Assignment 6B

Assignment 6B (Due Wed 11/13)

Read:

BG, Chapter 12, Sections 12.1 and 12.2. Pay special attention to the definition of dihedral angle.
Brown (Transformational Geometry): Read Sections 1.1, 1.2, 1.3. Most of this will be familiar from calculus or linear algebra. Pay particular attention to the definition of a transformation of the plane. Also read carefully 1.4. Be able to explain why a line reflection and point reflection are isometries. Also read 1.6.

Problems related to Cube and Tetrahedron: Lengths from similar triangles

Problem 6B-1

Triangle ABC is a right triangle with AB = sqrt 3, CA = sqrt 2, BC = 1.

Let O be a point on AC so that for some R, AO = OB = R.

a) Find R using the Pythagorean theorem for BCO.

b) Also, compute AO/AC.

Problem 6B-2

In this figure, the triangle ABC is isosceles, with AB = AC = s, AN = h, and BN = g, where N is the midpoint of BC. Line MO is the perpendicular bisector of AB.

a) Show the right triangle ABN is similar to the triangle AOM.

b) What is the scaling factor K from ABN to AOM? (If one multiplies the side lengths of ABN by K one obtains the corresponding side lengths in AOM.)

c) Use your K to find the ratio AO/AN in terms of s, h, g.

d) Finally, suppose that s = sqrt 3, h = sqrt 2 and g = 1. Check that your answer for AO/AN is the same as the ratio computed in Problem 1.

Problem 6B-3

Use one of the problems above and your earlier work on altitudes of regular tetrahedra to find the radius of the circumsphere about the regular tetrahedron.

Problem 6B-4

Compute a dihedral angle of a regular tetrahedron. (All the angles are the same.)

Problem 6B-5

Compute all the dihedral angles of a pyramid with a square base and 4 equilateral triangles as faces. (How many different angles are there?)

Transformations

Read Brown and answer the following questions:

Problem 6B-6

Prove that reflection in a point O is an isometry.

Problem 6B-7

If T is an isometry and A'B'C' is the image of triangle ABC by mapping T, explain why A'B'C' is congruent to ABC.

Problem 6B-8

Prove that the composition of two isometries is an isometry.