Chapter 11.3, Problem 71E

Bond AngleConsider a regular tetrahedron with vertices $(0,0,0),(k,k,0),(k,0,k)$ and $(0,k,k)$, where $k$ is a positive real number.

(a) Sketch the graph of the tetrahedron.

(b) Find the length of each edge.

(c) Find the angle between any two edges.

(d) Find the angle between the line segments from the centroid $(k/2,k/2,k/2)$ to two vertices. This is the bond angle for a molecule, such as ${\text{CH}}_4$ (methane) or ${\text{PbCl}}_4$ (lead tetrachloride), where the structure of the molecule is a tetrahedron.