Translations, Reflections, and Rotations

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1 Translations, Reflections, and Rotations The Marching Cougars Lesson 9-1 Transformations Learning Targets: Perform transformations on and off the coordinate plane. Identif characteristics of transformations that are rigid motions and characteristics of transformations that are non-rigid motions. Represent a transformation as a function using coordinates, and show how a figure is transformed b a function. SUGGESTED LERNING STRTEGIES: Debriefing, Think-Pair-Share, Predict and Confirm, Self Revision/Peer Revision Mr. Scott directs the Marching Cougars, the band at Chavez High School. He uses the coordinate plane to represent the football field. For the band s first show, he arranges the band in a rectangle that is marchers wide and 9 marchers deep. 10 M Notes CTIVITY 9 DISCUSSION GROUP TIPS s ou work in groups, read the problem scenario carefull and eplore together the information provided. Discuss our understanding of the problem and ask peers or our teacher to clarif an areas that are not clear. 015 College oard. ll rights reserved. The band begins b marching down the grid in this formation. Then the marchers move apart from each other verticall, while keeping the same distance between marchers within the same row. The diagrams on the net page show the initial shape of the marchers, and the two transformations that the undergo. To describe and classif the transformations, compare the pre-image of a transformation to its image. 10 Transformation 1 10 Transformation 10 C MTH TERMS transformation is a change in the position, size, or shape of a figure. The pre-image of the transformation is the original figure. The image is the figure after the transformation Use our own words to describe Transformation 1. ctivit 9 Translations, Reflections, and Rotations 103

2 CTIVITY 9 Lesson 9-1 Transformations M Notes. Compare Transformation 1 and Transformation. How do the two transformations compare? MTH TERMS rigid motion is a transformation that preserves size and shape. 3. Model with mathematics. Transformation 1 is an eample of a rigid motion. rigid motion keeps the same distance between the points that are transformed (in this situation, the marchers of the band); the shape and size of the pre-image and image are the same. a. How does Transformation 1 affect the distance between an two marchers in the band? b. How does Transformation affect the distance between the marchers? Is Transformation a rigid motion? TECHNOLOGY TIP You can also use geometr software to represent transformations, including rigid motions and non-rigid motions.. Review Transformation 1. Each point in the pre-image is mapped to a point in the image. For this reason, the transformation can be epressed as a function. a. Complete the table to show the positions of the four corners of the rectangle when Figure is mapped onto Figure. Figure (pre-image) Figure (image) (1, 10) (1, ) (1, ) (, 10) (, ) b. For an given point, how does the transformation change the -coordinate and -coordinate? 015 College oard. ll rights reserved. REDING IN MTH The arrow ( ) in the notation that shows how a point is transformed means goes to. c. You can use the notation (1, 10) (1, ) to show how a point is transformed. When ou use this notation to show how a general point (, ) is transformed, ou are epressing the transformation as a function. Epress Transformation 1 as a function. 10 Springoard Mathematics Geometr, Unit Transformations, Triangles, and Quadrilaterals

3 Lesson 9-1 Transformations CTIVITY 9 5. Review Transformation. a. Complete the table to show the positions of the four corners of the rectangle when Figure is mapped onto Figure C. M Notes CONNECT TO LGER Figure (pre-image) Figure C (image) (1, ) (1, ) (1, ) You ve used functions etensivel in algebra. Recall that a function is a set of ordered pairs in which each -value is associated with one, and onl one, -value. b. For an given point, how does the transformation change the -coordinate and -coordinate? c. Can Transformation also be epressed as a function? Eplain wh or wh not. Write the function if it eists.. Draw each image on the graph to show how the pre-image is transformed b the function. Then classif the transformation as rigid or non-rigid. a. (, ) ( + 3, ) DISCUSSION GROUP TIPS s ou read and discuss the transformations, ask and answer questions to be sure ou have a clear understanding of not onl all the terminolog used, but also the link between the algebraic notation and the graphs. 015 College oard. ll rights reserved. b. (, ) (, ) 7. Write the numeral in the middle of each pre-image in Item. Describe how the numeral should appear in each image. ctivit 9 Translations, Reflections, and Rotations 105

