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 A Point and a Cube (Posted on 2004-07-16)
Before tackling this one, take a look at this one.
```     +-------------D
/|            /|
/ |           / |
/  |          /  |
/   |         /   |
C-------------+    |
|    |        |    |
|    B--------|----+
|   /         |   /
|  /          |  /
| /           | /
|/            |/
+-------------A
```
A, B, C and D are non-adjacent vertices of a cube. There is a point P in space such that PA=3, PB=5, PC=7, and PD=6. Find the distance from P to the other four vertices and find the length of the edge of the cube.
There are two answers, one with P outside the cube and one with P inside the cube.

 No Solution Yet Submitted by Brian Smith Rating: 4.0000 (5 votes)

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 My numbers were wrong. | Comment 21 of 23 |

When I did this problem previously, I failed to notice the change in PD, so here we go again.  Contrary to popular believe, one does not need to know the exact point in space we're looking at.  The question asks for distances and the length of a side.

My points in space:  P(x, y, z); A(k, -k, k); B(-k, -k, -k); C(-k, k, k); D(k, k, -k); E(-k, k, -k); F(k, k, k); G(-k, -k, k); H(k, -k, -k)

Writing out equations for the distances to each point in the form of T - A + B - C = 9, etc, we end up with four equations and four variables.  Solve the system of equations any way you want.  I found that T = 119 / 4, A = 29 / 4,  B = -51 / 4 and C = 3 / 4.  Add these together as determined by the sign of k (if finding the distance to (-k, k, -k), it would be T + A - B + C.) to find the missing distances.

I found the length of an edge to be equal to (deep breath) ¡î((119 ¡¾ 4¡î(238))/6).  Pay careful attention to the parenthesis.

The remaining distances are [¡î(101/2)], [¡î(69/2)], [¡î(47/2)], and [¡î(21/2)].  Check my answers for correctness.  This was all done by hand with a bunch of screaming children around, so there are no guarantees on accuracy.  But if I'm right, those answers are exact and not approximations.

 Posted by John on 2004-07-26 16:06:24

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