Half Turns and Quarter Turns Rotations of Figures on the Coordinate Plane
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1 Half Turns and Quarter Turns Rotations of Figures on the Coordinate Plane 5 WARM UP 1. Redraw each given figure as described. a. so that it is turned 10 clockwise Before: After: s D b. so that it is turned 90 counterclockwise Before: After: Et LEARNING GOALS Rotate geometric figures on the coordinate plane 90 and 10. Identif and describe the effect of geometric rotations of 90 and 10 on two-dimensional figures using coordinates. Identif congruent figures b obtaining one figure from another using a sequence of translations, reflections, and rotations. c. so that it is turned 90 clockwise Before: After: II You have learned to model rigid motions, such as translations, rotations, and reflections. How can ou model and describe these transformations on the coordinate plane? LESSON 5: Half Turns and Quarter Turns M1-7
2 Getting Started Jigsaw Transformations There are just two pieces left to complete this jigsaw puzzle. 1 A B 1. Which puzzle piece fills the missing spot at 1? Describe the translations, reflections, and rotations needed to move the piece into the spot. B rotate 100 slide up and left. Which puzzle piece fills the missing spot at? Describe the translations, reflections, and rotations needed to move the piece into the spot. A rotate 95 clockwise slide up and right M1- TOPIC 1: Rigid Motion Transformations
3 AC T I V I T Y 5.1 Modeling Rotations on the Coordinate Plane In this activit, ou will investigate rotating pre-images to understand how the rotation affects the coordinates of the image. E 1. Rotate the figure 10 about the origin. a. Place patt paper on the coordinate plane, trace the figure, and cop the labels for the vertices on the patt paper. K H s c G b. Mark the origin, (0, 0), as the center of rotation. Trace a ra from the origin on the -ais. This ra will track the angle of rotation. F ft Both coordinates opposite s are 10 0 G J H K B pl E A (, 1) B (, 3) C (, 5) D (, 5) E (, ) d. Compare the coordinates of the rotated figure with the coordinates of the original figure. How are the values of the coordinates the same? How are the different? Eplain our reasoning. C A Coordinates of Pre-Image c. Rotate the figure 10 about the center of rotation. Then, identif the coordinates of the rotated figure and draw the rotated figure on the coordinate plane. Finall, complete the table with the coordinates of the rotated figure. D B F F (5, ) G (5, 5) H (, ) J (5, ) K (5, 1) Coordinates of Image A'C B'c c'c P'c E C F'c 5J G'c 5 5 H'C Tk 5 K'C 5 1 t X LESSON 5: Half Turns and Quarter Turns M1-9
4 NOTES Now, let s investigate rotating a figure 90 about the origin.. Consider the parallelogram shown on the coordinate plane. A B D 0 C a. Place patt paper on the coordinate plane, trace the parallelogram, and then cop the labels for the vertices. b. Rotate the figure 90 counterclockwise about the origin. Then, identif the coordinates of the rotated figure and draw the rotated figure on the coordinate plane. M1-70 TOPIC 1: Rigid Motion Transformations
5 c. Complete the table with the coordinates of the pre-image and the image. Coordinates of Pre-Image I Coordinates of Image i to d. Compare the coordinates of the image with the coordinates of the pre-image. How are the values of the coordinates the same? How are the different? Eplain our reasoning. The and coordinates switched The new X coordinates are opposite 3. Make conjectures about how a counterclockwise 90 rotation and a 10 rotation affect the coordinates of an point (, ). 0 C C C X LESSON 5: Half Turns and Quarter Turns M1-71
6 You can use steps to help ou rotate geometric objects on the coordinate plane. Let s rotate a point 90 counterclockwise about the origin. Step 1: Draw a hook from the origin to point A, using the coordinates and horizontal and vertical line segments as shown. c,3 5 3 E 3, 5 A T( 1, ) me 1 l A (, 1) Step : Rotate the hook 90 counterclockwise as shown. Point A is located at (1, ). Point A has been rotated 90 counterclockwise about the origin.. What do ou notice about the coordinates of the rotated point? How does this compare with our conjecture? M1-7 TOPIC 1: Rigid Motion Transformations
7 ACTIVITY 5. Rotating An Points on the Coordinate Plane Consider the point (, ) located anwhere in the first quadrant. (, ) 1. Use the origin, (0, 0), as the point of rotation. Rotate the point (, ) as described in the table and plot and label the new point. Then record the coordinates of each rotated point in terms of and. If our point was at (5, 0), and ou rotated it 90, where would it end up? What about if it was at (5, 1)? Original Point Rotation About the Origin 90 Counterclockwise Rotation About the Origin 90 Clockwise Rotation About the Origin 10 (, ) C X C C C 1, C 1 l 1 LESSON 5: Half Turns and Quarter Turns M1-73