4 CTIVITY 9 Lesson 9-1 Transformations M Notes Check Your Understanding Use the tet and diagram to answer Items and 9. The rectangle undergoes the transformation described b the function (, ) (, +1) Complete the table to show the coordinates of the image and pre-image for the four corners of the rectangle. Pre-image Image (1, 3) (1, 7) MTH TIP rigid motion can be modeled b sliding, rotating, or flipping a figure. non-rigid motion often involves stretching or compressing the figure. 9. Graph the transformation of the figure. Is the transformation a rigid motion or non-rigid motion? Eplain how ou know. 10. rectangle is transformed as shown College oard. ll rights reserved Springoard Mathematics Geometr, Unit Transformations, Triangles, and Quadrilaterals

5 Lesson 9-1 Transformations CTIVITY 9 a. Which function describes the transformation? b. Classif the transformation as rigid or non-rigid. Eplain wh ou classified the transformation that wa. M Notes LESSON 9-1 PRCTICE For Items 11 and 1, consider the following: rectangle undergoes the transformation described b the function (, ),. MTH TIP Man different transformations can transform a pre-image to the same image. Consider sliding, flipping, and turning College oard. ll rights reserved. 11. Graph the transformation of the figure. Is the transformation a rigid motion? Eplain. 1. Reason abstractl. Draw a plus sign (+) in the middle of the image. Describe how the transformation would change the plus sign. 13. ttend to precision. Use the graph of the rectangle to help ou classif each of the following transformations. a. Draw the image of the rectangle under the transformation described b the function (, ) (, ). Classif the transformation as rigid or non-rigid. b. Draw the image of the rectangle under the transformation described b the function (, ) (, + ). Classif the transformation as rigid or non-rigid. ctivit 9 Translations, Reflections, and Rotations 107

6 CTIVITY 9 Lesson 9- Translations M Notes Learning Targets: Perform translations on and off the coordinate plane. Predict the effect of a translation on a figure. SUGGESTED LERNING STRTEGIES: Visualization, Create Representations, Predict and Confirm, Think-Pair-Share Maria marches with the band. t the start of the halftime show, her position is (, ) on the coordinate plane. Then Mr. Scott tells everone to move ards to the right on the field and ards up the field. The band s transformation is shown in the diagram, and Maria s position is marked with an X MTH TERMS translation is a rigid motion in which ever point is moved the same distance and in the same direction. directed line segment is the distance and direction of the translation. This tpe of transformation is called a translation. On the coordinate plane, a translation is described b a function of the form (, ) ( + a, + b), in which a and b are positive or negative constants. In the eample above, a = and b =. You can think of a translation as a figure sliding up or down, left or right, or diagonall. During the translation, ever point of the figure moves the same distance and in the same direction. This distance and direction is called a directed line segment. In the diagram, the directed line segment of the translation is shown b each of the arrows. 1. Complete the table to show how the translation affects four of the Marching Cougars. Marcher Pre-image Image 015 College oard. ll rights reserved. Maria (, ) Joe (3, 7) lfredo (1, ) LeJaa (7, 11) 10 Springoard Mathematics Geometr, Unit Transformations, Triangles, and Quadrilaterals

7 Lesson 9- Translations CTIVITY 9 Mr. Scott arranges the band in a rectangle. Then he directs the band member at (, 10) to move to (, 13). The numbers on the coordinate plane show ards of the football field. M Notes For the band to move in a translation, how should each band member move? MTH TIP translation alwas preserves the length and width of a figure. 3. Critique the reasoning of others. Maria was positioned at point (, ). Her friend tells her that Maria s new position is described b the function ( + 3, ) since her new position is (7, 0). Is her friend correct? Eplain. 015 College oard. ll rights reserved. Check Your Understanding Draw the image of the figure under the translation described b the function.. (, ) ( +, ) 5. (, ) ( + 3, ) ctivit 9 Translations, Reflections, and Rotations 109

8 CTIVITY 9 Lesson 9- Translations MTH TERMS M Notes rhombus is a parallelogram with four congruent sides. Translations can also be uuu vdefined without the coordinate plane. For the directed line segment, a translation maps point P to point P so that the following statements are true: uuuv PP is parallel to PP = uuu v PP is in the same direction as. The epression Tuuu v( P) describes the translation of a given point P b the uuu v directed line segment. In the above eample, Tuuu v( P) = P. Items through refer to rhombus CD shown below. Point P is in the center of the rhombus. C D P. Draw the translation of the rhombus described b directed line segment. Include P = T uuu v ( P). 7. Which part of the rhombus maps onto C?. Identif a translation of the rhombus that would map eactl one point of the rhombus onto another point of the rhombus. 9. Draw the following translations. Show the pre-image and image, and label corresponding points in each. a. square CD, translated four inches to the right b. right triangle C, translated three inches up c. right triangle C, translated b Tuuu v 015 College oard. ll rights reserved. 110 Springoard Mathematics Geometr, Unit Transformations, Triangles, and Quadrilaterals

