1. Unit 10 Transformations

2. Lesson 10.1 Dilations

3. Lesson 10.1 Objectives Define transformation (G3.1.1) Differentiate between types of transformations (G3.1.2) Define dilation (G3.2.1) Identify the characteristics of a dilation (G3.2.2) Calculate the magnitude of a dilation (G3.2.2) Define transformation (G3.1.1) Differentiate between types of transformations (G3.1.2) Define dilation (G3.2.1) Identify the characteristics of a dilation (G3.2.2) Calculate the magnitude of a dilation (G3.2.2)

4. Definition of Transformation A transformation is any operation that maps, or moves, an object to another location or orientation.

5. Transformation Terms When performing a transformation, the original figure is called the pre-image. The new figure is called the image. Many transformations involve labels typically using letters of the alphabet. The image is named after the pre-image, by adding a prime symbol (apostrophe) When performing a transformation, the original figure is called the pre-image. The new figure is called the image. Many transformations involve labels typically using letters of the alphabet. The image is named after the pre-image, by adding a prime symbol (apostrophe) A A’

6. Types of Transformations There are 4 basic transformations: 1. A dilation. 2. A reflection in a line. 3. A rotation about a point. 4. A translation. There are 4 basic transformations: 1. A dilation. 2. A reflection in a line. 3. A rotation about a point. 4. A translation.

7. Example 10.1 Identify the following transformations: Rotation Reflection Translation

8. Dilation A dilation is a transformation that will increase or decrease the size of the figure while following the rules of similarity. A dilation has a center which is used much like a focal point to enlarge or reduce every figure from. All dilations have the following properties: 1. If point P is not at the center C, then the image P’ lies on ray CP. 2. If point P is at the center, then P = P’. A dilation is a transformation that will increase or decrease the size of the figure while following the rules of similarity. A dilation has a center which is used much like a focal point to enlarge or reduce every figure from. All dilations have the following properties: 1. If point P is not at the center C, then the image P’ lies on ray CP. 2. If point P is at the center, then P = P’. C P P’

9. Scale Factor of a Dilation The scale factor of a dilation is a positive number found by taking the distance from the center to the image divided by the distance from the center to the pre-image. It is basically a multiplier of how much the figure has been reduced or enlarged. We use the letter “k” to stand for the scale factor, and it can be found by: The scale factor of a dilation is a positive number found by taking the distance from the center to the image divided by the distance from the center to the pre-image. It is basically a multiplier of how much the figure has been reduced or enlarged. We use the letter “k” to stand for the scale factor, and it can be found by: C P P’ 3 12

10. Reduction or Enlargement A reduction is when the image is smaller than the pre- image. The scale factor will be a number between 0 and 1. 0 < k < 1 A reduction is when the image is smaller than the pre- image. The scale factor will be a number between 0 and 1. 0 < k < 1 An enlargement is when the image is larger than the pre-image. The scale factor will be a number greater than 1. k > 1 C C D D’ G G’

11. Example 10.2 Tell whether the figure shows an enlargement or a reduction. And also find the scale factor. 1. Tell whether the figure shows an enlargement or a reduction. And also find the scale factor. 1. 2. Enlargement Reduction

12. Example 10.3 1. What type of dilation has occurred? Enlargement 2. Find the scale factor. 3. Find x.

13. Scale Factor with Coordinates Centered at the Origin When the origin on a coordinate plane is used as the center of dilation, the scale factor is simply distributed to both the x and y values of each coordinate. So multiply k to both the x and y coordinates. This will not work directly for a dilation centered at any other location. When the origin on a coordinate plane is used as the center of dilation, the scale factor is simply distributed to both the x and y values of each coordinate. So multiply k to both the x and y coordinates. This will not work directly for a dilation centered at any other location.

