Transformations Rotations Reflections Dilations Symmetry Return - PDF document

Slide 1 / 168 Slide 2 / 168 New Jersey Center for Teaching and Learning 8th Grade Math Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of 2D Geometry: students and teachers. These materials may not be used for any commercial purpose without the written Transformations permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning 2013-12-09 community, and/or provide access to course materials to parents, students and others. www.njctl.org Click to go to website: www.njctl.org Slide 3 / 168 Slide 4 / 168 Table of Contents Click on a topic to go to that section · Transformations · Translations Transformations · Rotations · Reflections · Dilations · Symmetry Return to · Congruence & Similarity Table of · Special Pairs of Angles Contents Common Core Standards: 8.G.1, 8.G.2, 8.G.3, 8.G.4, 8.G.5 Slide 5 / 168 Slide 6 / 168 Any time you move, shrink, or enlarge a figure you make a The image can also be labeled with new letters as shown below. transformation. If the figure you are moving (pre-image) is labeled with letters A, B, and C, you can label the points on Triangle ABC is the pre-image to the reflected image triangle XYZ the transformed image (image) with the same letters and the prime sign. B Y B B' A X A A' w o h s n o i Pull t Pull a m r o f s n a r t r o f n image image pre-image pre-image C Z C C'

Slide 7 / 168 Slide 8 / 168 There are four types of transformations in this unit: There are four types of transformations in this unit: · Translations · Translations · Rotations · Rotations · Reflections · Reflections · Dilations · Dilations The first three transformations preserve the size and shape of the The first three transformations preserve the size and shape of the figure. They will be congruent. Congruent figures are same size figure. and same shape. In other words: In other words: If your pre-image is a trapezoid, your image is a congruent If your pre-image is a trapezoid, your image is a congruent trapezoid. trapezoid. If your pre-image is an angle, your image is an angle with the same If your pre-image is an angle, your image is an angle with the same measure. measure. If your pre-image contains parallel lines, your image contains If your pre-image contains parallel lines, your image contains parallel lines. parallel lines. Slide 9 / 168 Slide 10 / 168 Translations Return to Table of Contents Slide 11 / 168 Slide 12 / 168 This shows a translation of pre-image ABC to image A'B'C'. Each point in the pre-image was moved A translation is a slide that moves a figure to a different right 7 and up 4. position (left, right, up or down) without changing its size or shape and without flipping or turning it. You can use a slide arrow to show the direction and distance of the movement.

Slide 13 / 168 Slide 14 / 168 To complete a translation, move each point of the pre-image and label the new point. Click for web page Example: Move the figure left 2 units and up 5 units. What are the coordinates of the pre-image and image? PULL A' B' C' D' A B C D Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved? Both the pre-image and image are congruent. Slide 15 / 168 Slide 16 / 168 Translate pre-image ABCD 4 right and 1 down. Translate pre-image ABC 2 left and 6 down. What are the coordinates of the image and pre-image? What are the coordinates of the image and pre-image? A PULL A PULL C B D B C Are the line segments in the pre-image and image the same Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved? length? In other words, was the size of the figure preserved? Both the pre-image and image are congruent. Both the pre-image and image are congruent. Slide 17 / 168 Slide 18 / 168 A rule can be written to describe translations on the Translate pre-image ABCD 5 left and 3 up. coordinate plane. Look at the following rules and What are the coordinates of the image and pre-image? coordinates to see if you can find a pattern. PULL 2 Left and 5 Up 2 Left and 6 Down A (3,-1) A' (1,4) A (-2,7) A' (-4,1) B (8,-1) B' (6,4) B (-3,1) B' (-5,-5) C (7,-3) C' (5,2) C (-6,3) C' (-8,-3) A D (2, -4) D' (0,1) B C 5 Left and 3 Up 4 Right and 1 Down A (3,2) A' (-2,5) A (-5,4) A' (-1,3) D B (7,1) B' (2,4) B (-1,2) B' (3,1) C (4,0) C' (-1,3) C (-4,-2) C' (0,-3) D (2,-2) D' (-3,1) D (-6, 1) D' (-2,0) Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved? Both the pre-image and image are congruent.

