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GBK Geometry

GBK Geometry. Jordan Johnson. Today’s plan. Greeting Hand in Problem Write-up Lesson: Isometries, Congruence, and Symmetry Study Guide Clean-up. On parallelograms. Roger Shepard, 1981. Isometries / Rigid Motions.

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GBK Geometry

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  1. GBK Geometry Jordan Johnson

  2. Today’s plan • Greeting • Hand in Problem Write-up • Lesson: Isometries, Congruence, and Symmetry • Study Guide • Clean-up

  3. On parallelograms

  4. Roger Shepard, 1981

  5. Isometries / Rigid Motions • A transformation is an isometry (or rigid motion) iff it preserves distances and angles.

  6. Example Isometry Proof:Rotation Preserves Distance

  7. Example Isometry Proof:Rotation Preserves Distance

  8. Example Isometry Proof:Rotation Preserves Distance

  9. Example Isometry Proof:Rotation Preserves Distance 1 2 3

  10. Definitions • Transformations  definition of congruence: • Two figures are congruent iff there is an isometry in which one is the image of the other.

  11. Glide Reflections

  12. Definition • A glide reflection is the composite of a translation and a reflection in a line parallel to the direction of the translation. • In other words: Reflect, then translate, or vice versa.

  13. (Reflection) symmetry

  14. 2-fold Rotational Symmetries

  15. 3-fold Rotational Symmetry Flag of the Isle of Man

  16. Translational Symmetry Other neat examples: autologlyphs and ambigrams.

  17. A Tale of Two Fats Saturated Unsaturated

  18. Translational Symmetry

  19. …that form blocks…

  20. …like Lard.

  21. Unsaturated Fats

  22. Summary • Each isometry has an associated symmetry: • Reflection symmetry with respect to a line: • Figure coincides with its reflection image through the line • Rotation symmetry: • Figure coincides with its image when rotated less than 360° • n-fold symmetry, if the smallest such rotation is 360°⁄n • Translation symmetry: • Figure coincides with its translation image

  23. Homework • Problem for final • Study guide • Any make-up portfolio work • Recommended review for Unit 5: • Proof problems: • Two reflections across parallel lines= translation perpendicular to the lines of reflection. • Prove that the 3 transformations are isometries. • Asgs #37-38.

  24. Clean-up / Reminders • Pick up all trash / items. • Push in chairs (at front and back tables). • See you tomorrow!

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