14. Example 10.4 Draw the dilation of ABCD under a scale factor of k = ½ centered at the origin. A’B’ C’ D’

15. Lesson 10.1 Homework Lesson 10.1 – Dilations Due Tomorrow Lesson 10.1 – Dilations Due Tomorrow

16. Lesson 10.2 Reflections

17. Lesson 10.2 Objectives Define an isometry (G3.1.2) Define a reflection (G3.1.1) Identify characteristics of a reflection Define line of symmetry Utilize properties of reflections to perform bank shots Define an isometry (G3.1.2) Define a reflection (G3.1.1) Identify characteristics of a reflection Define line of symmetry Utilize properties of reflections to perform bank shots

18. Reflections A transformation that uses a line like a mirror is called a reflection. The line that acts like a mirror is called the line of reflection. When you talk of a reflection, you must include your line of reflection A reflection in a line m is a transformation that maps every point P in the plane to a point P’, so that the following is true 1. If P is not on line m, then m is the perpendicular bisector of PP’. 2. If P is on line m, then P = P’. A transformation that uses a line like a mirror is called a reflection. The line that acts like a mirror is called the line of reflection. When you talk of a reflection, you must include your line of reflection A reflection in a line m is a transformation that maps every point P in the plane to a point P’, so that the following is true 1. If P is not on line m, then m is the perpendicular bisector of PP’. 2. If P is on line m, then P = P’.

19. Example 10.5 Give the image of the following reflections: 1. K(4,5) in the y-axis 2. W(2,-7) in the x-axis 3. A(7,3) in the y-axis 4. L(-3,5) in the x-axis 5. I(-1,-3) in the y-axis 6. G(-4,-2) in the x-axis 7. N(5,1) over the line y = x. Give the image of the following reflections: 1. K(4,5) in the y-axis 2. W(2,-7) in the x-axis 3. A(7,3) in the y-axis 4. L(-3,5) in the x-axis 5. I(-1,-3) in the y-axis 6. G(-4,-2) in the x-axis 7. N(5,1) over the line y = x.

20. Reflection Formula There is a formula to all reflections. It depends on which type of a line are you reflecting in. The formulas below will map the original coordinates of (x,y) to: There is a formula to all reflections. It depends on which type of a line are you reflecting in. The formulas below will map the original coordinates of (x,y) to: Vertical: y-axis ( -x, y) Horizontal: x-axis ( x, -y) y = x ( y, x) ( x, y) x = a ( -x + 2a, y) y = a ( x, -y + 2a)

21. Example 10.6 Give the image of the following reflections: 1. O(3,0) over the line y = x 2. L(-3,5) over the line y = x 3. C(-2,6) over the line x = 1 4. E(-4,-6) over the line x = -1 5. D(2,4) over the line y = 1 6. R(-1,-5) over the line y = -2

22. Isometry An isometry is a transformation that preserves the following: length angle measures parallel lines distance between points An isometry is also called a rigid transformation. An isometry is a transformation that preserves the following: length angle measures parallel lines distance between points An isometry is also called a rigid transformation.

23. Theorem 7.1: Reflection Theorem A reflection is an isometry. That means a reflection does not change the shape or size of an object! A reflection is an isometry. That means a reflection does not change the shape or size of an object! m

24. Example 10.7 Use the diagram to name the image of  1 after the reflection 1. Reflection in the x-axis 1.  4 2. Reflection in the y-axis 2.  2 3. Reflection in the line y = x 3.  3 4. Reflection in the line y = -x 4.  1 5. Reflection in the y-axis, followed by a reflection in the x-axis. 5.  3 Use the diagram to name the image of  1 after the reflection 1. Reflection in the x-axis 1.  4 2. Reflection in the y-axis 2.  2 3. Reflection in the line y = x 3.  3 4. Reflection in the line y = -x 4.  1 5. Reflection in the y-axis, followed by a reflection in the x-axis. 5.  3

25. Line of Symmetry A figure in the plane has a line of symmetry if the figure can be mapped onto itself by a reflection in a line. What that means is a line can be drawn through an object so that each side reflects onto itself. There can be more than one line of symmetry, in fact a circle has infinitely many around. A figure in the plane has a line of symmetry if the figure can be mapped onto itself by a reflection in a line. What that means is a line can be drawn through an object so that each side reflects onto itself. There can be more than one line of symmetry, in fact a circle has infinitely many around.

26. Example 10.8 Determine the number of lines of symmetry each figure has. 1. Determine the number of lines of symmetry each figure has. 1. 2. 3. 4. 5. 6. One Two Four None Three One

27. Bank Shots In Physics, the Law of Reflection states that any object that strikes a flat surface will bounce off at the same angle. FYI: That angle is measured to a normal line, which is drawn perpendicular to the surface. That concept can be confirmed by utilizing the definition of reflection and applying it with an everyday task such as a bank shot. Thus the Geometry of a bank shot can be accomplished using the following process: 1. Reflect the target location over the wall to be used for the bank shot. 2. Draw a segment that connects the ball and the image of the target after reflection. 3. The point of intersection of the segment and the wall should be the aiming point to perform the desired bank shot. In Physics, the Law of Reflection states that any object that strikes a flat surface will bounce off at the same angle. FYI: That angle is measured to a normal line, which is drawn perpendicular to the surface. That concept can be confirmed by utilizing the definition of reflection and applying it with an everyday task such as a bank shot. Thus the Geometry of a bank shot can be accomplished using the following process: 1. Reflect the target location over the wall to be used for the bank shot. 2. Draw a segment that connects the ball and the image of the target after reflection. 3. The point of intersection of the segment and the wall should be the aiming point to perform the desired bank shot.

28. Geometry of a Bank Shot

29. Double Bank Shot

30. Lesson 10.2 Homework Lesson 10.2 – Reflections Due Tomorrow Lesson 10.2 – Reflections Due Tomorrow

31. Lesson 10.3 Rotations

32. Lesson 10.3 Objectives Define a rotation (G3.1.1) Identify characteristics of rotation Define rotational symmetry Recognize patterns for rotations around the origin of a coordinate plane Define a rotation (G3.1.1) Identify characteristics of rotation Define rotational symmetry Recognize patterns for rotations around the origin of a coordinate plane

33. Definitions of Rotations A rotation is a transformation in which a figure is turned about a fixed point. The fixed point is called the center of rotation. The amount that the object is turned is the angle of rotation. The rotation with either be in a clockwise direction or a counterclockwise direction. At higher levels of math/science, a clockwise rotation is said to be negative in direction. So, a counterclockwise rotation is positive in direction. A rotation is a transformation in which a figure is turned about a fixed point. The fixed point is called the center of rotation. The amount that the object is turned is the angle of rotation. The rotation with either be in a clockwise direction or a counterclockwise direction. At higher levels of math/science, a clockwise rotation is said to be negative in direction. So, a counterclockwise rotation is positive in direction.

34. Theorem 7.2: Rotation Theorem A rotation is an isometry.

35. Example 10.9 Solve for x and y. 1. Solve for x and y. 1. 2.

36. Rotating About the Origin Recall that a reflection in the line y = x will swap x and y. (x, y)  (y, x) Now, recall what happens when you perform a reflection in the y-axis. (x, y)  (-x, y) Rotating about the origin in 90 o counterclockwise turns is like reflecting in the line y = x and in the y-axis at the same time! (x, y)  (y, x)  (-y, x) To rotate 180 o would require you to do that process twice. Rotating about the origin in 90 o clockwise turns will do the same, only the second reflection would be in the x-axis. Recall that a reflection in the line y = x will swap x and y. (x, y)  (y, x) Now, recall what happens when you perform a reflection in the y-axis. (x, y)  (-x, y) Rotating about the origin in 90 o counterclockwise turns is like reflecting in the line y = x and in the y-axis at the same time! (x, y)  (y, x)  (-y, x) To rotate 180 o would require you to do that process twice. Rotating about the origin in 90 o clockwise turns will do the same, only the second reflection would be in the x-axis.

37. Example 10.10 Rotate the figure the given number of degrees around the origin. List the coordinates of the image. 1. 90 o CCW Rotate the figure the given number of degrees around the origin. List the coordinates of the image. 1. 90 o CCW 2. 180 o CCW 3. 270 o CCW

38. Rotational Symmetry A figure in a plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180° or less. Notice, the rotation of 180° or less could go either clockwise or counterclockwise. The rotation must occur around the center of the object. A figure in a plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180° or less. Notice, the rotation of 180° or less could go either clockwise or counterclockwise. The rotation must occur around the center of the object. This figure has 90 o and 180 o rotational symmetry

39. Example 10.11 Determine if the figure has rotational symmetry. If so, describe any rotations that would map the figure onto itself. 1. Determine if the figure has rotational symmetry. If so, describe any rotations that would map the figure onto itself. 1. 2. 3.

40. Lesson 10.3 Homework Lesson 10.3 – Rotations Due Tomorrow Lesson 10.3 – Rotations Due Tomorrow

41. Lesson 10.4 Translations

42. Lesson 10.4 Objectives Define a translation (G3.1.1) Describe a translation using coordinate notation Define a translation (G3.1.1) Describe a translation using coordinate notation

43. Translation Definition A translation is a transformation that maps an object by shifting, or sliding, the object and all of its parts in a straight line. A translation must also move the entire object the same distance. A translation is a transformation that maps an object by shifting, or sliding, the object and all of its parts in a straight line. A translation must also move the entire object the same distance.

44. Coordinate Form Every translation has a horizontal amount of movement and a vertical amount of movement. A translation can be described in coordinate notation. Every translation has a horizontal amount of movement and a vertical amount of movement. A translation can be described in coordinate notation. A translation in the horizontal direction will change x. A translation in the vertical direction will change y. Adding values to the x-coordinate will move the object to the right Subtracting values from the x-coordinate will move the object to the left Subtracting values from the y-coordinate will move the object to down Adding values to the y-coordinate will move the object to up

45. Example 10.12 Describe the translation in words, and using coordinate notation. 1. Describe the translation in words, and using coordinate notation. 1. 2. 3. Right 1 and Up 3Left 5 and Up 3Right 1 and Down 2

46. Example 10.13 Find the coordinates of the image after performing the given translation. 1. R(3,5) (x,y)  (x+2, y+7) 1. R’(5,12) 2. A(-1,-6) (x,y)  (x+4, y-3) 2. A’(3,-9) 3. C(2,-4) (x,y)  (x-8, y+1) 3. C’(-6,-3) 4. E(4,-2) (x,y)  (x-4, y-5) 4. E’(0,-7) Find the coordinates of the image after performing the given translation. 1. R(3,5) (x,y)  (x+2, y+7) 1. R’(5,12) 2. A(-1,-6) (x,y)  (x+4, y-3) 2. A’(3,-9) 3. C(2,-4) (x,y)  (x-8, y+1) 3. C’(-6,-3) 4. E(4,-2) (x,y)  (x-4, y-5) 4. E’(0,-7)

47. Example 10.14 Find the coordinates of the image of each vertex of the polygon by performing the given translation. 1. (x,y)  (x+3, y-2) Find the coordinates of the image of each vertex of the polygon by performing the given translation. 1. (x,y)  (x+3, y-2) 2. (x,y)  (x-5, y+6)

48. Theorem 7.4: Translation Theorem A translation is an isometry.

49. Lesson 10.4 Homework Lesson 10.4 – Translations Due Tomorrow Lesson 10.4 – Translations Due Tomorrow

50. Lesson 10.5 Vectors

51. Lesson 10.5 Objectives Utilize vectors to perform translations (L1.2.3) Identify properties of vectors (L1.2.3) Perform vector addition/subtraction (L1.2.3) Utilize vectors to perform translations (L1.2.3) Identify properties of vectors (L1.2.3) Perform vector addition/subtraction (L1.2.3)

52. Vectors Another way to describe a translation is to use a vector. A vector is a quantity that shows both direction and magnitude, or size. It is represented by an arrow pointing from pre-image to image. The starting point at the pre-image is called the initial point. The ending point at the image is called the terminal point. Another way to describe a translation is to use a vector. A vector is a quantity that shows both direction and magnitude, or size. It is represented by an arrow pointing from pre-image to image. The starting point at the pre-image is called the initial point. The ending point at the image is called the terminal point.