Slide 19 / 168 Slide 20 / 168 Translating left/right changes the x-coordinate. Translating left/right changes the x-coordinate. · Left subtracts from the x-coordinate Translating up/down changes the y-coordinate. · Right adds to the x-coordinate 2 Left and 5 Up 2 Left and 6 Down A (3,-1) A' (1,4) A (-2,7) A' (-4,1) B (8,-1) B' (6,4) B (-3,1) B' (-5,-5) Translating up/down changes the y-coordinate. C (7,-3) C' (5,2) C (-6,3) C' (-8,-3) · Down subtracts from the y-coordinate D (2, -4) D' (0,1) · Up adds to the y-coordinate 5 Left and 3 Up 4 Right and 1 Down A (3,2) A' (-2,5) A (-5,4) A' (-1,3) B (7,1) B' (2,4) B (-1,2) B' (3,1) C (4,0) C' (-1,3) C (-4,-2) C' (0,-3) D (2,-2) D' (-3,1) D (-6, 1) D' (-2,0) Slide 21 / 168 Slide 22 / 168 Write a rule for each translation. A rule can be written to describe translations on the coordinate plane. 2 Left and 5 Up 2 Left and 6 Down A (3,-1) A' (1,4) A (-2,7) A' (-4,1) B (8,-1) B' (6,4) B (-3,1) B' (-5,-5) C (7,-3) C' (5,2) C (-6,3) C' (-8,-3) 2 units Left … x-coordinate - 2 D (2, -4) D' (0,1) y-coordinate stays click to reveal click to reveal (x, y) (x-2, y-6) (x, y) (x-2, y+5) click to reveal rule = (x - 2, y) 5 Left and 3 Up 4 Right and 1 Down 5 units Right & 3 units Down… x-coordinate + 5 A (3,2) A' (-2,5) A (-5,4) A' (-1,3) y-coordinate - 3 click to reveal B (7,1) B' (2,4) B (-1,2) B' (3,1) rule = (x + 5, y - 3) C (4,0) C' (-1,3) C (-4,-2) C' (0,-3) D (2,-2) D' (-3,1) D (-6, 1) D' (-2,0) (x, y) (x+4, y-1) click to reveal (x, y) (x-5, y+3) click to reveal Slide 23 / 168 Slide 24 / 168 1 What rule describes the translation shown? 2 What rule describes the translation shown? E' A (x,y) (x, y - 9) A (x,y) (x - 4, y - 6) D' F' B (x,y) (x, y - 3) B (x,y) (x - 6, y - 4) E E P P u u l l l l C D (x,y) (x - 9, y) D C (x,y) (x + 6, y + 4) F F P P u u l l l l G' D (x,y) (x - 3, y) D (x,y) (x + 4, y + 6) G G E' D' F' G'

Slide 25 / 168 Slide 26 / 168 3 What rule describes the translation shown? 4 What rule describes the translation shown? A (x,y) (x - 3, y + 2) A (x,y) (x + 8, y - 5) P u l l P u l l P u l l B (x,y) (x + 3, y - 2) P u l l B (x,y) (x - 5, y - 1) E' E D' C (x,y) (x + 2, y - 3) D C (x,y) (x + 5, y - 8) F F' E' D' D (x,y) (x - 2, y + 3) D (x,y) (x - 8, y + 5) E F' D G G' F G' G Slide 27 / 168 Slide 28 / 168 5 What rule describes the translation shown? A (x,y) (x - 3, y + 2) P P u u l l l l B (x,y) (x + 3, y - 2) E' D' F' C (x,y) (x + 2, y - 3) Rotations E D (x,y) (x - 2, y + 3) D F G' G Return to Table of Contents Slide 29 / 168 Slide 30 / 168 A rotation (turn) moves a figure around a point. This point can be on the figure or it can be some other point. This point is called the point of rotation. P

Slide 31 / 168 Slide 32 / 168 When you rotate a figure, you can describe the rotation by giving the direction (clockwise or counterclockwise) and the angle that the figure is rotated around the point of rotation. Rotations are counterclockwise unless you are told otherwise. Describe each of the rotations. Rotation Hint A The person's finger is the B point of rotation for each figure. This figure is rotated This figure is rotated Click for answer Click for answer 180 degrees 90 degrees clockwise counterclockwise about point B. about point A. Slide 33 / 168 Slide 34 / 168 The following descriptions describe the same rotation. How is this figure What do you notice? rotated about the B A In order to Can you give your own example? origin? determine the angle, draw two rays (one In a coordinate from the point D C of rotation to plane, each pre-image B' quadrant point, the C' other from the represents point of rotation to the A' image point). D' Measure this angle. This figure is rotated 270 degrees clockwise Click to Reveal about the origin or 90 degrees counterclockwise about the origin. Check to see if the pre-image and image are congruent. Slide 35 / 168 Slide 36 / 168 The sum of the two rotations (clockwise and counterclockwise) 6 How is this figure rotated about point A? (Choose more is 360 degrees. than one answer.) If you have one rotation, you can calculate the other by C subtracting from 360. A clockwise D B B' P P u u l l l l B counterclockwise A, A' E C 90 degrees C' D 180 degrees E' D' E 270 degrees Check to see if the pre-image and image are congruent.