8 . Graph ABC b plotting the points A (3, ), B (, 1), and C (, 9). 0 Use the origin, (0, 0), as the point of rotation. Rotate ABC as described in the table, graph and label the new triangle. Then record the coordinates of the vertices of each triangle in the table. Original Triangle ABC Rotation About the Origin 90 Counterclockwise c A9B9C9,3 A (3, ) B (, 1) C 1, C (, 9) f 9, M1-7 TOPIC 1: Rigid Motion Transformations Rotation About the Origin 90 Clockwise C Rotation About the Origin 10 CX A0B0C0 A09B09C A C 3 B C bi D 9 c c 9
9 Let s consider rotations of a different triangle without graphing. 3. The vertices of DEF are D (7, ), E (5, 5), and F (1, ). a. If DEF is rotated 90 counterclockwise about the origin, what are the coordinates of the vertices of the image? Name the rotated triangle. NOTES O C X D C 7, D 7 T EEE I Esis's l b. How did ou determine the coordinates of the image without graphing the triangle? 0 C c. If DEF is rotated 90 clockwise about the origin, what are the coordinates of the vertices of the image? Name the rotated triangle. D C 7,,7 E C 5,5 5,5 F C I C l LESSON 5: Half Turns and Quarter Turns M1-75
10 NOTES d. How did ou determine the coordinates of the image without graphing the triangle? o C e. If DEF is rotated 10 about the origin, what are the coordinates of the vertices of the image? Name the rotated triangle. D C 7, 7 E C 5,5 C 5 5 F L l 1 f. How did ou determine the coordinates of the image without graphing the triangle? M1-7 TOPIC 1: Rigid Motion Transformations
11 ACTIVITY 5.3 Verifing Congruence Using Rigid Motions more than one transformations Describe a sequence of rigid motions that can be used to verif that the shaded pre-image is congruent to the image. 1. of 0 pre Rotate go Counterclockwise Translate up 0. 0 Translate lefts down X l LESSON 5: Half Turns and Quarter Turns M1-77
12 O 3. 0 Transl at 5 left 5 op gts O. 0 Rotate 10 Translate right 1 down M1-7 TOPIC 1: Rigid Motion Transformations
13 TALK the TALK NOTES Just the Coordinates Using what ou know about rigid motions, verif that the figures represented b the coordinates are congruent. Describe the sequence of rigid motions to eplain our reasoning. 1. QRS has coordinates Q (1, 1), R (3, ), and S (, 3). Q9R9S9 has coordinates Q9 (5, ), R9 (, ), and S9 (7, 3).. Rectangle MNPQ has coordinates M (3, ), N (5, ), P (5, ), and Q (3, ). Rectangle M9N9P9Q9 has coordinates M9 (0, 0), N9 (, 0), P9 (, ), and Q9 (0, ). LESSON 5: Half Turns and Quarter Turns M1-79
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15 Assignment Write In our own words, eplain how each rotation about the origin affects the coordinate points of a figure. a. a counterclockwise rotation of 90 b. a clockwise rotation of 90 c. a rotation of 10 Remember A rotation turns a figure about a point. A rotation is a rigid motion that preserves the size and shape of figures. Practice 1. Use JKL and the coordinate plane to answer each question. K L J 0 a. List the coordinates of each verte of JKL. b. Describe the rotation that ou can use to move JKL onto the shaded area on the coordinate plane. Use the origin as the point of rotation. c. Determine what the coordinates of the vertices of the rotated J9K9L9 will be if ou perform the rotation ou described in our answer to part (b). Eplain how ou determined our answers. d. Verif our answers b graphing J9K9L9 on the coordinate plane.. Determine the coordinates of each triangle s image after the given transformation. a. Triangle ABC with coordinates A (3, ), B (7, 7), and C (, 1) is translated units left and 7 units down. b. Triangle DEF with coordinates D (, ), E (1, 5), and F (, 1) is rotated 90 counterclockwise about the origin. c. Triangle GHJ with coordinates G (, 9), H (3, ), and J (1, ) is reflected across the -ais. d. Triangle KLM with coordinates K (, ), L (, 7), and M (3, 3) is translated units right and 9 units up. e. Triangle NPQ with coordinates N (1, 3), P (1, ), and Q (9, 0) is rotated 10 about the origin. LESSON 5: Half Turns and Quarter Turns M1-1
16 Stretch 1. Rotate Trapezoid GHJK 90 clockwise. Rotate ABC 135 clockwise around point G. around point C. B H J G K A C Review Given a triangle with the vertices A (1, 3), B (, ), and C (5, ). Determine the vertices of each described transformation. 1. A reflection across the -ais.. A reflection across the -ais. 3. A translation 5 units horizontall.. A translation units verticall. Rewrite each epression using properties. 5. ( 1 ) 3( 5). ( 7) M1- TOPIC 1: Rigid Motion Transformations
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