Find the top vertex coordinate of a regular tetrahedron

[jwxie]

Homework Statement

A regular tetrahedron has the vertices of its base A(1,1,0) B(3,1,0) C(2,1+(3^(1/2),0). Find coordinate of vertex S?

Homework Equations

The Attempt at a Solution

If this is a tetrahedron

Then we know the length by caclulating the distance formula, which gives length of 2.

My game plan is to find the height in order to determine the z-coordinate.
I thought I could just get length of the apothem, which is given by a = [sqrt(3)/6]*s, where s is 2 in this case. this gives us sqrt(3)/3

Then I tried to calculate the length of BH... so (sqrt(3)/3))^2 + (1)^2 = BH^2
and i have BH = 2/sqrt(3), or 2*sqrt(3)/3

anyway. this leads to caclulate the height AOH, and i had [2/sqrt(3)]^2 + (2)^2 = AOH^2, where 2^2 comes from the length of AB...
and AOH is 4/sqrt(3)

I don't have any solution to this problem, but upon googling someone attempted the problem.
http://answers.yahoo.com/question/index?qid=20100903153810AAmVNTY

I am not sure where I did wrong, if that solution is corrected.
Just by looking at the z-value, I would have 4/sqrt(3)... which is different from whatever was solved on yahoo answer.

Can anyone please help me on this? Thanks!

 

[lanedance]

first find the centre, H in the diagram, of the base, if you think it through you should be able to find something in terms of vectors like:
H = A + AB/2 + (AC + BC)*(1/2)*(1/3)

then you if S = (x,y,z), and now you know x & y

to find z just consider a side length