INSTRUCTIONS: This test is in two parts. The questions in Part A require multi-step calculations and/or proofs. The questions in Part B are SHORT and should require very short calculations, answers by inspection, or short verbal answers.
1. Dihedral Angle (15 points)
Consider a pyramid ABCP with base an equilateral triangle ABC and a vertex P so that the other 3 triangular faces are isosceles right triangles with right angles at P. (Another way to describe this pyramid is that it can be produce by cutting off a corner from a cube.)
Problem: Calculate the exact dihedral angle (possibly as an inverse
trig function) between the faces triangle ABP and triangle ABC. Show your work.
2. Parallelograms (20 points)
(a) State the Definition of a parallelogram (the standard definition, not an equivalent).
(b)
Using the basic facts about parallels and transversals (but not anything about
parallelograms beyond the definition), prove that the diagonals of a parallelogram
ABCD bisect each other.
3. Barycentric coordinates (25 points)
Given a triangle ABC and points A', B', C' on the sides, as shown, and lines AA', BB', CC' concurrent at P.
Assume that AC'/C'B = 1/3 and CB'/B'A = 2/3. THE FIGURE IS NOT TO SCALE!
(a) Find the barycentric coordinates (x, y, z) of P with respect to triangle ABC.
(b) Find the ratio BA'/A'C.
(c) Find the ratio PA'/PA.
(d)
What are the barycentric coordinates of C'?
4. Polyhedral Volume (10 points)
The setup:
Suppose T is a tetrahedron ABCD with volume (T) = 1 cm^3.
Form a truncated polyhedron O from T by cutting off 4 small tetrahedra, one at each vertex, according to the following recipe:
At vertex A, let B1, C1, D1 be the points on edges AB, AC, AD, respectively, with AB1/AB = AC1/AC = AD1/AD = 1/3. Cut off the tetrahedron AB1C1D1.
Use the same method (and the same ratio 1/3) to cut off a tetrahedron at each of B, C, D.
Question
· What is volume of the polyhedron O? ____________________
· Show reasoning.
5. Describing a Platonic Solid (15 points)
Describe the regular dodecahedron, specifically:
(a) What is the shape of the faces? _____________________________
(b) How many faces meet at a vertex? ____________________
(c) How many faces are there in total? ___________________
(d) How many edges? ____________________
(e) How many vertices? ____________________
(f) What is the dual of the dodecahedron? ____________________________
6. Plotting Points Given Barycentric Coordinates (15 points)
Given their barycentric coordinates (x, y, z) with respect to triangle ABC in the figure, indicate the location of points P, Q, and R. If the point is an existing point in the figure, circle it and label. If it is not an existing point, draw it in (approximately by eyeballing, not constructing) and label.
(a) P is the point with coordinates (1/6, 3/6, 2/6)
(a) Q is the point with coordinates (1/2, 1/4, 1/4)
(b) R is the point with coordinates (1, -2/3, 2/3)
