MA1004: Geometry
School | Cardiff School of Mathematics |
Department Code | MATHS |
Module Code | MA1004 |
External Subject Code | 100405 |
Number of Credits | 10 |
Level | L4 |
Language of Delivery | English |
Module Leader | Dr Timothy Logvinenko |
Semester | Autumn Semester |
Academic Year | 2018/9 |
Outline Description of Module
This module gives an introduction to elementary plane Euclidean geometry. We present this material in a way which emphasises axiomatic approach, logical thinking and rigorous proofs, as well as careful use of diagrams as an aid to understanding problems and finding solutions. In the latter half of the module we also introduce basic notions of spherical geometry, emphasising the differences between it and Euclidean geometry.
Free Standing Module Requirements: A pass in A-Level Mathematics of at least Grade A
On completion of the module a student should be able to
- Understand basic geometrical notions of lines, circles, triangles, distances, angles, tangents, bisects, isometries, similarities, etc.
- Prove geometrical figures to be congruent
- Being able to carry out classical geometrical constructions such as those of an inscribed and a circumscribed triangle
- Visualise non-Euclidean geometries using the geometry on a sphere as an example
- Compute areas and sums of internal angles of basic geometrical figures
How the module will be delivered
27 fifty-minute lectures
5 fifty-minute tutorial classes
Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.
Students are also expected to undertake at least 50 hours private study including preparation of worked solutions for tutorial classes.
Skills that will be practised and developed
Constructing logical proofs based on a set of axioms and previously established results
Rigorous use of diagrams in the course of the proof
Developing geometrical intuition and visualisation in 2 and 3 dimensions
How the module will be assessed
Formative assessment is by means of regular tutorial exercises. Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures and tutorial classes.
The major component of summative assessment is the written examination at the end of the module. This gives students the opportunity to demonstrate their overall achievement of learning outcomes. It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.
The examination paper has two sections of equal weight. Section A contains a number of compulsory questions of variable length but normally short. Section B has a choice of two from three equally weighted questions.
Assessment Breakdown
Type | % | Title | Duration(hrs) |
---|---|---|---|
Exam - Autumn Semester | 100 | Geometry | 2 |
Syllabus content
- Introduction: Euclid's axioms. Distances and angles
- Isometries and congruences.
- Triangle congruences: SAS, ASA and SSS. Isosceles triangles.
- Perpendicular bisects. Distance from a point to a line.
- Bisectors. Inscribed and circumscribed circles of a triangle.
- Spherical geometry: intro. Lines and angles.
- Parallel postulate. Sum of angles of a triangle / n-gon.
- Area. Ceva's Theorem.
- Spherical geometry: triangles on a sphere.
- Similarity. Pythagoras theorem.
Essential Reading and Resource List
Note on "Introduction to Geometry" – To be made available via Learning Central at the start of the module
Background Reading and Resource List
Jennings, G.A. 1994. Modern geometry with applications, Springer-Verlag
Coxeter, H.S.M. Introduction to Geometry, Wiley Classics Library, John Wiley and Sons
Euclid, Elements, Books I, III and IV, various publishers (the original treatise on the subject)