Geometry Transformed: Euclidean Plane Geometry Based on Rigid Motions

✍ Scribed by James R. King (author)

• Publisher: American Mathematical Society
• Year: 2021
• Tongue: English
• Leaves: 284
• Series: Pure and Applied Undergraduate Texts
• Category: Library

✦ Synopsis

Many paths lead into Euclidean plane geometry. Geometry Transformed offers an expeditious yet rigorous route using axioms based on rigid motions and dilations. Since transformations are available at the outset, interesting theorems can be proved sooner; and proofs can be connected to visual and tactile intuition about symmetry and motion. The reader thus gains valuable experience thinking with transformations, a skill that may be useful in other math courses or applications. For students interested in teaching mathematics at the secondary school level, this approach is particularly useful since geometry in the Common Core State Standards is based on rigid motions. The only prerequisite for this book is a basic understanding of functions. Some previous experience with proofs may be helpful, but students can also learn about proofs by experiencing them in this book--in a context where they can draw and experiment. The eleven chapters are organized in a flexible way to suit a variety of curriculum goals. In addition to a geometrical core that includes finite symmetry groups, there are additional topics on circles and on crystallographic and frieze groups, and a final chapter on affine and Cartesian coordinates. The exercises are a mixture of routine problems, experiments, and proofs. This book is published in cooperation with IAS/Park City Mathematics Institute.

✦ Table of Contents

Cover
Title page
Copyright
Contents
Introduction
Transformations and Secondary Geometry
Advice for Students and Less Experienced Geometers
Advice for Students and Less Experienced Geometers
Information for More Experienced Geometers
Information for More Experienced Geometers
A Chapter Guide for Instructors and Others
A Chapter Guide for Instructors and Others
Acknowledgments
Acknowledgments
Chapter 1. Congruence and Rigid Motions
Rigid Motions
Informal Preview of a Problem Solution
Sameness Properties of Congruence
Exercises and Explorations
Chapter 2. Axioms for the Plane
Incidence Axiom
Distance and the Ruler Axiom
Protractor Axiom and Angles
Plane Separation
Rigid Motions and Lines
The Other Axioms
Exercises and Explorations
Chapter 3. Existence and Properties of Reflections
Deducing the Properties of Reflections
Isosceles Triangles and Kites
Circles and Lines
Light, Angles, and Reflections
Paper Folding and Tools for Construction
Exercises and Explorations
Chapter 4. Congruence of Triangles
Triangle Congruence Tests
Applications of Triangle Congruence
Properties of Rigid Motions
Midpoint Triangle and Angle Sum
Exercises and Explorations
Chapter 5. Rotation and Orientation
Rotations and Double Reflections
Rotation-Reflection Relations
Symmetry at a Point
Orientation of a Plane
Orientation-Preserving and Orientation-Reversing Transformations
Exercises and Explorations
Chapter 6. Half-turns and Inequalities in Triangles
Half-turn Properties
Inequalities with Angles
Circles and Distance to Lines
Reflections and the Triangle Inequality
Exercises and Explorations
Chapter 7. Parallel Lines and Translations
The Euclidean Parallel Postulate
Transversals and Parallel Lines
Parallelograms
Rectangles
Midpoint Figures
Generalizing Parallelograms
Translations as Half-turn Products
Products of Translations
Direction from Translation
Direction and Rotation from Polar Angle
Vectors
Exercises and Explorations
Chapter 8. Dilations and Similarity
Similarity Theorems for Triangles
Right Triangles
The Regular Pentagon and Its Ratios
Ratios, Signed Ratios, and Scale Factors
Transversals of Parallels and Ratios
Parallel Segments and Centers of Dilation
Construction by Scaling Models
Harmonic Division
Composition of Dilations
Circles, Angles, and Ratios
Radical Axis, Intersections, and Triangle Existence
Centers of Dilation and the Midpoint Triangle
Exercises and Explorations
Chapter 9. Area and Its Applications
Areas of Triangles and Parallelograms
Area Proofs of the Pythagorean Theorem
Area and Scaling
Area and the Circle
Affine Relationships and Area
Exercises and Explorations
Chapter 10. Products and Patterns
Products of Rotations
Symmetry and 90-Degree Rotations
Triangles and 60 Degrees of Rotation
Translations and Symmetry
Tessellations and Symmetric Wallpaper Designs
Translations and Frieze Symmetry
Triple Line Reflection Products
Exercises and Explorations
Chapter 11. Coordinate Geometry
Axes and Coordinates
Midpoints, Half-turns, and Translations
Lines, Dilations, and Equations
Euclidean Geometry and Cartesian Coordinates
Perpendicular Lines in the Coordinate Plane
Graphs and Transformations
Unit Circle and Rotation Formula
Complex Numbers and Transformations of the Plane
Barycentric Coordinates
Vectors and Affine Transformations
Axioms and Models
Conclusion
Exercises and Explorations
Bibliography
Index
Back Cover