Chapter 9.4, Problem 50E

Central Angle of a Tetrahedron A tetrahedron is a solid with four triangular faces, four vertices, and six edges, as shown in the figure. In a regular tetrahedron the edges are all of the same length. Consider the tetrahedron with vertices $A\left(1,0,0\right)$, $B\left(0,1,0\right)$, $C\left(0,0,1\right)$, and $D\left(1,1,1\right)$.

(a) Show that the tetrahedron is regular.

(b) The center of the tetrahedron is the point $E\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$ (the “average” of the vertices). Find the angle between the vectors that join the center to any two of the vertices (for instance, $\angle AEB$). This angle is called the central angle of tetrahedron.

[Note: In a molecule of methane $\left({\text{CH}}_{\text{4}}\right)$ the four hydrogen atoms form the vertices of a regular tetrahedron with the carbon atom at the center. In this case chemists refer to the central angle as the bond angle. In the figure, the tetrahedron in the exercise is shown, with the vertices labeled $H$ for hydrogen and the labeled $C$ for carbon.]