9 Lesson 9- Translations CTIVITY 9 Check Your Understanding M Notes 10. Mr. Scott arranges the band in the shape of triangle C, shown below. Point P is in the interior of the triangle. He plans three transformations of the triangle, in which point P is mapped onto P, P, and P, respectivel. P P' P" P C a. Which of the three transformations are translations? Eplain our answer b appling the definition of a translation. b. Maria stands at point P of triangle C. To move to point P, her instructions are to move eight steps to the right. Do these instructions appl to all the other band members? Eplain. LESSON 9- PRCTICE The figure shows heagon CDEF undergoing a translation to the right. C TECHNOLOGY TIP 015 College oard. ll rights reserved. F D E 11. Which part of the pre-image is translated onto CD? 1. Epress the translation as a function of two points of the heagon. 13. Make use of structure. appling this translation man times, ou could create a row of heagons. What additional translation could be repeated to fill the page with heagons? Is it Tuuu v C, Tuuu v D, or Tu v E uu? You can use drawing or painting software to model translations. First draw a figure, then make a cop of it, and then drag the cop of the figure anwhere on the screen. No matter where ou place the cop, ou have created a translation of the original figure. Tr the translation of the heagon described here. ctivit 9 Translations, Reflections, and Rotations 111

10 CTIVITY 9 Lesson 9-3 Reflection M Notes Learning Targets: Perform reflections on and off the coordinate plane. Identif reflectional smmetr in plane figures. SUGGESTED LERNING STRTEGIES: Visualization, Create Representations, Predict and Confirm, Think-Pair-Share Mr. Scott plans another transformation for the Marching Cougars. The sign of the -coordinate of each marcher changes from positive to negative. Maria, whose position is shown b the X in the diagram, moves from point (, ) to point ( ) This tpe of transformation is called a reflection. Reflections are sometimes called flips because the figure is flipped like a pancake. On the coordinate plane, eamples of reflections are defined b the functions (, ) (, ), which is a reflection across the -ais. The eample shown above is described b (, ) (, ), which is a reflection across the -ais. The function (, ) (, ) defines a reflection across the line =. Ever reflection has a line of reflection, which is the line that the reflection maps to itself. In the above diagram, the line of reflection is the -ais. 1. Complete the table for the two reflections. 015 College oard. ll rights reserved. Pre-image Image (, ) (, ) Image (, ) (, ) (1, 10) (1, 10) (, 10) (1, ) ( 1, ) (, ) 11 Springoard Mathematics Geometr, Unit Transformations, Triangles, and Quadrilaterals

11 Lesson 9-3 Reflection CTIVITY 9 The figure shows an arrow in the coordinate plane. The tip of the arrow is located at point (, 10). M Notes 1 1. Predict the direction of the arrow after these reflections: a. across the -ais b. across the -ais 3. Draw the two reflections. Were our predictions correct? 015 College oard. ll rights reserved.. What reflection maps the arrow to a downward arrow that also has its tip at the point (, 10)? 5. Could a reflection map the arrow so it points to the right or to the left (i.e., parallel to the -ais)? If es, describe the line of reflection. ctivit 9 Translations, Reflections, and Rotations 113

12 CTIVITY 9 Lesson 9-3 Reflection M Notes Mr. Scott arranges the tuba plaers in an arrow formation, shown below. Then the tuba plaers undergo a reflection that is described b the function (, ) (, ). MTH TIP Cut out paper shapes to model transformations. To show a reflection, flip the shape across the line of reflection.. Draw the reflection. Identif the line of reflection. 7. Which tuba plaers travel the longest distance during the reflection? Identif this distance.. Which tuba plaer does not travel an distance during the reflection? 9. Eplain wh the reflection does not change the distance between an given tuba plaer and the point (, 5). 015 College oard. ll rights reserved. 11 Springoard Mathematics Geometr, Unit Transformations, Triangles, and Quadrilaterals

