Transformations 3

1. Transformations 3 Translations Reflections Rotations

2. Objectives Students will be able to: Identify the translation on a given figure. Predict the results a transformation will have on a figure. Show that the rigid transformation preserves congruence.

3. Standard • MAFS.912.G-CO.2.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

4. Vocabulary • Rigid Motion • Transformation • Pre-image • Image • Translation • Rotation • Reflection • Isometry • Vector

5. Rigid Motion No geometrical object - no point, no line, no polygon, nothing - is fixed in place. We may move any geometrical object as we wish. One special kind of motion - the kind most important to us - is rigid motion. Let us first illustrate and name the types of rigid motion. After we'll give the official definition of rigid motion.

6. Type One Name: Translation

7. Type Two Name: Reflection

8. Type Three Name: Rotation

9. Preimage and Image Notice that in each example, we had a figure before or pre transformation and a figure after or post transformation. The pre-move transformation figure we call the preimage. The post-transformation figure we call the image. Notice too that each point of the preimage corresponds to a point in the image. Typically we use the prime (') to show correspondence. Thus if A is a point on the preimage, A' will be the point on the image to which it corresponds.

10. Prime Correspondence Illustrated

11. Translation Defined A translation "slides" an object a fixed distance in a given direction.  The original object and its translation have the same shape and size, and they face in the same direction.  A translation creates a figure that is congruent with the original figure and preserves distance (length) and orientation (lettering order).  A translation is a direct isometry. 

12. Illustration of Translation

13. Teacher should work translation problems 1-4 on transformations worksheet 1 on board while students work the same problems on their paper or whiteboard.

14. Reflection Defined Reflection is a rigid motion, meaning an object changes its position but not its size or shape. In a reflection, you create a mirror image of the object. There is a particular line that acts like the mirror. In reflection, the object changes its orientation (top and bottom, left and right). Depending on the location of the mirror line, the object may also change location.

15. Thus Thus after a reflection, the line of reflection is the perpendicular bisector of the segments which connect points on the preimage with the image points to which they correspond. I mean: the line of reflection is perpendicular to each line segment which connects preimage and image points which correspond, and the mirror cuts each such segment in half.

16. Reflection over a vertical line

17. Reflection over a Horizontal line

18. Reflection over the line y=x

19. Reflection over a line

20. Teacher should work Reflection problems 1-4 on Transformation worksheet 1 on the board while students work along on their papers or whiteboards.

21. Rotation Defined Rotation is a rigid motion, meaning an object changes its position but not its size or shape. In a rotation, an object is turned about a "center" point, through a particular angle. (Note that the "center" of rotation is not necessarily the "center" of the object or even a point on the object.) In a rotation, the object changes its orientation (top and bottom). Depending on the location of the center of rotation, the object may also change location.

22. Rotation 180 degrees about point D

23. Rotation 270 degrees counter clockwise about point F.

24. Rotation 90 degrees counter clockwise about point F inside the figure.

25. Teacher should work Rotation questions 1-6 on Transformation worksheet 1 on board while students work the same problems on their papers or whiteboards.

26. Rigid Motion Defined We have named and defined the three kinds of rigid motion. They are translation, reflection, and rotation. We now say that a rigid motion is one that consists of a sequence of reflections, rotations, and translations. Do note that the sequence can have as many (or few) steps as you like. It can be one reflection, one rotation, or one translation. It can be any number of reflections, rotations, and translations in any order. In each of these cases the size and shape of the figure remains the same.

27. Isometry What did you notice about the pre-image and image in each transformation? A transformation in which the pre-image and image are congruent is an Isometry. Translations, Reflections, and Rotations are all Isometries.

28. Illustration

29. Vector This is a Vector A vector has magnitude (how long it is) and direction (the way the arrow points). Vectors are used in many ways, one way is to translate a figure.

30. How to Describe a Translation When we describe a translation, we must name both the object that we slide and the vector along which we slide. Thus to describe the transformation below, we say: slide triangle ABC along the vector v.

31. How to Describe a Reflection When we describe a reflection, we must name both the object reflected and the line over which it is reflected. Thus to describe the transformation below, we say: reflect triangle ABC over line m.

32. How to Describe a Rotation When we describe a rotation, we must name both the object rotated and the point about which it is rotated. If possible, we must say whether the rotation was clockwise or counterclockwise. If it is not stated it assumed to be counterclockwise. If possible, we must give the number of degrees turned. Thus to describe the diagram of the next slide, we can say: triangle ABC was rotated counterclockwise about point D.

33. The Rotation

34. A Curious Feature of Rotations For a translation, there can be no doubt about its direction. For a reflection, there can be no doubt about its line of reflection. But for a turn, we can always say either ‘clockwise’ or ‘counterclockwise’. For instance, a clockwise turn of 120° corresponds to a counterclockwise turn of 240°.

35. A Final Note: Terminology We have chosen the names ‘translation’, ‘reflection’ and ‘rotation’ for the three kinds of rigid motions. Others use ‘slide’, ‘flip’ and ‘turn’. If it isn’t clear what’s what: Translation = Slide Reflection = Flip Rotation = Turn