Grades 11-12 Video Solutions 2013-Levels 11&12 Video Solutions 2013 problem23

Video Transcription

Question number 23.  The solid cube in the figure is cut by a plane passing through the three neighboring vertices D, E, and B of the vertex A. Similarly, the cube is cut  by planes passing through the three neighboring vertices of all other seven corners.  What will the piece containing the center of the cube look like?  So first, let us locate the center of the cube. If we find the point here on the intersection of the diagonals of the top face and the side face, we can say that the center  of the cube is somewhere over here.  And let's make our first cut by connecting the vertices D, E, and B. And we see that our cuts pass through the midpoints  of the intersecting diagonals here.  Now, repeating the procedure here on the top face corner H would give us the same diagonal cut in the upper face.  And repeating the procedure on vertices D or E would give us a cut connecting vertices H and A.  And those are exactly the diagonals of the top face.  So they would intersect here at the point we already marked. And similarly, on the two faces shown,  we would have cuts like that.  And the points of intersection of those cuts have to be connected all throughout. So with the four planes cutting the four upper corners,  we would have cuts in blue connecting these points  like so.  And then one with the back face, which we can't see very well.  And one with the left face, which we can't see very well.  And then along those cuts, we have a plane that contains the center.  And so we see that that's the upper half of our resulting solid after cutting up the cube.  The lower half would look exactly the same by symmetry.  And so what we see here is exactly the tetrahedron in A.

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