To show rhombus,
All lengths are equal,
Show AB = BC = CD = DA
Length AB = square root ( (4-10)^2 + (-2-0)^2)
= square root (36 + 4)
= square root(40)

Length BC = square root( (10-12)^2 + (0-6)^2)
= square root(40)

Length CD = .....
= square root(40)

Length DA = ....
= square root(40)

Since all lengths are SqRt (40) and equal, hence ABCD is a rhombus //

B) difference between rhombus and square is that,
Square also ensures 90deg between each interior angle.
To show in coordinate:
Consider Angle ABC
We will show AB is NOT perpendicular to BC
(We will just show 1 side)

In other words,
Grad of AB * Grad of BC = -1
If and ONLY if they are perpendicular.
If they are not perpendicular, the value will not be -1
Grad of AB = (0--2) / (10-4)
= 2/6
= 1/3

Grad of BC = (0-6) / (10-12)
= -6/ -2
= 3

Grad of AB * Grad of BC
= 1 NOT equal to -1
Hence they are not perpendicular.

Hence it's clear ABCD is not a square//