Geometry Transformations: Concepts and Applications

1. Unit 3 Transformations This unit addresses transformations in the Coordinate Plane. It includes transformations, translations, dilations, reflections, rotations, symmetry (including line/plane and rotational), compositions of reflections in parallel lines, glide reflections, and the fundamental theorem of Isometries. It also includes tessellations, symmetry in tessellations, and proportions of dilations.

2. Definitions • Transformations -A change in a geometric figure’s position, shape, or size. • Preimage -This is the original figure before it is transformed • Image -This is the resulting image after it is transformed • Isometry -A transformation in which the preimage and the image are congruent. Examples of transformations which would result in isometry would include: flips, slides, and turns

3. Examples • Flip - • Slide - • Turn -

4. Transformation Maps • A transformation maps a figure onto its image and may be described with an arrow notation: • Prime notation: (‘) is sometimes used to identify image points. In this diagram, K’ is the image of K (K K’) K’ K J Q Q’ J’

5. Reflections • A reflection -or flip- is an isometry in which a figure (preimage) and its image have opposite orientations. Thus, a reflected image in a mirror appears backwards. • Here, triangle ABC is reflected along a line to produce triangle A’B’C’. Since the reflection is isometry, triangle ABC is congruent to triangle A’B’C’ A A’ B C C’ B’

6. Properties of Reflections • Below is a reflection in line r, where the following properties are true: • If a point A is on line r, then the image of A (remember, the “after transformation” creates the image) is A itself -that is, A = A’ In other words, the reflection of a point is the same point • If a point B is not on line r, then line r is the perpendicular bisector of the line segment B/B’ Line of reflection B r A=A’ B’

7. Quiz (50 Points) • What is a Transformation? • What is a Pre-Image? • What is an Image? • What is Isometry? • Give an example of Isometry • What is the name for this symbology? __’__ • What is a reflection? • What is one property concerning reflections and points? • What is the second property concerning reflections and points? • Are a Flip and a Slide the same type of Transformation?

8. In Class Assignment • Worksheet on Reflections • May work in pairs • 10 to 15 minutes

9. Translation • A translation (or slide) is an isometry that maps all points of a figure the same distance in the same direction. Thus you can use a vector to describe a translation. • Mr K’s definition of a vector (for translations): an ordered pair (x,y) that tells me where to go from where I am

10. Translation Example • What if you had a point A, at coordinate (-3,-1), and you were told to use the vector (6,6) to find the image of A (we’ll call it A’). What is the coordinate of A’? • Use the vector in the same manner you use slope. So you would go right 6, and up 6 (rise over run). • The new coordinates would thus be (3,5) A’ (3,5) A (-3,-1)

11. Another way to work with Translations • You may see a pre-image, and an image, and be asked to identify the vector. Again, this is no different that determining slope. • From A (-3,3) to A’ (3,5) you must go right 6, and up 2, so the vector would be (6,2) A’ A (3,5) (-3,3)

12. Homework • Page 128 # 1 - 34

13. Rotations • A rotation occurs around a point -like the center of a circle. These hold true: • You need to know the center of the rotation -we’ll call this a point such as r. After the rotation (the image after the pre-image) this point gets renamed -such as r’ • You need to know the angle of rotation -this will be a positive number of degrees- • You need to know whether the rotation is clockwise or counterclockwise • Unless stated otherwise, rotations in the book are counterclockwise

14. Rotations Continued • Having this information a rotation is conducted by: • Rotating x degrees about a point (such as r) where a transformation occurs such that: • The image of point r is itself (that is, r = r’) • For any point (such as V), rV’ = rV (the distance of each segment) and the measure of VrV’ = x degrees V’ X degrees r and r’ V

15. Drawing a Rotation • Example: Draw the image (after picture) of triangle LOB for a 100 degree rotation about C. • Mark a point (i.e.) C. Draw a line from C to a point on triangle LOB (i.e. O). • Measure 100 degrees on the protractor from that line, and draw another line

16. Drawing a Rotation Continued • Use a compass to measure the distance from C to the point on the line. • Use the compass to mark the same distance on the second line • Do the same for each succeeding corner (mark a 100 degree angle and mark the distance) • It looks like this….

17. Draw a Rotation :) C 100 degrees O B’ O’ L’ L B

18. Assignment • Page 650, 10 - 16 • Page 651 35, 37, 38

19. Do Now • Draw two parallel lines • Draw a triangle • Reflect the triangle about the first line • Then reflect that image about the second line • What transformation that you’ve studied had the same effect as the composition of two reflections in parallel lines?

20. Reflect the triangle about line 3 and then reflect this image about line 4 2) What transformation that you’ve studied has the same effect as the composition of two reflections in intersecting lines?

