Pre-Calculus Help!!!?

Question

A researcher for a pharmaceutical firm has graphed the structure of a molecule with atoms having positions A=(2,0,0), B=(1,sqr(3),0) and C=(1,1/3,(2sqr(2))/3)). She needs to have every atom in this molecule equidistant from each other. Has she achieved this goal? Justify your answer.

Help!!!

Answer 1

Distance = sqrt(delta-x^2 + delta-y^2 + delta-z^2)

To avoid taking square roots, we can just compare the square of the distances.

The square of the distance from A to B is:
(2-1)^2 + (0 - sqrt(3))^2 + (0-0)^2 =
1 + 3 + 0 =
4

The square of the distance from B to C is:
(1-1)^2 + (sqrt(3) - 1/3)^2 + (0 - 2sqrt(2)/3)^2 =
0 + (3 - 2/3 sqrt(3) + 1/9) + 8/3 =
5 + 7/9 - 2/3 sqrt(3) =
approximately 4.62

Since the square of the distance from A to B is not the same as the square of the distance from B to C, the three points are not all equidistant.