Geometry Transformed: Euclidean Plane Geometry Based on Rigid Motions

Description

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. 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