Assignments for Week 7

Exercise 12 (Due Friday, 11/12)

The models should be made carefully from manila folder of other cardboard that you can write on.

E 12.1 -- Nesting a tetrahedron in a cube.

E 12.2 -- Cross Sections

Use the same models, buit use a different color of pen or pencil from the segments above.

Assignment 7. Due Monday, 11/15. (75 Points)

Reading

Begin reading Brown. Read from the beginning through the introduction to line reflections and rotations.

You are strongly urged to make models to help with the section on polyhedra.

7.1 Tetrahedron (20 points)

A plane of symmetry of a regular tetrahedron divides the polyhedron into two congruent parts.

  1. 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.)
  2. 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.
  3. Prove that there is a point P which is equidistant from the vertices, A, B, C, D. Find the distance from P to A.
  4. Prove that there is a point P which is equidistant from each face. Find the distance from P to a face.
  5. Find the dihedral angles between the faces of the regular tetrahedron.

7.2 Cube (15 points)
  • 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.
  • Make a model of one of these pyramids with sides the same as your cube from Friday.
  • Explain how the centers of the faces of the cube form an octahedron. What are the lengths of the sides of the octahedron?

    • 7.3 Similar triangles with a common angle. (15 points)

     Suppose we have two lines, AB and AC. Let X be a point on line AB and Y be a point on line AC.
    • Suppose that triangle ABC is similar to triangle AXY. Show that there is a constant k so that AX/AB = k and AY/AC = k. Consider the possibility that the ratios can be negative as well as positive.
    • Show that in this case line BC is parallel to line XY. We call this figure a (parallel) Thales figure.
    • Suppose that triangle ABC is similar to triangle AYX. Show that there is a constant k so that AX*AB = AY*AC = k. Consider the possibility that k can be negative if A is between B and X, etc.
    • Show that in this case line BC is parallel to the reflection of line XY in the angle bisector of BAC. We can call this case "anti-parallel" or "anti-Thales".

    7.4 Double line reflections in parallel lines. (15 points)

    In the (x,y) plane, let line m be given by the equation x = a. Also let line n have equation x = b, and line p have equation x = c. Let M. N, P denote the transformations of reflection in m, n, and p respectively.

    • If Q = (x,y) find the formula for Q' = reflection of Q in line m.
    • Continuing with Q', find the formula for Q'' = reflection of Q' in line m. Then Q'' = N(Q') = NM(Q) so you now have the formula for NM. Use this formula to conclude that NM is a translation. What is the translation vector of this translation?
    • Use similar reasoning to find a formula for MN. What is the translation vector for this translation? What is the inverse of MN?
    • Now let Q''' be the reflection of Q'' in p, so Q''' = PNM(Q). Write a formula for PNM(Q), when Q=(x,y). What kind of isometry is this? (Namely, from the list of kinds of isometries, we know, which is it?)

    7.5 Double line reflections in intersecting lines. (10 points)

    Given m = line AB and n = line AC, reflect point Q in m to get Q'. Then reflect Q' in n to get Q''. Thus Q'' = NM(Q).

    • Prove that angle QAQ'' is the same whatever Q is (the lines stay fixed) and is double the angle BAC. (Hint: This one would be hard with coordinates, so try using triangles. What kind of triangle is QAQ', Q'AQ'', etc.?
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