Mathematics 138
Review questions for the midterm exam

Disclaimer: these are not all as carefully crafted as real exam questions should be.
The question numbers match the items in the "things you should know how to do" document.
A1. The "theorem of Pappus" axiom for a finite affine plane is needed to prove
a. more than one of the following.
b. addition is commutative.
c. multiplication is commutative.
d. addition is associative
e. multiplication is associative.
A2 and A3. Here is a diagram showing some points in an affine plane

Use it to prove that multiplication of points A and C gives the same result whether B or B' is used as an auxiliary point.  In your proof, point out carefully where you use the uniqueness of parallels and where you use the Desargues axiom.
A2. Using the listing for the large affine faculty senate, take A = Alice, B = Helen, C = Terry
A' = Emily, B' = Wally C' = Kevin
a. By listing committees that are concurrent or parallel, as appropriate, verify that the hypotheses of Desargues' Theorem are satisfied.  
b. State in terms of committees the conclusion of Desargues' Theorem for this case, and verify that it holds according to the faculty senate data.
A4 In the small faculty senate, the education committee is chosen as the x axis, with Eva as additive identity 0 and Hal as multiplicative identity 1.  The library committee is chosen as y axis, again with Eva as additive identity 0 and Cal as multiplicative identity 1.
a. What are the coordinates of Fay?
b. What instructor has coordinates (1,2)?c. What committee satisfies the equation x + 2y = 1 (coefficients are integers mod 3)
d. What equation is satisfied by the budget committee?

B1 a. What name is given to a geometry that uses hyperbolic measure of angle and parabolic measure of length? 
b. Consider a line m of such a geometry and a point P that does not lie on m.  How many points on m cannot be joined to P by a line of the geometry?  How many lines through P fail to intersect with m?
B2 Consider a model of a Cayley-Klein geometry whose "points" are the points of the Euclidean plane with y > 0 and whose lines are either Euclidean vertical lines or Euclidean semicircles with center on the x axis.  Analyze this model in terms of incidence properties of points and lines, and thereby determine what sort of measure it must use for distance and angle.
B3 A geometry in which the length of a line cannot exceed p and angles are defined in terms of cross-ratio would be called
a. Galilean
b. hyperbolic
c. doubly hyperbolic
d. cohyperbolic
e. Minkowskian
B4 For the pair of rays that intersect at (0,0) in the diagram, show what areas would be used to define the elliptic, parabolic, and hyperbolic measure of the angle between the rays. (You will have to draw some extra curves on the diagram.)

B5 What is the numerical value of the cross-ratio CR(A, B, I, J) for the four points shown in the diagram? 

B6 In the diagram, E is the midpoint of AC and AD = 2 BD.  Prove directly from the Law of Sines (applied to six triangles)that CF = 2 BF. (Hint: call the angles at P a, b, and g ,and use them along with the angles at D, E, and F.

C1. Write down a 2x2 matrix A that represents a reflection in the line y = 4x/3 and a matix B that represents reflection in the line x = 0.  Compute the product AB and interpret it as an isometry of the Euclidean plane.

C2. Construct a 3x3 matrix that represents a 90 degree counterclockwise rotation of the Euclidean plane about the point (2,1).  State how this isometry could be expressed as the product of two reflections.

C3. What single reflection is the same as reflection in the line x = 2, followed by reflection in the line x = -1, followed by reflection in the line x = 0?

C3. Describe verbally, in full detail, the isometry of the Euclidean plane represented by the matrix below. Is it a translation, rotation, reflection, or glide reflection?  Describe a sequence of three or fewer reflections that leads to this isometry.

C4. Suppose that a counterclockwise rotation of 45 degrees about the origin is followed by a clockwise rotation of 45 degrees about (2,2).  What are the fixed points and fixed lines (if any) of the resulting isometry?
C5. Write down a 3x3 matrix for a Euclidean isometry (not the identity) that holds fixed every point on the line 
y = 1 + 2x. 
Describe this isometry in words, and state what other lines are held fixed.
C6. Consider an isosceles right triangle ABC whose vertices are A = (1,0), B = (0,1), C = (0,0)
a. Show that the effect of reflection in the y axis, followed by reflection in the x axis, followed by reflection in the line x + y = 1, is a glide reflection.  Draw a diagram showing the axis of this glide reflection.
b. If this glide reflection is followed by another reflection in the y axis, what is the resulting isometry?  How does this illustrate what you know about sequences of Euclidean reflections in general?
C7. By using the definition of Euclidean angle in terms of scalar products, show that the ray through the origin along the direction (2,1) bisects the angle between the ray through the origin along (1,0) and the ray through the origin along (4,3)
C8. A half- turn about A is followed by a 45-degree counterclockwise rotation about B.
Determine graphically what point is held fixed. 

D1 Given that an affine transformation that holds fixed three noncollinear points is the identity, prove that the affine transformation T that carries triangle ABC into triangle A'B'C' is unique.
D2 Construct the 3x3 matrix representing the affine transformation that maps (1,0) into (0,4), (3,0) into (0,2), and (3,1) into (2,2).  (Hint: a diagram makes this easier)
D3 Suppose that point C lies on the segment AB and that AC = 2 BC.  An affine transformaton carries A into A' and B into B'.  Prove that it carries C into a point C' that is collinear with A'B' and that A'C' = 2 B'C'.
D4. Suppose that f(s+t) = f(s)f(t) and that f(s+t) = f(s)+f(t).  Prove that f(-2) = -2.
D5 Given the two congruent triangles shown in the diagram, describe a sequence of reflections that moves ABC to the position of A'B'C'

E1. State in physical terms (using motion along a straight line) the equivalent of the statement that any two points in the Galilean plane determine a unique line.
E2. For which geometry (Euclidean or Galilean) does the matrix
represent an isometry? Determine the fixed points and fixed lines, if any, of this isometry, then describe it in geometrical terms as simply as you can (for full credit, do not just describe it as a sequence of two isometries)
E3 With the aid of a diagram, define the distance between a point and a line in the Galilean plane.

E5 The vertices of a Galilean triangle are A = (0,1) B = (2,3), C = (3,10).  Where does the bisector of the angle at A intersect the special line x = 1?
E4 Prove that the measure of the largest angle of a Galilean triangle equals the sum of the measure of the other two angles. State the dual of this theorem.