p42_037 - 37. The description in the problem statement...

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37. The description in the problem statement implies that an atom is at the centerpoint C of the regular tetrahedron, since its four neighbors are at the four vertices. The side length for the tetrahedron is given as a = 388 pm. Since each face is an equilateral triangle, the “altitude” of each of those triangles (which is not to be confused with the altitude of the tetrahedron itself) is h 0 = 1 2 a 3 (this is generally referred to as the “slant height” in the solid geometry literature). At a certain location along the line segment representing “slant height” of each face is the center C 0 of the face. Imagine this line segment starting at atom A and ending at the midpoint of one of the sides. Knowing that this line segment bisects the 60 angle of the equilateral face, then it is easy to see that C 0 is a distance AC 0 = a/ 3. If we draw a line from C 0 all the way to farthest point on the tetrahedron (this will land on an atom we label B ), then this new line is the altitude
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