Theorem 1.5: If   (x1, y1)   and   (x2, y2)   are points on the line whose equation is   y = mx + b   then the distance between them is

Proof: By Theorem 1.1, the slope between them is

so, clearing denominators gives us

y2 - y1 = m(x2 - x1)

If we substitute this into the distance formula,

we get

Factor out an (x2 - x1)2

which equals

next theorem (1.6)