13 Lesson 9-3 Reflection CTIVITY Use appropriate tools strategicall. Use geometr software to eplore reflections. First, use the software to draw a pentagon. Then draw a line of reflection that passes through one side of the pentagon. Use the software to reflect the pentagon over this line of reflection. M Notes 11. What happens under this reflection to points that lie on the line of reflection? 015 College oard. ll rights reserved. 1. Use the software to eplore how a point not on the line of reflection is related to its image. a. Measure the distance of a point to the line of reflection. Then measure the distance of the point s image to the line of reflection. What do ou find? b. Draw the segment that connects a point and its image. How is this segment related to the line of reflection? ctivit 9 Translations, Reflections, and Rotations 115

14 CTIVITY 9 Lesson 9-3 Reflection M Notes Like other transformations, reflections can be defined independentl of the coordinate plane. reflection is a transformation that maps P to P across line l such that: If P is not on l, then l is the perpendicular bisector of PP. If P is on l, then P = P. To describe reflections, we will use the notation r l (P) = P, in which r l is the function that maps point P to point P across line l, the line of reflection. 13. The diagram shows pentagon CDE and the reflection r l. E D C a. Draw line l. b. Label the points rl ( ) =, r ( ) = l, and so on for the five vertices. 1. Quadrilateral DC was constructed b drawing scalene triangle C, then drawing its reflection across line C. Point P is the intersection of C and D. P C a. Prove that C is perpendicular to D. D 015 College oard. ll rights reserved. b. Eplain wh = D. 11 Springoard Mathematics Geometr, Unit Transformations, Triangles, and Quadrilaterals

15 Lesson 9-3 Reflection CTIVITY 9 Item 15 is related to the Perpendicular isector Theorem. The theorem states that if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. The converse is also true. If a point is equidistant from two points and, then it must lie on the perpendicular bisector of. Look again at quadrilateral DC. It was constructed b a reflection of a figure, which means that it has reflectional smmetr. Line segment C is the line of smmetr of the figure. line of smmetr can be an actual line in the figure, or it ma be imaginar. figure ma have an number of lines of smmetr, including an infinite number. 15. For each figure shown below, draw all of the lines of smmetr. M Notes 1. Triangle C has three lines of smmetr, labeled l 1, l, and l 3 in the figure below. What can ou conclude about the triangle? Eplain. 015 College oard. ll rights reserved. l 1 D E l F C l 3 ctivit 9 Translations, Reflections, and Rotations 117

16 CTIVITY 9 Lesson 9-3 Reflection M Notes Check Your Understanding 17. Match the figures to their number of lines of smmetr. rhombus isosceles triangle square circle a. one line b. two lines c. four lines d. infinitel man lines The figure shows scalene triangle C, which b definition has three unequal sides. Point D is on C, and D is perpendicular to C. D C 015 College oard. ll rights reserved. 1. Eplain wh D is not a line of smmetr for the triangle. 19. Does triangle C have an lines of smmetr? Eplain. 11 Springoard Mathematics Geometr, Unit Transformations, Triangles, and Quadrilaterals

17 Lesson 9-3 Reflection CTIVITY 9 LESSON 9-3 PRCTICE Use the figure for Items 0 and 1. M Notes 0. Draw the reflection of the arrow described b each of these functions, and identif the line of reflection. a. (, ) (, ) b. (, ) (, ) c. (, ) (, ) 1. Reason abstractl. Describe a reflection that would map the arrow onto itself. Irregular pentagon CDE has eactl one line of smmetr, which passes through point. 015 College oard. ll rights reserved. E D. What is the image of point C under a reflection across the line of smmetr? 3. Does an point remain fied under a reflection across the line of smmetr? Eplain.. Which sides of the pentagon must be congruent? C ctivit 9 Translations, Reflections, and Rotations 119

