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Geometry

Geometry. 7.1 Rigid Motion in a Plane. Goals. Identify the three basic rigid transformations. Use correct terminology in transformations. Transformations. Reflections Rotations Translations. Transformations. The “original” object is the preimage . The copy is the image. preimage.

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Geometry

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  1. Geometry 7.1 Rigid Motion in a Plane

  2. Goals • Identify the three basic rigid transformations. • Use correct terminology in transformations. Geometry 7.1 Rigid Motion in a Plane

  3. Transformations • Reflections • Rotations • Translations Geometry 7.1 Rigid Motion in a Plane

  4. Transformations • The “original” object is the preimage. • The copy is the image. preimage image Geometry 7.1 Rigid Motion in a Plane

  5. Translation Preimage Image Geometry 7.1 Rigid Motion in a Plane

  6. Reflection Preimage Image Line of Reflection Geometry 7.1 Rigid Motion in a Plane

  7. Rotation Images Center of Rotation Preimage Geometry 7.1 Rigid Motion in a Plane

  8. Labels • If a preimage point is A, then the corresponding image point is A’. • A’ is read “A prime”. • Preimage ABC is transformed to image A’B’C’. Geometry 7.1 Rigid Motion in a Plane

  9. Reflection A A’ C C’ B B’ Preimage Image Geometry 7.1 Rigid Motion in a Plane

  10. Mapping • A transformation is a mapping. • If RST is translated to XYZ, we say “RST is mapped onto XYZ”. • We write RST  XYZ. Geometry 7.1 Rigid Motion in a Plane

  11. Write the mapping for this diagram. B 90° V T 60° 90° 30° 60° A C 30° ABC  ______ VTN N This mapping is a rotation. Geometry 7.1 Rigid Motion in a Plane

  12. CD  AB  D V T C DA  B S R A VR TV RS ABCD  RSTV SRVT BADC  This mapping is a reflection. Geometry 7.1 Rigid Motion in a Plane

  13. Isometry • Iso = The same • Metry = Measure • Isometry means “The same measure”. Geometry 7.1 Rigid Motion in a Plane

  14. Isometries preserve… • segment length • angle measure • parallel lines • distances • Transformations that are isometries are called Rigid Transformations. Geometry 7.1 Rigid Motion in a Plane

  15. Translations, Reflections, and Rotations are ISOMETRIES. Geometry 7.1 Rigid Motion in a Plane

  16. A A' B C B' C' Are all transformations also isometries? Segment length not preserved. Angle measures not preserved. Orientation is preserved. NO! ABC  A'B'C' Geometry 7.1 Rigid Motion in a Plane

  17. The rest of this chapter will be done by construction methods. Geometry 7.1 Rigid Motion in a Plane

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