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Math Help - Need to find the coordinates of the top points of a cube, A, B, C, and D

  1. #1
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    Need to find the coordinates of the top points of a cube, A, B, C, and D

    Given information: A(13,30,9) C(2,5,17) B(bx,by,13) D(dx,dy,13), hence the height of points B and D are the same .
    Need to determine bx,by,dx,dy.
    Segment AC and BD are the diagonals.

    Thanks
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  2. #2
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    Re: Need to find the coordinates of the top points of a cube, A, B, C, and D

    Hello, Ants!

    I have a game plan . . .


    A(13,30,9),\;B(x_b,y_b,13),\lC(2,5,17),\;D(x_d,y_d  ,13)\,\text{ form a square.}
    \text{Segments }AC\text{ and }BD\text{ are the diagonals.}
    \text{Determine }x_b,y_b,x_d,y_d.
    Code:
                      (2,5,17)
                          ♥ C
                      *    *
              D   *         *
    (xd,yd,13) ♥              *
               *              *
                *              *
                 *              ♥ (xb,yb,13)
                  *         *   B
                   *    *
                  A ♥
                (13,30,9)
    We have four unknowns; we will need four equations.


    The sides of the square are equal.

    . . \begin{array}{ccccc} \overline{AB}^2 &=& (x_b-13)^2 + (y_b-30)^2 + (13-9)^2  \\ \overline{BC}^2 &=& (x_b-2)^2 + (y_b-5)^2 + (13-17)^2 \\ \overline{CD}^2 &=& (x_d-2)^2 + (y_d-5)^2+(13-17)^2 \\ \overline{DA}^2 &=& (x_d-13)^2 + (y_d-30)^2 + (13-9)^2 \end{array}

    These give us three equations.


    The diagonals are equal.

    . . \overline{AC}^2 \:=\:11^2 + 25^2 + 8^2 \:=\:810 \quad\Rightarrow\quad AC \:=\:9\sqrt{10}

    . . \overline{BD}^2 \:=\:(x_d-x_b)^2 + (y_d-y_b)^2 + (13-13)^2

    This gives us the fourth equation.


    Can you take it from here?

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