1 OBJECTIVE The student will learn the basic concepts of translations, rotations and glide reflections.

2 Reflections are the building blocks of other transformations. We will use the material from the previous lesson on line reflections to create new transformations. IMPORTANT

3 A translation, is the product of two reflections R (l) and R (m) where l and m are parallel lines. DEFINITION

4 That is R (l) l d 2d m R (m)

5 Direction Translation Distance

6 If you draw a figure on a piece of paper and then slide the paper on your desk along a straight path, your slide motion models a translation. In a translation, points in the original figure move an identical distance along parallel paths to the image. In a translation, the distance between a point and its image is always the same. A distance and a direction together define a translation.

7 an isometry, and is a direct transformation, and has no fixed points. A translation is Theorems Proof: Use the components of a translation.

8 Given two parallel lines you should be able to construct the translation of any set of points and describe that translation.

9 Play Time Consider the y-axis and the line x = 2 as lines of reflection. Find the image of the  ABC if A (- 3, -1), B (- 3, 3) and C (0, 3) reflected in the y-axis and then x = 2.

10 Play Time Consider the y-axis and the line x = 2 as lines of reflection. Find the image of the  ABC if A (- 3, -1), B (- 3, 3) and C (0, 3) reflected in the x-axis and then x = 2. x = 2 A B C A’ B’ C’ A” B” C” Write this transformation with algebraic notation. i.e. x’ = f (x, y)y’ = f (x, y) x y

11 Given a translation you should be able to construct the two lines whose reflections produce the necessary transformation. They are not unique.

12 Play Time Consider the point P and its image P”, find two lines l and m so that the reflection in l followed by the reflection in m moves P to P”. P P”

13 Play Time Consider the point P and its image P”, find two lines l and m so that the reflection in l followed by the reflection in m moves P to P’. P P” l l is any arbitrary line perpendicular to PP’. m m is a line parallel to l and the distance from l to m is ½ the distance from P to P”. P’ H

14 You should be able to do the previous construction using a straight edge and a compass.

15 Rotations θ O

16 A rotation is the product of two line reflections R (l) R (m), where l and m are not parallel. The center of the rotation is O = l  m. The direction of the rotation is about O from l toward m, and the angular distance of rotation is twice the angle from l to m. Definition

17 That is R (O, θ ) = R (l) R (m) l m θ θ /2

18 an isometry, and is a direct transformation, and has one fixed points. A rotation is Theorems Proof: Use the components of a rotation.

19 Given two lines that are not parallel you should be able to reflect a set of points in the first line and then again in the second line.

20 Play Time Reflect  ABC in l and then in m. l m θ

21 produce the necessary transformation. They are not unique. Given a rotation you should be able to construct the two lines whose reflections

22 m θ/2 l H

23 You should be able to do the previous construction using a straight edge and a compass.

24 Lemma The only isometry that has three noncollinear fixed points is the identity mapping e, that fixes all points. What about three collinear points?

25 Lemma: The only isometry with three fixed points is the identity mapping e. 25 Three fixed points; A, B, C.Prove: T = e. (1) T(A)=A’, T(B)=B’, T(C)=C’ Given. (2) AP = AP’, BP = BP’, CP = CP’,Def of isometry (3) P = P’Congruent triangles. (4) T is the identity map e.Def of identity map. What is given?What will we prove? Why? QED A = A’ P C = C’ B = B’ Let P be any point in the plane. We will show P’ = P Note: You need all three of these distances. Why?

26 Glide Reflections

27 Definition A glide reflection, G (l, AB) is the product of a line reflection R (l) and a translation T (AB) in a direction parallel to the axis of reflection. That is, AB ‖ l.

28 This combination of reflection and translation can be repeated over and over: reflect then glide, reflect then glide, reflect then glide, etc. An example of this is the pattern made by someone walking in the sand. This calls for a field trip to Ocean City. The same line of reflection is used to reflect each figure to a new position followed by a glide of a uniform distance.

29

30 Theorems A glide reflection is an isometry, and is an opposite transformation, and there are no invariant points under a glide reflection. Proof: Use the components of a glide reflection.

31 * Transformations in General * Theorem. Given any two congruent triangles, ΔABC and ΔPQR, there exist a unique isometry that maps one triangle onto the other. AB C PQ R

32 Given three points A, B, and C and their images P, Q, and R, there exist a unique isometry that maps these points onto their images. Theorem 1 Proof: It will suffice to show you how to find this isometry. It will be the product of line reflections which are all isometries.

33 Focus – You will need to do this for homework and for the test.

34 If it is a direct transformation then it is a translation or a rotation. We can use the methods shown previously in “Rotation” or “Translation”.RotationTranslation Theorem 1 If it is an indirect transformation the following method will always work.

35 Prove that given three points A, B, and C and their images P, Q, and R, there exist a unique isometry that maps these points onto their images. A B C P Q R B’ C’ A’ Theorem 1 Do you see that it is an indirect transformation?

36 The previous proof shows that every isometry on the plane is a product of at most three line reflections; exactly two if the isometry is direct and not the identity. Theorem 2: Fundamental Theorem of Isometries.

37 A nontrivial direct isometry is either a translation or a rotation. Corollary

38 Corollary: A nontrivial direct isometry is either a translation or a rotation. What do we know? It is a direct isometry. It is a product of two reflections. If the two lines of reflection meet then it is a rotation! If the two lines of reflection do not meet (they are parallel) it is a translation!

39 A nontrivial indirect isometry is either a reflection or a glide reflection. Corollary

40 Corollary: A nontrivial indirect isometry is either a reflection or a glide reflection. What do we know? It is an indirect isometry. It is a product of one or three reflections. If one it is a reflection! If the three line reflection then it could be either a reflection or a glide reflection!

41 Assignment Bring graph paper for the next three classes.

42 Assignments T3 & T4 & T5