

I have changed the numbers so you're not giving me the answers.

Let A (1, 1), B (5,2), C (2, 4) and D (x, y) be the vertices of a parallelogram ABCD.

What are the coordinates of vertex D?

-As I am struggling on how to do this, I would appreciate it if you could show me an example of how to do it? Using my random numbers or your own?

Answer 1

HINT:

We know the diagonals of parallelogram with vertices $(x_i,y_i)$ (where $i=1,2,3,4$ in order) bisect each other (proof)

$$\implies\frac{x_1+x_3}2=\frac{x_2+x_4}2$$

Answer 2

Hint

$$\vec{BD}= \vec{BC}+\vec{CD}= \vec{BC}+\vec{BA}$$

added

You get

$$(x-5,y-2)=(-3,2)+(-4,-1)$$