Grades 11-12 Video Solutions 2012-2012_11-12

Video Transcription

Three vertices of a cube, not all on the same face, are P, 3, 4, 1, Q, 5, 2, 9, and R, 1, 6, 5. Which point is the center of the cube?  All right, so let's look at a cube right here. We have three kinds of lengths on this cube that  are interesting to us, that are like between vertices. We have this side length right here,  which is like the shortest. We have the length between opposite vertices on the same face, which is like a medium length. And we finally have the longest of the lengths,  which is the distance between opposite corners of our cube. So if we can figure out which is the  longest length, and then the midpoint between these two vertices has to be the center of our  cube. So let's just start finding the distances between these three points. So the distance  between P and Q is, let's just use the Pythagorean theorem, 5 minus 3 squared plus 2 minus 4 squared  plus 9 minus 1 squared, and then expand it out and we get square root of 72. We do the same thing  for Q and R to get root 48. We do the same thing for R and P to get root 24. That means this guy  is the short side, this guy is the medium side, this guy is the long side length. And so the  midpoint of P and Q has to be the center of our cube. So let's just average these two points  to get the midpoint. So 3 plus 5 over 2 is 4, 4 plus 2 over 2 is 3, and 1 plus 9 over 2 is 5. So the point 4, 3, 5 is our answer.

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