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Math Help - find the top vertex coordinate of a regular tetrahedron

  1. #1
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    find the top vertex coordinate of a regular tetrahedron

    1. The problem statement, all variables and given/known data

    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?

    2. Relevant equations

    3. 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 dont have any solution to this problem, but upon googling someone attempted the problem.
    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 coord of vertex S? - Yahoo! Answers

    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!
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  2. #2
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    regular tetrahedron

    Hi,
    Aregular tetrahedron has four faces each face an equilateral triangle.Not long ago there was a problem answered by earboth entitled find the height of the pile in the geometry forum. If you go back you will get a nice diagram showing how to calculate the altitude given one side

    bjh
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  3. #3
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    http://www.mathhelpforum.com/math-he...tml#post598112
    Thanks for the direction.
    Attached Thumbnails Attached Thumbnails find the top vertex coordinate of a regular tetrahedron-equilateral_triangle_median_m_equation.png  
    Last edited by jwxie; February 16th 2011 at 06:58 PM.
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  4. #4
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    this is my work
    a = 2 (length of the side)
    a = c



    i dont have an answer to check against with
    can someone plz verify this is the z-coordinate?

    thanks!
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  5. #5
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    Hello, jwxie!

    h \,=\,\dfrac{2\sqrt{6}}{3}

    i dont have an answer to check against with
    Can someone plz verify this is the z-coordinate?

    Yes . . . Good work!

    I get: . D\,\!\left(2,\:1\!+\!\frac{\sqrt{3}}{3},\: \frac{2\sqrt{6}}{3}\right)

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