Assignment 6A (Due Wed 11/6) Build and Analyze a Tetrahedron

Handout: A net for a regular tetrahedron ABCD. We denote by s the length AB.

This tetrahedron can be cut into two congruent "halves" by a plane. Each "half" is itself a tetrahedron, but not regular.

1. Construct (with straightedge and compass or Sketchpad) a net for a half-tetrahedron ABCM. Make the edge length AB the same size in your net as the AB in the original net.

2. Compute all the edge lengths (exact, theoretical) of this "half" if the edge lengths of the original regular tetrahedron are all s. (The answer will be an expression in s.)

3. Use this model to figure out the slant height and the altitude of the original regular tetrahedron. (These lengths are lengths of segments on a face of your "half". Draw the segments on your model and then use plane geometry to figure out the lengths. The answer will be an expression in s)

4. Use the model to prove that all 4 altitude segments of the regular tetrahedron intersect at a point O (i.e., the altitudes are concurrent).

5. Tell the exact distance from O to any of the vertices ABCD of the original. (The answer will be an expression in s.)

6. Tell the exact distance from O to any of the faces (such as ABC) of the original. (The answer will be an expression in s.)