Mathematics 138
What you should know how to do for the midterm exam

I use this list to construct the midterm.  First I write four free-response questions (one each from general topics A, C, D, and E, then I write 12 multiple-choice items on the topics that were not covered by these problems.

A. Finite affine geometry
1. State, recognize, or explain the axioms for an affine plane.
2. Draw or apply diagrams to show how the axioms of Desargues and Pappus apply to a specified "affine faculty senate," especially to proving key results about addition or multiplication.
3. Prove the following theorems about affine planes, given either a diagram to illustrate the proof, or a proof that requires a diagram, or a proof with key steps omitted:
a. The result of addition or multiplication does not depend on the choice of "auxiliary instructor" in the special case where the two auxiliary instructors are on the same committee as the appropriate identity.
b. Addition or multiplication is associative.
c. Addition or multiplication is commutative.
d. The equation AC = E or A+C = E can be solved uniquely for A or C, given the other, as long as division by 0 is not required.
4. Given a list of instructors and committees for an "affine senate" and coordinates for the members of two intersecting committees, determine
a. the equation satisfied by a specified committee.
b. the committee that satisfies a specified linear equation.
c. the coordinates of a specified instructor.
d. the instructor with specified coordinates.

B. Incidence properties of Cayley-Klein geometries
1. Fill in all or part of the 3x3 grid enumerating the nine possible Cayley-Klein geometries in terms of whether the measurement of distance and angle is elliptic, parabolic, or hyperbolic and assigning a name to each geometry.  For each geometry, given a line m and a point P not on m, specify how many points on m cannot be joined to P by a line and how many lines through P do not intersect m.
2. Given a model of one of the nine Cayley-Klein plane geometries, use incidence properties (number of lines through a given point parallel to a given line and number of points on a line that cannot be joined to a specified point) to determine the nature of the geometry and give it a name. (You are not responsible for knowing the details of the models that were listed in the handout.)
3. Given the definition of distance or measure of angle in a geometry, identify it as elliptic, parabolic, or hyperbolic.
4. Explain or apply the connection between the elliptic, parabolic, or hyperbolic measurement of angle and the measurement of the area of an appropriate portion of a circle, triangle, or hyperbola.
5. State and apply the definition of cross-ratio for the segments determined by four points on a line or the angles determined by four rays with a common vertex, and use the law of sines to prove that cross-ratio is the same for any line cutting four rays with a common vertex.
6. Use the law of sines to prove theorems of Euclidean geometry such as the theorems of Menelaus and Ceva. (You are not responsible for memorizing the statements of these specific theorems.)

C. Isometries of the Euclidean plane
1. Write or identify the 2x2 matrix for a specified reflection about a line through the origin or rotation or rotation about the origin.
2. Write the 3x3 matrix that represents a specified reflection, translation, rotation, or glide reflection.
3. Given a Euclidean isometry, represent it as the product of three or fewer reflections, and prove or apply theorems that are used to accomplish this, especially
  a. The product of three reflections in lines through a single point equals a reflection in a suitably chosen line though that point.
  b. The product of three reflections in parallel lines equals a reflection in a suitably chosen line parallel to the given line.
  c. The product of two half-turns is a translation.
  d. A glide reflection followed by a reflection equals a translation or a rotation.
4. Given an isometry that is specified by a 3x3 matrix or verbally as a composition of isometries, determine its fixed points and fixed lines
5. Construct an isometry with specified fixed points or lines, either by writing down a 3x3 matrix or by giving a verbal description of the isometry.
6. Given a matrix or a sequence of isometries that specifies a glide reflection, determine the axis of the glide reflection.
7. Construct or identify the direction vector for the bisector of a specified angle in the Euclidean plane.
8. Construct graphically the result of performing successive Euclidean isometries (by expressing them cleverly as products of reflections).

D. Affine transformations of the Euclidean plane.
1. Prove that an affine transformation that fixes three noncollinear points must be the identity and show that there is a unique affine transformation that carries three specified noncollinear points into three other specified noncollinear points.
2. Given two triangles in the plane, construct the affine transformation that maps one into the other.
3. Prove that any affine transformation is a collineation.
4. Carry out steps of the proof that any collineation of the Euclidean plane is an affine transformation, and indicate how this proof can fail to apply to other affine planes.
5. Determine what isometries are required in order to make specified triangles congruent in the Euclidean plane.

E. Galilean geometry
1. Interpret the "points" and "lines" of Galilean geometry in terms of events and uniform motions along a line, and provide a physical interpretation of the isometries of Galilean geometry.
2. Write down or identify the 3x3 matrices that represent specified Galilean isometries.
3. Determine ordinary or special distances or angles for points or lines in the Galilean plane.
4. State, prove, and apply the Galilean “triangle equality” and the Galilean Law of Sines.
5. Construct the bisector of the Galilean angle between two lines.
6. Given a theorem of Galilean geometry, formulate its dual theorem.  Given the proof of the theorem, construct a proof for its dual. 
Note: area and co-area, doublets, Galilean dilatations and compressions, parabolas and cyclic rotations will be added to this list for the final exam, but they are not relevant to the midterm.