Math 444 Assignment 13 (Due Wed, 12/9) (60 points)
13.1. The Regular Tetrahedron (30 points)
Make a model of half a Regular Tetrahedron which has been cut by a plane of symmetry. (You will find this helpful for the next parts.)
If the length of an edge of the Regular Tetrahedron ABCD is S, find the height of the tetrahedron when it is resting on its base ABC.
Prove that there is a point P which is equidistant from the vertices, A, B, C, D. Find the distance from P to A.
Prove that there is a point P which is equidistant from each face. Find the distance from P to a face.
Find the dihedral angles between the faces of the regular tetrahedron.
13.2. The Cube (20 points)
Make a model of a cube. Show by drawing on the cube that there is a regular tetrahedron EFGH whose vertices are also vertices of the cube. If the length of the side of the cube is C, what is the length of the side of the tetrahedron.
If the base of the cube is the square ABCD with A'B'C'D' opposite, explain why the cube can be broken into three congruent pyramids with square bases such as A'ABCD. Find the dihedral angles of this pyramid.
Explain how the centers of the faces of the cube form an octahedron. What are the lengths of the sides of the octahedron?
13.3. Dodecahedron (10 points)
Make a model of a dodecahedron and draw on the model a cube whose vertices are also vertices of the dodecahedron.