18 CTIVITY 9 Lesson 9- Rotations M Notes Learning Objectives: Perform rotations on and off the coordinate plane. Identif and distinguish between reflectional and rotational smmetr. SUGGESTED LERNING STRTEGIES: Debriefing, Think-Pair-Share, Predict and Confirm, Self Revision/Peer Revision For the big finish of the Marching Cougars performance, Mr. Scott arranges the band in a triangle formation. Then the triangle appears to rotate clockwise around a point, like a pinwheel. The center of the rotation is the verte located at (0, 0) on the coordinate plane MTH TERMS rotation is a transformation that maps points across a circular arc. It preserves the distance of each point to the center of the rotation. This spinning tpe of transformation is called a rotation. It is defined b the center of the rotation, which is mapped to itself, and the angle of the rotation. The center of rotation ma be part of the figure being rotated, as is the case shown above. Or it ma be a point outside the figure. Functions that describe rotations include (, ) (, ), which is a clockwise rotation of 90. Repeat the function twice to produce (, ) (, ), which is a rotation of 10. The function (, ) (, ) is a clockwise rotation of 70, or 90 counterclockwise. 1. Complete the table for the two rotations shown above. Pre-image Image (, ) (, ) (3, 10) (10, 3) ( 3, 10) (0, 0) Image (, ) (, ) 015 College oard. ll rights reserved. 10 Springoard Mathematics Geometr, Unit Transformations, Triangles, and Quadrilaterals

19 Lesson 9- Rotations CTIVITY 9 Use the figure for Items and 3. M Notes Predict the direction of the arrow after these rotations. The center of each rotation is (0, 0). a. 90 degrees clockwise b. 10 degrees c. 90 degrees counterclockwise 3. Draw the three rotations. Were our predictions correct?. Describe the rotation that maps figure onto figure. (Hint: First identif the center of rotation.) 015 College oard. ll rights reserved. a. b. ctivit 9 Translations, Reflections, and Rotations 11

20 CTIVITY 9 Lesson 9- Rotations M Notes The angle of rotation can be used to describe the image of ever point of the figure. Consider the rotation shown below. Rectangle CO is rotated 90 clockwise, and maps onto rectangle C O. From the diagram, it appears that O O and that CO C O. If ou measure O, ou will find that it, too, equals 90. C ' ' O C' The following figure is a square that is centered around the origin O. Point is one verte of the square. O 5. Draw a rotation of the square 5 clockwise about the origin. Label the point that is the image of point.. What is the measure of O? 015 College oard. ll rights reserved. 1 Springoard Mathematics Geometr, Unit Transformations, Triangles, and Quadrilaterals

21 Lesson 9- Rotations CTIVITY 9 7. Use appropriate tools strategicall. Use geometr software to eplore rotations. First, use the software to draw a quadrilateral. Then plot a point P that will serve as the center of rotation. Use the software to rotate the quadrilateral 10 counterclockwise around point P. M Notes P P. Choose a point of the pre-image and the corresponding point of the image. Use the software to measure the distance of each point from point P. What do ou find? 015 College oard. ll rights reserved. 9. Use the software to measure the angle formed b a point, its image, and point P, with point P as the verte of the angle. What do ou notice about this measure? 10. Use the software to change the shape of the quadrilateral b dragging one or more of the points that form the quadrilateral. Do the results from Items and 9 remain true no matter the shape of the quadrilateral? Eplain. ctivit 9 Translations, Reflections, and Rotations 13

22 CTIVITY 9 Lesson 9- Rotations M Notes Like translations and reflections, rotations can be defined independentl of the coordinate plane. rotation that maps P to P and Q to Q has the following properties. Let point O be the center of rotation. PO = P O, and QO = Q O POP QOQ To describe rotations, we will use the notation R = O, m ( P) P, in which R O m is the function that maps point P to point P with center of rotation at point O and through an angle of m. 11. The diagram shows isosceles triangle C, and point O inside it. Draw the following rotations. Label the images of the three vertices. a. R O, 90 b. R C,90, c. R,90 O 1. Describe three rotations that each map a square onto itself. C 015 College oard. ll rights reserved. 1 Springoard Mathematics Geometr, Unit Transformations, Triangles, and Quadrilaterals

23 Lesson 9- Rotations CTIVITY 9 Review our answer to Item 1. Figures such as squares have rotational smmetr, meaning that a rotation of less than 30 can map the shape onto itself. The smallest such angle is the angle of rotational smmetr. 13. Identif whether these figures have rotational smmetr, reflectional smmetr, or both. lso identif the angles of rotational smmetr and lines of reflectional smmetr. a. isosceles triangle M Notes b. equilateral triangle 015 College oard. ll rights reserved. c. rhombus d. regular heagon MTH TIP Each side of a regular polgon has the same measure, and each angle has the same measure. ctivit 9 Translations, Reflections, and Rotations 15

