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

1. ## 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

2. ## 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 . . .

$\displaystyle A(13,30,9),\;B(x_b,y_b,13),\lC(2,5,17),\;D(x_d,y_d ,13)\,\text{ form a square.}$
$\displaystyle \text{Segments }AC\text{ and }BD\text{ are the diagonals.}$
$\displaystyle \text{Determine }x_b,y_b,x_d,y_d.$
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.

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

These give us three equations.

The diagonals are equal.

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

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

This gives us the fourth equation.

Can you take it from here?