Question

How can you prove that the points `A(2, 2), B(5, 7), C(-5, 13),` and `D(-8,8)` are the vertices of a rectangle?

Answer 1

`ABCD` is a rectangle.

Explanation:

Given four vertices `A(2,2),B(5,7),C(-5,13)` and `D(-8,8)`, the figure is a rectangle, if the two diagonals are equal in size and they bisect each other . The latter means the midpoints of two diagonals are same.

Let us check the latter first. The midpoint of `AC` is `((2+(-5))/2,(2+13)/2)` or `(-3/2,15/2)`. The midpoint of `BD` is `((5+(-8))/2,(7+8)/2)` or `(-3/2,15/2)`. As midpoint is same, diagonals bisect each other and `ABCD` is atleast a parallelogram.

Further `AC=sqrt((2-(-5))^2+(2-13)^2)=sqrt(49+121)=sqrt170`

and `BD=sqrt((5-(-8))^2+(7-8)^2)=sqrt(169+1)=sqrt170`

as two diagonals are equal and we already know, it is a parallelogram,

it is a rectangle.