Expert: Sherman D. - 4/23/2009

Question
Determine the center of the cube that have three of its vertices (2,3,0), (0,5,4), and (4,1,8 ).


Answer
first you need to find the distance between each point.

(2,3,0) and (0,5,4)
D = sqrt((0 - 2)^2 + (5 - 3)^2 + (4 - 0)^2)
D = sqrt(4 + 4 + 16)
D = sqrt(24)
D = 2sqrt(6)

(2,3,0) and (4,1,8)
D = sqrt((4 - 2)^2 + (1 - 3)^2 + (8 - 0)^2)
D = sqrt(4 + 4 + 64)
D = sqrt(72)
D = 6sqrt(2)

(0,5,4) and (4,1,8)
D = sqrt((4 - 0)^2 + (1 - 5)^2 + (8 - 4)^2)
D = sqrt(16 + 16 + 16)
D = 4sqrt(3)

if you notice, none of the distance are the same and sqrt(24)^2 + sqrt(48)^2 = sqrt(72)^2, this can only mean one thing, two of the points from the corner of the cube, two of the points forms the diagonal of one of the squares, and the other two points forms the diagonal of the cube.

since all we need is the diagonal of the cube. all we have to do is find the midpoint of that diagonal to find the center of the cube.

(2,3,0) and (4,1,8)
M = ((2 + 4)/2), ((3 + 1)/2), ((8 + 0)/2)
M = (6/2),(4/2),(8/2)
M = (3,2,4)

so the center of the cube is at (3,2,4)