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Motion Geometry Part I

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Motion Geometry Part I
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Motion Geometry Part I

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  1. Motion GeometryPart I Solve Problems Organize Analyze Geometry Reason Model Compute Measure Communicate

  2. Transformations

  3. Transformations A transformation is a change in position, shape, or size of a figure.

  4. Example • Putting together a jigsaw puzzles is an example motion geometry in action and can be used to illustrate transformations.

  5. How does it work? • When you get a new jigsaw puzzle, you dump all the pieces out of the box onto a table.

  6. What do you do next? • You probably turn the pieces over so that they are all face up. • You might adjust the angle of the pieces. • You might slide a piece across the table. Each of these represents a transformation of the piece.

  7. Each of these translations has a special name. • Flipping the piece over is an example of a reflection (or flip). • Changing the angle of the piece is an example of a rotation (or turn). • Moving the piece across the table is an example of a translation (or slide).

  8. Isometry If a figure and the figure formed by transforming it are congruent, the transformation is called an isometry. If a transformation is an isometry, the sizeand shape of the figure remains the same and only the position of the figure changes.

  9. Fact • In an isometry distance is also preserved. Since the figures before and after the transformation are congruent, the distance between corresponding points does not change.

  10. What do you think? • Is flipping a puzzle piece an isometry?

  11. Solution • Yes. • The image and object are congruent. • Shape, size, and distance are preserved.

  12. What do you think? • Is turning or rotating a puzzle piece an isometry?

  13. Solution • Yes. • The image and object are congruent. • Shape, size, and distance are preserved.

  14. What do you think? • Is sliding a puzzle piece across the table an isometry?

  15. Solution • Yes. • The image and object are congruent. • Shape, size, and distance are preserved.

  16. Orientation The orientation of an object refers to the order of its parts as you move around the object in a clockwise or a counter-clockwise direction.

  17. Example:What is the orientation of the giraffe’s nose, ears, and tail starting with the nose and going clockwise?

  18. Solution • Nose – Ears – Tail

  19. What do you think? • If the giraffe is slid to a new position, does its orientation change?

  20. Solution No. It is still nose – ears – tail.

  21. What do you think? • If the giraffe is turn or rotated, does its orientation change?

  22. No. The orientation of the giraffe does not change. In both cases the order is nose – ears – tail.

  23. What do you think? • If the giraffe is reflected, does its orientation change?

  24. Yes the orientation changes in a reflection. Starting at the nose and going clockwise, its orientation is now: nose – tail – ears.

  25. Translations

  26. Translations  A translation is a transformation that moves all points of a figure the samedistancein the same direction.

  27. Translations In order to translate a figure you need to know two things. • How far will it be translated? • In what direction will it be translated?

  28. Fact A translation (or slide) preserves • size, • shape, • distance, and • orientation.

  29. Terminology In a transformation, the given figure is called the preimage and the transformed figure is called the image. Points on the image that correspond to points on the preimage are labeled similarly but with primes. A transformation is said to map a figure onto its image.

  30. Try It • Choose one of your attribute pieces. • Draw an arrow on your paper. • Place your attribute piece at the end of the arrow. Trace around it. • Use the arrow (vector) to represent the direction and distance, translate your attribute piece. Trace around it.

  31. Did you align a vertex or a side at the foot of the arrow? Foot of arrow Foot of arrow

  32. B' u image B preimage C' A' C A • It is more difficult to translate using a vertex than a side. You can slide the side along the arrow. BUT

  33. u B A C Do not rotate as you slide. If you rotate with the vertex alone it is difficult not to rotate as well as slide the figure.

  34. Translating Polygons by Construction • A polygon can be translated by translating its vertices and then connecting these points. • So it is only necessary to know how to translate a point in order to know how to translate a polygon.

  35. Example • Translate point A according to the given vector.

  36. Plan • Construct a parallelogram with the ends of the arrow (vector) and point A as three of its vertices. • The fourth vertex will be, the required image.

  37. Use your compass to measure the length of the vector • Copy this length from point A in the general direction of the arrow.

  38. Using your compass measure the distance from the end of the arrow to point A. • Copy this distance from the head of the vector. • The intersection of arcs is the fourth vertex.

  39. Try It • Draw a line segment on your paper and a vector (arrow) near it. • Transform the segment according to the vector.

  40. Solution:

  41. Application of a translation

  42. Frieze Patterns • A frieze pattern is a pattern that repeats itself along a straight line. The pattern may be mapped onto itself with a translation. • Wallpaper borders are practical applications of frieze patterns. • Frieze patterns can be found around the eaves of some old buildings.

  43. Translation with dot paper • Translations on dot paper can be accomplished using the slope of the translation vector.

  44. Try It

  45. 4 3 Solution image preimage

  46. Mathematical Notation of a Translation • A translation, T, that moves an object h units to the right or left and k units up or down is T(h,k). • This may also be written using the following notation. T: (x, y) (x + h, y + k) • If h is positive, the object moves to the right. • If h is negative, the object moves to the left. • If k is positive, the object moves up and • If k is negative, the object moves down.

  47. Try It Where would the point (2, -3) be located after the translation described by T(-5,7)?

  48. Solution The point moves left 5 and up 7 so T(-5, 7) (2,-3) (2 - 5,-3 + 7) The point moves from (2, -3) to (-3, 4) under this translation.

  49. Try It • Translate the triangle using T(3, -4).

  50. image Solution