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2 Chapter 8 Smmetr GOLS You will be able to identif and appl line smmetr identif and appl rotation smmetr relate smmetr to transformations solve problems b using diagrams Smmetr is often seen in art. What smmetr do ou see in this art? NEL 363

3 CHPTER 8 YOU WILL NEED cardboard scissors a ruler Getting Started Square Smmetr Zachar said, if ou rotate a square, ou can onl make four-sided and eight-sided shapes with lines of smmetr.? Can ou make shapes with other numbers of sides b rotating a square?. Cut out a square.. Rotate the square about its centre point 180 clockwise. What shape do ou get? 364 Chapter 8 Smmetr NEL

4 C. Rotate the original square about the bottom right verte 90 cw. What shape do ou get if ou combine the new shape with the original one to make one large shape? D. How do ou know our combined shape from part C has a line of smmetr? E. Tr rotating the square around different points, using different numbers of degrees. How man sides do our combined shapes have? F. Can ou make shapes that do not have four sides or eight sides or do not have lines of smmetr? Eplain. WHT DO You Think? Decide whether ou agree or disagree with each statement. Eplain our decision. 1. parallelogram is not smmetrical. 2. line that separates a 2-D figure into two equal parts is a line of smmetr. 3. Shapes with more sides have more lines of smmetr. 4. If a shape has smmetr, it will fit in the same outline after ou rotate it. NEL Getting Started 36

5 8.1 Line Smmetr YOU WILL NEED coloured pencils a ruler scissors square dot paper triangle dot paper GOL Use line smmetr to classif polgons and complete shapes. INVESTIGTE the Math Zachar has a quarter of a piece of tile left from his grandmother s tiled floor. He wants to see what the whole tile looked like. He decides that the original tile probabl had smmetr.? What did the original tile probabl look like?. Draw the shape on square dot paper, as shown. vertical line of reflection horizontal line of reflection. Reflect the shape from part across the vertical line of reflection shown. Measure the distance of each point from the line to check our drawing. C. Reflect the combined shape from part across the horizontal line of reflection shown. Measure the distance of each point from the line to check our drawing. Reflecting D. How did drawing the shape on the dot paper help ou complete it? 366 Chapter 8 Smmetr NEL

6 WORK WITH the Math EXMPLE 1 Completing a shape using a line of smmetr Zachar drew half of the pattern for his stained glass window as shown. He asked Francis to finish it. What will the complete pattern look like? Francis s Solution I drew the half of the pattern on dot paper. The red line is the line of smmetr. I drew each point of the image to be the same perpendicular distance from the line of smmetr as the original point. The stained glass window will look like this. EXMPLE 2 Determining if a line is a line of smmetr Is D a line of smmetr? Viktor s Solution D C E D C D is not a line of smmetr. I drew the line C. If D were a line of smmetr, then C would be its perpendicular bisector. E and EC would have equal lengths and /E would be a right angle. I measured with a ruler. E and EC are not equal. I measured /E with a protractor. It is not 90. NEL 8.1 Line Smmetr 367

7 EXMPLE 3 Classifing designs b number of lines of smmetr Sort these designs according to the number of lines of smmetr the have... C. D. E. F. G. H. I. J. Rani s Solution.. C. 1 line 2 lines 2 lines D. E. F. 4 lines 0 lines 1 line G. H. I. J. 6 lines 3 lines 3 lines 2 lines I drew each design on dot paper. I located lines of smmetr b measuring, and then I drew them. I knew that a point on one side of a line of smmetr would be the same perpendicular distance from the line as the corresponding point on the other side. I decided to consider both colour and shape to decide whether there reall was a line of smmetr. Number of Lines of Smmetr Shape 0 E 1, F 2, C, G 3 I, J I sorted the designs according to the number of lines of smmetr the had. Some shapes would have had more lines of smmetr if the hadn t had coloured sections. 4 D 6 H 368 Chapter 8 Smmetr NEL

8 EXMPLE 4 Identifing reflection smmetr in a tessellation Identif lines of smmetr in this tessellation. Luc s Solution I thought about the design in columns and rows. I drew a horizontal line between rows. The design above the line was reflected below the line. That means the line is a line of smmetr. I drew a vertical line between columns. I could see that the column of design on the left of the line was reflected on the right. When I drew a second line between columns, I could see that the columns of design on left and right were again reflected. ecause tessellations go on forever, an line I draw between columns in this design will be a line of reflection. I can draw horizontal and vertical lines of smmetr in this tessellation. In Summar Ke Ideas line of smmetr is a line that separates a figure or a shape into two congruent halves so that each point of one half is at the same perpendicular distance from the line of smmetr as the corresponding point in the other half. In a design, each corresponding point must be the same colour. Need to Know You can classif a polgon b the number of lines of smmetr it has. You can complete a smmetrical design if ou are given half of it and the line of smmetr. If ou can identif line smmetr in one part of a tessellation, ou will be able to find the same smmetr elsewhere in the tessellation. NEL 8.1 Line Smmetr 369

