Q). Show that the four points are angular points of a rectangle$ (0,-1) (4,-3) (8,5) (4,7)$.

Question

I started to solve the question by taking the sides of rectangle ABCD then added a midpoint in the rectangle and divided the rectangle in diagonal then found out the midpoint of diagonals AC and BD (Which was the most possible thing i could do in the question).But at the page where all the answers are written rather than (4,2) since both diagonals bisect each other is written [Hence by converse of Pythagoras' theorem

Answer 1

The vector from $(0,-1)$ to $(4,-3)$ is $(4,-2)$, as is the vector from $(4,7)$ to $(8,5)$. You can check the other two sides are $(4,8)$ This establishes we have a parallelogram. Then since $(4,-2)\cdot (4,8)=0$ they are perpendicular-a rectangle.