Question

Calculate the distance between the pairs of coordinates, and classify them according to the distance between them. 

Answer 1

The true key to geometry is when we stop being concerned with distance and do everything with squared distances. The square roots magically melt away.

The squared distance between two points (a,b) and (c,d) is given by the Pythagorean Theorem:

$D^2 = (a-c)^2 + (b-d)^2$

Apparently some of our squared distances will be 10 and some will be 29. We can pretty much eyeball which is which, but let's work them out:

(3,4),(2,1): $\quad D^2=(3-2)^2+(4-1)^2=1+9=10$

(3,7),(5,2): $\quad D^2=(3-5)^2+(7-2)^2=4+25=29$

(5,2),(3,3): $\quad D^2=(5-2)^2+(3-2)^2=10$

(-2,3),(1,2): $\quad D^2=(-2 - 1)^2 + (3-1)^2 = 10$

(-4,-2),(-3,1): $\quad D^2=(-4 - -3)^2 + (-2 - 1)^2=10$

(4,-1),(-1,1): $\quad D^2=(4 - -1)^2 + (-1 - 1)^2 = 29$

With integer coordinates we'll always get a squared distance that's the sum of two squares, but it won't always be 10 or 29.