Download Math 371 Modern Geometries Exam Info Winter 2013 The exam is

Math 371 Modern Geometries
Exam Info Winter 2013
The exam is Friday, April 19, in Annex 23A, 9-11:30 (but you can have extra
time if needed, up to 12). There will be two questions that you haven’t
seen before, but which use ideas that we have seen. Other questions are
similar to in-class examples, assignment, test or old exam questions (see
website for old exams). You can bring in a formula sheet (regular letter
size paper, both sides)
Sections on the exam: 1.1-1.8, 6.1, 7.1-7.4, 7.6-7.12, fractals and topology
(see class notes)
In the following list P means with proof, NP means don’t worry about the
proof.
Chapter 1:
 Pythagoras’s Theorem (P)
 The converse of Pythagoras’s theorem (P)
 The five axioms of Euclidean geometry- especially the fifth one
 The definition of distance, line segment, line
 Definition of isometry
 Definition of congruence
 You should know the extra 3 axioms, or at least that there exist
translations, rotations and reflections
 You should know the three congruent triangle theorems- SSS (P), SAS
(P), ASA (NP)- so how to phrase the theorem. Also need to know
the lemma that two circles intersect in zero, one or two points (NP)
 Definitions of direct isometry, translation, rotation and reflection, to
show direct isometry it is enough to show orientation of one triangle
is preserved, also if the orientation of one triangle is changed then
the orientation of all triangles is changed
 In Euclidean geometry, the sum of the angles of a triangle is equal
to 180 (P), Pons asinorum (P), also the converse (P)
 Star Trek Lemma (P), tangential form (P), bowtie lemma (P)
 Similar triangles- def is AAA, but enough to have AA or side ratioangle-side ratio
 Power of the point (P)- for points inside or outside the circle,
tangential version (P)
 The golden ratio, golden rectangle and golden triangle
Chapter 7 (and 6.1):
 Stereographic Projection (how to find images of points)
 Arclength element in PUHP
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Vertical lines in PUHP (NP)
Definition of isometry and how to check using arclength elements
Inversion in the circle: know how to find images of points/objects,
inversions preserve angles (NP)
How to draw line segments between arbitrary points
Using Fractional Linear Transformations (FLTs), how to determine if
one corresponds to an isometry
How to use cross-ratios to find isometries with certain properties
How to find translations, and how to check if it is parabolic or
hyperbolic
How to find a rotation or the center of a rotation
How to find an isometry for a reflection
How to find the length between 2 points
The axioms for Hyperbolic Geometry- especially the fifth
Fractals and Topology
 What it means for an object o be self-similar, how to find the selfsimilarity dimension
 Be able to draw the first few stages of a fractal given a rule and an
initial object
 The Cantor set and its relationship to ternary numbers
 How to find the first few stages of a Sierpinski relative given the three
maps (the maps would be explained to you- ex cde means first
map includes rotation by 180, second is rotation by 270 and third is
horizontal reflection)
 Be able to classify objects as being topologically equivalent or not
(like the alphabet exercise)