21. Compositions of Reflections • Theorem 3-1 - A translation or rotation is a composition of two reflections • Theorem 3-2 - A composition of reflections in two parallel lines is a translation • Theorem 3-3 - A composition of reflections in two intersecting lines is a rotation

22. Composition of Reflections in Parallel Lines • This image is reflected on line l. Then it is reflected on line m. The result is a a translation (slide). l m

23. Composition of Reflections in Intersecting Lines • This image is reflected in line A, then line B. The result is a rotation. a b c

24. Fundamental Theorem of Isometries • Theorem 3.4 - In a plane, one of two congruent figures can be mapped onto the other by a composition of at most three reflections. • Isometry Classification Theorem: • Theorem 3-4 - There are only four Isometries. They are: • Reflection • Translation (slide) • Rotation • Glide Reflection

25. Example of a Glide Reflection • Remember -First you glide, then reflect • Find the image of a triangle TEX for a glide reflection where the glide vector is (0,-5) and the reflection line is x = 0. T First, use the vector, and move (glide) down -5. Then, create a reflection based on the x = 0 line. (Numbers have been omitted for clarity of drawing). X E T’ X’ E’

26. Homework • Page 147 #s 1-28

27. Symmetry • A figure has symmetry (is symmetrical) if there is an isometry that maps the figure onto itself. • If the isometry is the reflection of a plane figure, the figure has reflectional symmetry or line symmetry. • This is something most people already know (the idea of symmetry), they just haven’t applied it to geometry. • For example, a face has symmetry, if you draw a line of reflection down the middle -through the nose, chin and so on

28. Symmetry • A figure can have more than one line of symmetry. • Consider a hexagon: How many lines of symmetry are there in a hexagon? It looks like a lot, but there are actually 6

29. Symmetry • How many lines of symmetry are there in this rectangle? There are only 2. Why aren’t there any for the corners?

30. Rotational Symmetry • Rotational Symmetry: Where a figure is it’s own image (after transformation) for some rotation of 180 degrees or less. For example, this equilateral triangle. The angle of rotation is 120 degrees. -if you rotate the triangle 120 degrees, you get the exact same image (isometry). 120 degrees

31. Rotational Symmetry • Would a square have rotational symmetry? If so, what would the angle of rotation be? • Would a rectangle have rotational symmetry? If so, what is the angle of rotation? • Would a hexagon have rotational symmetry? What would the angle of rotation be? • Does this figure have rotational symmetry? If so, what is the angle of rotation?

32. Point Symmetry • A figure that has point symmetry has 180 degree rotational symmetry. • Thinking back to the examples given (Square, Rectangle, Equilateral Triangle, unusual polygon), which had Point symmetry? • NOTE: 3 dimensional objects can have various types of symmetry, including rotational symmetry about a line, and reflectional symmetry in a plane. • A pencil could have rotational symmetry about a line. • A house reflected in a lake of water could have reflectional symmetry in a plane.

33. 3 Dimensional Examples • Imagine a ping pong paddle. • Does it have rotational symmetry? • Does it have reflectional symmetry? • How about a coffee mug? • How about an umbrella?

34. Classwork • Page 664 1- 19

35. Tessellations • Tessellation (or Tiling) occurs when you repeat a pattern of figures that completely covers a plane -without gaps or overlaps. • These tessellations can be created with translations, rotations, or reflections. • You see it in art, nature (honeycombs), and everyday life (tiled floors).

43. Example • Identify the transformation, and the repeating figure • It is a translation (slide) • The figure is a hexagon

44. Determining Figures that Tessellate • Because the figures in a tessellation do not overlap or leave gaps, the sum of the measures of the angles around any vertex must be 360 degrees. If the angles around a vertex are all congruent (they are) then the measure of each angle must be a factor of 360. • To determine whether a regular n-gon will tessellate, you must calculate one angle around it’s vertex. • Then you determine whether the angle is a factor of 360.

45. Determine the measure of one angle to determine Tessellation • A = (180(n-2))/n • For example, determine one angle for an 18-gon • A = (180(18-2))/18 • A = 160 • Is 160 a factor of 360 degrees? No. Therefore there is no pattern possible where you can tessellate an 18-gon, and have no overlaps or gaps

46. Examples • What about an equilateral triangle? • A = (180(3-2))/3 ---or 60. Thus, one interior angle of an equilateral triangle is 60 degrees. • 60 is a factor of 360, so equilateral triangles do in fact tessellate.

47. Theorems and Ideas • A figure does not have to be a regular polygon to tessellate. It should be noted, however, that the method we used to determine tessellation only works on regular polygons • Theorem 3-5 - Every Triangle (whether it is equilateral or not) tessellates • Theorem 3-6- Every quadrilateral tessellates (whether they are regular or not -it just has to be a 4 sided figure)

48. Symmetries in Tessellations • You will often find symmetries in Tessellations. • These would include • Line Symmetry • Rotational Symmetry • Glide Reflectional Symmetry • Translational Symmetry

49. Quasicrystals • A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry.