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9.1: Translations How can you make the word ZIP look like the word PIN? How about bob and pod? Transformations: Preimage: Image: Isometry: How do you identify an isometry? Example: Which of the following appear to be isometries? Why? 1) 2) 3) Mapping – description of a transformation into a diagram, tells you the preimage and image, along with corresponding parts (an arrow is used to show a mapping) Examples: In each diagram, the dashed-line figure is an image of the solid-line figure. (a) Choose an angle or point from the preimage and name its image. (b) List all pairs of corresponding sides. (c) Write the transformation as a mapping. 4. 5. Translations: transformations that map all points of a figure the same distance in the same direction Is a translation an isometry? Translation Rule: rule that is written like an ordered pair that describes how to move a figure (x , y) → (x + 4, y -3) x + 4 means add 4 to each x-value or mover right 4 y – 3 means to subtract 3 from each y-value or move down 3 Examples: Find the image of the given figure under the given translation. 6. (x, y) (x 2, y + 3) 7. (x, y) (x + 2, y 3) Identifying Translation Rules: Find a preimage point and its image point. What is being done to the x-values? What is being done to the y-values? Use these to write a translation rule using arrows…Ex: (x, y) (x + 2, y 3) Examples: What rule describes the following translations? 8. 9. 9.2: Reflections Definition: use a line like a mirror that reflects the image across it What is the original figure called in a transformation? What is the figure called after the transformation? Line of Reflection: How to Create a Reflection 1) Graph the preimage on a plane 2) Graph the line of reflection a. Remember “y =” is a HORIZONTAL LINE through the value on the y-axis b. “x=” is a VERTICAL LINE through the value on the x-axis c. y = mx+b is a diagonal line with slope = m and y-intercept = b 3) Count from the preimage to line of reflection and then across Example: Graph the following reflections in the given line. 1) W (-3, 3) in the y – axis 2) in the line x = 1 y 3) X (2,3) in the line y=x -2 y y x x Reflections in the Axes 1. If (x, y) is reflected in the x – axis, its image is (x , -y) 2. If (x, y) is reflected in the y – axis, its image is the point (-x, y) Examples: x 11. The reflection image of NO in line q is BC . Find the measure of BC if NO 6. D A q M L C B N O 12. a) Name the image of ▲ABC after a reflection in line b. b) Take the image found in part a and reflect it in line a. What is the image? M N L a J P K Q O I 13. The reflection image of OL in line q is b c G H D A E B C R F . D A C M B q L N O 9.3: Rotations Definition: Center of Rotation: Angle of Rotation: angle formed by the pre-image point, center and the image point after the rotation. Examples of rotations. 1) 2) Center of a Regular Polygon: point that is equidistant from the vertices The center and number of vertices will determine the number of congruent triangles in the figure. This can be used to determine the central angle and rotations in polygons. Examples: X is the center of the regular pentagon. 1) What is the image of E under a 144 counterclockwise rotation about X? Use this measurement to get a 144 rotation and answer the question. 2) Rectangle HIJK is rotated 90 clockwise about point O. Find the image of I J . [A] AB A [B] BC I [C] CD H [D] AD B J O K D C 3) Rectangle PQRS is rotated 90 clockwise about point O. Find the image of RS . A [A] PQ [B] AD [C] CD B Q R O P [D] PS S D C 4) Use the graph below to complete the sentence. Figure AB C D is the image of y figure ABCD 10 under a rotation _______ B [A] 180 about the origin. [B] 90 counterclockwise about the origin. C A A –10 [C] 270 counterclockwise about the origin. D B 10 x D C [D] 90 clockwise about the origin. 5) Use the graph below to complete the sentence. figure ABCD under a rotation _______ –10 Figure AB C D is the image of y 10 D C [A] 270 about the origin. [B] 90 clockwise about the origin. A B –10 [C] 180 about the origin. [D] 90 counterclockwise about the origin. 10 x B C D –10 A 9.4: Symmetry Symmetry: if an isometry maps a figure onto itself Types of Symmetry: 1) Line or Reflectional Symmetry a. when a reflection maps a figure onto itself b. Line of Symmetry: line of reflection dividing the figure into equal halves 2) Rotational Symmetry a. when a rotation of 180 or less maps the figure back onto itself 3) Point Symmetry a. When a figure has 180 rotational symmetry Identifying Lines of Symmetry Sketch lines on the figure that divide it in half The two parts should reflect across the line of reflection onto each other. Examples: Draw all lines of symmetry for the following figures. 1) 2) Examples: Do the following figures have rotational symmetry? 3) 4) 5) 9.5: Dilations Dilation: when a figure enlarges or reduces from its original shape Scale Factor: tells how to enlarge or reduce a figure Finding a Scale Factor 1) Identify as an enlargement or reduction 2) Identify corresponding values of the image and the preimage. 3) Write a ratio of the image to the preimage Enlargement: when the scale factor is greater than From the dashed to the solid figure is a with a scale factor of 4 2 Reduction: when the scale factor is between and From the dashed to the solid figure is a with a scale factor of 5 7 Example: Is J’K’L’M’ a reduction or enlargement of JKLM? What is the scale factor? 10. Draw the image of the figure for a dilation centered at the origin with scale factor 3. 9.6: Composition of Reflections Reflections across Two Parallel Lines: 1) These result in a translation of the pre-image. 2) The resulting distance of the translation is 2 (distance between the lines) Reflections across Intersecting Lines: 1) These result in a rotation about the point of intersection. 2) The angle of rotation is always equal to 2 (the acute angle formed by the lines) Examples: Find the image of each figure for a reflection first across line and then across line m. Is the resulting transformation a translation or a rotation? For a translation, describe the direction and distance. For a rotation, tell the center of rotation and the angle of rotation. 1) 2) Glide Reflections: a composition of a translation and reflection to make a transformation ALWAYS PERFORM THE TRANSLATION FIRST! Graph ∆ABC and its glide reflection image for the given translation and reflection line. 3. (x, y) (x 2, y); y = 0 4. (x, y) (x, y 3); x = 0