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Translation: slide Reflection: flip ( mirror) Rotation: turn Dialation: enlarge or reduce

Translation: slide Reflection: flip ( mirror) Rotation: turn Dialation: enlarge or reduce. Geometric Transformations:. Pre-Image: original figure Image : after transformation. Use prime notation. Notation:. A’. C. C ’. B. B’. A. Isometry. AKA: congruence transformation

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Translation: slide Reflection: flip ( mirror) Rotation: turn Dialation: enlarge or reduce

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  1. Translation:slideReflection: flip (mirror)Rotation:turnDialation:enlarge or reduce Geometric Transformations:

  2. Pre-Image: original figureImage: after transformation. Use primenotation Notation: A’ C C ’ B B’ A

  3. Isometry AKA: congruence transformation a transformation in which an original figure and its image are congruent.

  4. Theorems about isometries FUNDAMENTAL THEOREM OF ISOMETRIES Any two congruent figures in a plane can be mapped onto one another by at most 3 reflections ISOMETRY CLASSIFICATION THEOREM There are only 4 isometries. They are:

  5. TRANSLATION:moves all points in a planea given directiona fixed distance

  6. TRANSLATION VECTOR:(also known as translation rule)DirectionMagnitude PRE-IMAGE IMAGE

  7. Translate by the vector (rule) <x, y>

  8. x moves horizontaly moves vertical Translate by <3, 4>

  9. Different notationT(x, y) -> (x+3, y+4)

  10. Translations PRESERVE:SizeShapeOrientation

  11. Reflectionover a line (mirror)

  12. Properties of reflections PRESERVE • Size (area, length, perimeter…) • Shape CHANGE orientation (flipped)

  13. Reflect x-axis: (a, b) -> (a,-b)Change sign y-coordinate

  14. Reflect y-axis: (a, b) -> (-a, b)Change sign on x coordinate

  15. X-axis reflection

  16. Y-axis reflection

  17. Rotations have: Center of rotation Angle of rotation:

  18. Rotated 90 degrees counterclockwise

  19. ROTATIONS PRESERVE SIZE • Length of sides • Measure of angles • Area • Perimeter SHAPE ORIENTATION

  20. Rotations on a coordinate plane about the origin 90 (a, b) -> (-b, a) 180 (a, b) -> (-a, -b) 270 (a, b) -> (b, -a) 360 (a, b) -> (a, b)

  21. Coordinate Geometry rules Reflections x axis (a, b) -> (a, -b) y axis (a, b) -> (-a, b) y=x (a, b) -> (b, a) y=-x (a, b) -> (-b,-a) Rotations Counter Clockwise about the origin 90 (a, b) -> (-b, a) 180 (a, b) -> (-a, -b) 270 (a, b) -> (b, -a) 360 (a, b) -> (a, b)

  22. GLIDE REFLECTIONSYou can combine different Geometric Transformations…

  23. Practice: Reflect over y = x then translate by the vector <2, -3>

  24. After Reflection…

  25. After Reflection and translation…

  26. Symmetry Line Symmetry If a figure can be reflected onto itself over a line. Rotational Symmetry If a figure can be rotated about some point onto itself through a rotation between 0 and 360 degrees

  27. What kinds of symmetry do each of the following have?

  28. What kinds of symmetry do each of the following have? Rotational (180) Point Symmetry Rotational (90, 180, 270) Point Symmetry Rotational (60, 120, 180, 240, 300) Point Symmetry

  29. Dilations A dilation is a transformation (notation   ) that produces an image that is the same shape as the original, but is a different size. A dilation stretches or shrinks the original figure.    The description of a dilation includes the scale factor (orratio) and the center of the dilation.   The center of dilation is a fixed point in the plane about which all points are expanded or contracted.  

  30. Dilations & Scale Factor A dilation of scale factor k whose center of dilation is the origin may be written:  Dk (x, y) = (kx, ky). If the scale factor, k, is greater than 1, the image is an enlargement (a stretch). If the scale factor is between 0 and 1, the image is a reduction (a shrink). (It is possible, but not usual, that the scale factor is 1, thus creating congruent figures.)

  31. Dilations Preserve Properties Preserved (invariant) under a dilation: 1.  angle measures (remain the same)2.  parallelism (parallel lines remain parallel)3.  colinearity (points stay on the same lines)4.  midpoint (midpoints remain the same in each figure)5.  orientation (lettering order remains the same)---------------------------------------------------------------6.  distance is NOT preserved (NOT an isometry)     (lengths of segments are NOT the same in all cases      except a scale factor of 1)

  32. Dilations Create Similar Figures

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