9 Checking 1. Complete each shape using the red line of smmetr. a) b) 2. Sort the shapes into two groups: those with fewer than si lines of smmetr and those with si or more lines of smmetr... C. D. E. F. G. Practising 3. Sort the designs into two groups: those with fewer than si lines of smmetr and those with si or more lines of smmetr... C. 4. Multiple choice. Which of these tiles has line smmetr?.. C. D.. Multiple choice. Which line in this tessellation is a line of reflection? C D 370 Chapter 8 Smmetr NEL

10 6. Multiple choice. Which line in this tessellation is not a line of reflection? C D 7. Trace the shape at right and draw its other half using a ruler. The red line is a line of smmetr. 8. rectangle has four lines of smmetr. What must be true about it? 9. trapezoid has one line of smmetr. What must be true about it? Eplain wh. 10. Can a trapezoid have two lines of smmetr? Eplain. 11. Show that a heagon can have 0, 1, or 2 lines of smmetr. 12. Each shape is half of a smmetric design. Show four possible complete designs for each. Eplain our method. a) b) 13. Eplain how the number of lines of smmetr of a regular polgon relates to the number of sides it has. 14. How could ou move the red portions in this design so that the design has four lines of smmetr? Draw the new design on dot paper and show the lines of smmetr. NEL 8.1 Line Smmetr 371

11 1. How do ou know that a shape with two lines of smmetr has at least four sides? 16. a) Draw a shape or design and cut it along one of the lines of smmetr. b) sk another student to complete the design using one of the cut pieces and to eplain how he or she figured out the complete pattern. Closing 17. How is a line of reflection related to a line of smmetr? Give an eample. Etending 18. Create an artistic design, either using computer software or b hand, that has eactl si lines of smmetr. Identif the lines of smmetr in our design This shape is of a shape that has four lines of smmetr. 8 What might the shape look like? 20. a) Trace these shapes. Put them together to make a shape with line smmetr. b) Make our own puzzle like the one in part a). 372 Chapter 8 Smmetr NEL

12 8.2 Rotation Smmetr GOL Determine whether and how a shape can be turned to fit onto itself. INVESTIGTE the Math Francis is designing cand boes for an assembl line. Each bo will be filled with cand and covered with a lid. Francis has to choose one of these three designs for the boes. For the process to go as quickl as possible, he should choose the design whose lid can be attached the most quickl. square base parallelogram base rectangle base YOU WILL NEED cardboard or ristol board scissors a protractor? Which design should Francis choose?. Draw a square on a piece of cardboard. Mark one corner with an X. Draw the lines of smmetr of the square and etend them past the edges.. Determine the centre of the square. Describe how ou located it and how ou know it is the centre. C. Draw an outline of the square on paper. Rotate the square clockwise about the centre until it fits into its outline again. Draw the image. D. Through what angle was the square rotated? How can ou tell? E. Continue to rotate the square clockwise, stopping when it fits in its outline. Measure the angle of rotation each time. When the square is back in its original position, record the number of times ou rotated the square and the number of degrees that ou rotated it each time. F. Repeat parts,, and E for the rectangle and then for the other parallelogram. G. Which of these lids has rotation smmetr? Which lid has the greatest order of rotation smmetr? H. Which design should Francis choose for the bo? Wh? rotation smmetr When a rotating shape fits eactl over its original position with a turn of less than 360, the shape has rotation smmetr. order of rotation smmetr the number of times, within a 360 rotation about an internal point, that a shape will coincide with its original position; for eample, this shape has rotation smmetr of order 2 because, during a 360 rotation, it coincides with its original position twice. NEL 8.2 Rotation Smmetr 373

13 Reflecting I. How did ou locate all three shapes centres of rotation? J. Can a shape with line smmetr also have rotation smmetr? Eplain with an eample. K. Can a shape have rotation smmetr but not have line smmetr? Eplain with an eample. WORK WITH the Math EXMPLE 1 Determining order of rotation smmetr What is the order of rotation smmetr about the centre of a ield sign? What is its angle of rotation? Francis s Solution The ield sign is an equilateral triangle. I drew the triangle and marked one verte with a black dot. I drew the three lines of smmetr. The point where the intersect is the centre of rotation. I traced the triangle and rotated it about the centre clockwise until it fit into the tracing. I measured the angle of rotation to be 120. The shape looked the same, but now the marked verte was on the opposite side. I rotated it 120 again to fit into the tracing. Now the marked verte was on the bottom. I rotated the triangle again to fit in the tracing. gain, the angle of rotation was 120. This time the marked verte returned to the original position. I rotated the ield sign three times to return it to its original position. So, it has rotation smmetr of order 3. Each turn was 120, so the angle of rotation is Chapter 8 Smmetr NEL

14 EXMPLE 2 Rotating a shape about a verte Rotate this kite about a verte to create a shape with rotation smmetr. P Q R Erin s Solution S P Q 40 S R centre of rotation I cut out the kite and drew its outline. I decided to rotate it about the verte S. I measured with a protractor, and the internal angle of S is 40. I decided to rotate the kite b this amount so that there wouldn t be an spaces. 40 I rotated the cut-out kite 40 and drew its outline. 40 I rotated the cut-out kite a second time b 40. I drew the outline of the shape. I rotated the cut-out kite b 40 si more times, eight times in all. There were now nine copies of the kite. drawing each image, I got a design with rotation smmetr. This is m shape. It has rotation smmetr of order 9. The angle of rotation is 40. I noticed that I think I could have predicted the order of rotation smmetr b dividing 360 b the measure of the angle S at the centre of rotation. NEL 8.2 Rotation Smmetr 37

