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Transcript
Chapter 11: Extending Geometry 11.1 Transformations 11.1.1. Translations, Rotations, and Reflections 11.1.1.1. Slides or Translations 11.1.1.1.1. If each point P in a plane corresponds to a unique point in the plane, P’, such that directed segments PP” is congruent and parallel to the directed segment AB, then the correspondence is called the translation associated with the directed segment AB and is written TAB 11.1.1.1.2. TAB(P) = P’ A C D A' C' D' B B' 11.1.1.1.3. 11.1.1.2. Turns or Rotations 11.1.1.2.1. Consider a point C and an angle with a measure from 180 to 180. If each point P in the plane corresponds to a unique point in the plane, P’, such that mPCP’ = and PC = P’C, then the correspondence is called a rotation, with center C and angle , and is written RC, 11.1.1.2.2. RC,(P) = P’ E F E' H H' F' G 11.1.1.2.3. RG,-45o K J K' L L' J' I I' M 11.1.1.2.4. RM,-45o N O N' Q Q' R O' P' P RR,-60o 11.1.1.2.5. 11.1.1.3. Flips and Reflections 11.1.1.3.1. If each point on line l corresponds to itself, and each other point P in the plane corresponds to a unique point P’ in the plane, such that l is the perpendicular bisector of PP' , then the correspondence is called the reflection in the line l, and is written Ml 11.1.1.3.2. Ml (P) = P’ U T V T' U' V' S S' 11.1.1.3.3. 11.1.1.4. Identifying transformations – see example 11.1 p. 598 11.1.1.5. Combinations of motions 11.1.1.5.1. glide-reflection 11.1.1.5.1.1. Slide + Flip 11.1.1.5.1.2. Translation + Reflection X W Z Y W'' Y '' Z'' X'' 11.1.1.5.1.3. 11.1.1.6. Transformations and congruence 11.1.1.6.1. no change in size or shape 11.1.1.6.2. all objects translated are congruent 11.1.1.6.3. congruence transformation 11.1.1.6.3.1. preserves distance 11.1.1.6.3.2. other characteristics 11.1.1.6.3.3. also called an isometry 11.1.1.6.4. Two figures are congruent if and only if there exists a translation, rotation, reflection, or glide-reflection that sets up a correspondence of one figure as the image of the other 11.1.2. Connecting Transformations and Symmetry 11.1.2.1. Symmetry in the plane 11.1.2.1.1. Reflectional symmetry 11.1.2.1.2. line of symmetry 11.1.2.1.3. n rotational symmetry 11.1.2.1.4. center of rotational symmetry 11.1.3. Transformations that Change Size 11.1.3.1. size transformations 11.1.3.1.1. If point O corresponds to itself, and each other point P in the plane corresponds to a unique point on OP such that OP’ = r(OP) for r > 0, then the correspondence is called the size transformation associated with center O and scale factor r, and can be written S O, r 11.1.3.1.2. SO, r (P) = P’ 11.1.3.2. Size transformations and similarity 11.1.3.2.1. Two figures are similar if and only if there exists a combination of an isometry and a size transformation that generates one figure as the image of the other 11.1.3.2.2. 11.1.4. Transformations that Change Both Size and Shape 11.1.4.1. Topological transformations 11.1.4.1.1. shrinking 11.1.4.1.2. bending 11.1.4.1.3. stretching 11.1.4.1.4. If one figure can be transformed to another by a topological transformation then they are topologically equivalent 11.1.5. Problems and Exercises p. 669 11.1.5.1. Home work: 7-10, 11, 12, 16, 17, 20, 22, 23b, 25, 37