24 CTIVITY 9 Lesson 9- Rotations M Notes Check Your Understanding 1. The figure shows parallelogram CD (shaded) and its rotation. a. Name the rotation. b. Label the vertices ',, and C of the rotated figure. c. If DC = 30, find DC. 15. Identif an eample of a figure that meets these sets of properties. a. rotational smmetr with angle of rotational smmetr of 90 b. rotational smmetr with angles of rotational smmetr of 7 c. rotational smmetr with angle of rotational smmetr of 10, two lines of reflectional smmetr D C d. rotational smmetr with angle of rotational smmetr of 10, no lines of reflectional smmetr LESSON 9- PRCTICE 1. Describe the rotation that would move the arrow to these positions. a. pointing up, with the tip at (, 7) b. pointing down, with the tip at (0, 3) c. pointing up, with the tip at (3, 0) 17. Identif the direction of the arrow and the position of the tip after these rotations. a. 90 counterclockwise about (, 3) b. 10 about (, 3) c. 90 clockwise about (, 0) 1. Describe a figure that has an angle of rotational smmetr of Construct viable arguments. Do ou think it is possible for a figure to have an angle of rotational smmetr of 37? If so, describe the figure. If not, eplain wh not. 015 College oard. ll rights reserved. 1 Springoard Mathematics Geometr, Unit Transformations, Triangles, and Quadrilaterals

25 Translations, Reflections, and Rotations The Marching Cougars CTIVITY 9 CTIVITY 9 PRCTICE Write our answers on notebook paper. Show our work. Lesson 9-1 For Items 1, a square is drawn in the coordinate plane, with vertices as shown in the diagram. Then the square undergoes a rigid motion. 1. The function that describes the rigid motion could be (, ). (, ).. ( 3, ). C. (, ). D. (, 3 ). Lesson 9- For Items 5 and, a down arrow is drawn with its tip at (, 0). Then it undergoes a translation described b the directed line segment ( 3, 1). 5. The image of the tip of the arrow is at point. (, 1).. ( 1, 1). C. (, 3). D. ( 3, ).. The image of the arrow points in which direction?. down. left C. right D. diagonal 7. The diagram shows regular pentagon CDE. 015 College oard. ll rights reserved.. If the point (0, 3) is mapped to (0, 0), what could be the image of (3, 0)?. (0, 0). (0, 3) C. (3, 3) D. ( 3, 3) 3. The length of the diagonal of the square is 3. Can ou determine the length of the diagonal of the image of the square? Eplain.. Draw the transformations of the square described b these functions. Classif each as rigid or non-rigid. a. (, + ) b. ( + 3, 3) c. 1 (, ) E D a. Draw the translation Tu v E uu. Label the images of each verte. b. Is it possible for a translation to map more than one point of the pentagon onto another point of the pentagon? Eplain. C ctivit 9 Translations, Reflections, and Rotations 17

26 CTIVITY 9 Translations, Reflections, and Rotations The Marching Cougars Lesson 9-3 For Items 10, a square is drawn in the coordinate plane, with vertices as shown in the diagram. Then the square is reflected across the -ais. Lesson 9- For Items 1 and 13, a down arrow is drawn with its tip at (, 0). Then it undergoes a clockwise rotation of 90.. The function that describes the reflection is (, ). (, 3 ).. (, ). C. (, ). D. (, ). 9. n up-pointing arrow is drawn inside the square. In the image, the arrow points in which direction?. up. down C. left D. right 10. Describe a reflection of the square that would map it onto itself. 11. Right triangle C is reflected across line l, which is parallel to but distinct from, one of the legs of the triangle. Prove that points, C,, and C are collinear. ' 1. If the tip of the arrow moves to (0, ), what is the center of rotation?. (, ). (, ) C. (0, ) D. (0, 0) 13. How man possible centers of rotation will produce an image of a left-pointing arrow?. zero. one C. two D. infinite 1. Describe the rotational and reflectional smmetr of these shapes. a. ellipse b. right isosceles triangle c. letter S d. plus sign S College oard. ll rights reserved. C l ' C' MTHEMTICL PRCTICES Construct Viable rguments and Critique the Reasoning of Others 15. Figure is a square that is centered at the origin O. student claims that the reflection r ( = 0) and the rotation R O,10 transform the square in the same wa. Critique this claim. 1 Springoard Mathematics Geometr, Unit Transformations, Triangles, and Quadrilaterals