53. Component Form of Vectors Component form of a vector is a way of combining the individual horizontal and vertical movements of a vector into a more simple form. Component form works the same as coordinate notation Naming a vector is the same as naming a ray : Component form of a vector is a way of combining the individual horizontal and vertical movements of a vector into a more simple form. Component form works the same as coordinate notation Naming a vector is the same as naming a ray :

54. Example 10.15 Name the given vector and give the component form of the vector. 1. Name the given vector and give the component form of the vector. 1. 2. 3. 4.

55. Magnitude & Direction of a Vector Recall that every vector has a magnitude and direction. The magnitude measures the size of the vector. For translations, the magnitude will measure the distance between the pre-image and the image. To find the magnitude of a vector, you must use Pythagorean Theorem with the component form of the vector. Recall that every vector has a magnitude and direction. The magnitude measures the size of the vector. For translations, the magnitude will measure the distance between the pre-image and the image. To find the magnitude of a vector, you must use Pythagorean Theorem with the component form of the vector. The direction of the vector is measured as the angle made at the initial point of the vector. This is typically the angle made with the positive or negative x- axis. To find that angle, you must use inverse trigonometry to find the angle made at the initial point of the vector.

56. Example 10.16 Find the magnitude and direction of each vector. 1. Find the magnitude and direction of each vector. 1. 2. 3.

57. Vector Addition Imagine a translation that maps a figure by moving 4 units to the right, 5 units up, 6 units to the right, and 2 units up. How many units will horizontally has the figured moved? 10 How many units vertically has the figured moved? 7 So the resulting translation would be described as adding the vectors and by adding like terms. = The same can be done when subtracting vectors as well. Imagine a translation that maps a figure by moving 4 units to the right, 5 units up, 6 units to the right, and 2 units up. How many units will horizontally has the figured moved? 10 How many units vertically has the figured moved? 7 So the resulting translation would be described as adding the vectors and by adding like terms. = The same can be done when subtracting vectors as well.

58. Example 10.17 Perform the stated vector operation. 1. Perform the stated vector operation. 1. 2. 3. 4. 5.

59. Lesson 10.5 Homework Lesson 10.5 – Vectors Due Tomorrow Lesson 10.5 – Vectors Due Tomorrow

60. Lesson 10.6 Glide Reflections and Compositions

61. Lesson 10.6 Objectives Perform compositions of transformations (G3.1.3) Define a glide reflection (G3.1.1) Perform compositions of transformations (G3.1.3) Define a glide reflection (G3.1.1)

62. Compositions of Transformations When two or more transformations are combined to produce a single transformation, the result is called a composition.

63. Order is Important The order of compositions is important! A rotation 90 o CCW followed by a reflection in the y-axis yields a different result when performed in a different order. The order of compositions is important! A rotation 90 o CCW followed by a reflection in the y-axis yields a different result when performed in a different order.

64. Example 10.18 Perform the following compositions. 1. Reflection in x-axis (x,y)  (x+3,y+5) Perform the following compositions. 1. Reflection in x-axis (x,y)  (x+3,y+5) 2. 90 o CCW Rotation about the origin Reflection over the line y=x

65. Example 10.19 Describe the following compositions. Remember to describe all important information about every composition. 1. Describe the following compositions. Remember to describe all important information about every composition. 1. 2.

66. Theorem 7.6: Composition Theorem The composition of two (or more) isometries is an isometry.

67. Glide Reflection A glide reflection is a transformation in which a reflection and a translation are performed one after another. The translation must be parallel to the line of reflection. As long as this is true, then the order in which the glide reflection is performed does not matter! A glide reflection is a transformation in which a reflection and a translation are performed one after another. The translation must be parallel to the line of reflection. As long as this is true, then the order in which the glide reflection is performed does not matter!

68. Example 10.20 Perform the following glide reflections. 1. Perform the following glide reflections. 1.

69. Lesson 10.6 Homework Lesson 10.6 – Glide Reflections & Compositions Due Tomorrow Lesson 10.6 – Glide Reflections & Compositions Due Tomorrow