15 EXMPLE 3 Identifing rotation smmetr in a tessellation Determine the order of rotation smmetr and angle of rotation of the shape upon which this tessellation is based. Zachar s Solution I thought the angle of rotation should be 60, because I thought I saw si equilateral triangles making a heagon in the design. I rotated the design 60. The flowers in the design matched one another, so it seems that I was right. I continued to rotate the design b 60. It returned to its original position after si turns. There are si flowers around the central flower, so the basic shape, heagon, was rotated si times. The complete circle is 360. Since the pattern has a rotation smmetr of order 6, each angle will be 60. The order of rotation smmetr is 6. The angle of rotation is Chapter 8 Smmetr NEL

16 In Summar Ke Ideas shape ma have rotation smmetr and line smmetr: lines of smmetr: 4 order of rotation smmetr: 4 It ma have line smmetr but not rotation smmetr: lines of smmetr: 1 order of rotation smmetr: 1 It ma have rotation smmetr but not line smmetr: lines of smmetr: 0 order of rotation smmetr: 2 It ma have neither tpe of smmetr: lines of smmetr: 0 order of rotation smmetr: 1 Need to Know shape with rotation smmetr of order 1 has no rotation smmetr. The order of rotation smmetr and the angle of rotation for that smmetr are factors of 360. For eample, the square above has rotation smmetr of order 4, so the angle of rotation for that smmetr must be If a shape has more than one line of smmetr, then its centre of rotation is located where those lines intersect. If ou identif rotation smmetr around one point in a tessellation, ou will be able to identif the same rotation smmetr around ever like point. Checking 1. For each shape, determine the number of lines of smmetr, its order of rotation smmetr, and the angle of rotation. a) b) c) NEL 8.2 Rotation Smmetr 377

17 Practising 2. Joceln cut her apple in half crosswise. She sas that the section shows an eample of rotation smmetr of order. Ramon sas it shows line smmetr. Who is correct? Eplain. 3. Multiple choice. Choose the correct description for this Haida mask, not including the face.. 1 line of smmetr, rotation smmetr of order 1, no angle of rotation. 2 lines of smmetr, rotation smmetr of order 2, 180 angle of rotation C. 4 lines of smmetr, rotation smmetr of order 4, 90 angle of rotation D. 8 lines of smmetr, rotation smmetr of order 8, 4 angle of rotation 4. Multiple choice. Which statement is false?. n octagon can have rotation smmetr of order 2.. n isosceles triangle has rotation smmetr of order 2. C. shape can have an order of smmetr that is 4 less than its number of sides. D. parallelogram can have rotation smmetr of order 4.. For each object, determine the number of lines of smmetr the design has, its order of rotation smmetr, and the angle of rotation. a) b) c) 378 Chapter 8 Smmetr NEL

18 6. Does this portion of a tessellation have rotation smmetr? If so, what are the angle and order of rotation smmetr? 7. Complete the other half of each shape using the indicated line of smmetr. What is the order of rotation smmetr of the full shape? a) b) c) 8. a) Which of these items have rotation smmetr but not line smmetr? b) Identif the order of rotation smmetr of each item... C. D. E. F. G. H. I. 9. Which parallelogram has a higher order of rotation smmetr? Eplain our answer. NEL 8.2 Rotation Smmetr 379

19 Reading Strateg Monitoring Comprehension How can ou appl what ou know about polgons and interior angles? 10. a) For each regular polgon, determine its order of rotation smmetr and its angle of rotation... C. D. E. F. G. b) Look at our answers to part a). Eplain how a regular polgon s order of rotation smmetr relates to the number of sides it has and wh that makes sense. 11. Which of these screw heads have the same order of rotation smmetr? Eplain... C. D. E. F. 12. What is the order of rotation smmetr for each washer? a) b) 13. leandra drew a square that was 4 cm on each side. Then, she lengthened each side b 2 cm. Will the new shape still have rotation smmetr? Eplain. 380 Chapter 8 Smmetr NEL

20 14. a) These polgons have no equal sides. Do the have rotation smmetr about their centres? Eplain. b) Each of these polgons has at least two equal sides. Do the have rotation smmetr about their centres? Eplain. 1. Show that the purple shape has rotation smmetr. 16. Use this equilateral triangle to create a new combined shape with different rotation smmetr. 17. Find a photograph or drawing of each item. a) a 2-D shape with line smmetr and rotation smmetr b) a 2-D shape with line smmetr but without rotation smmetr c) a 2-D shape with rotation smmetr but without line smmetr 18. shape has rotation smmetr of order infinit. What is the shape and what does that mean? Closing 19. In what was is rotation smmetr like line smmetr? In what was is it different? Etending 20. Tape two mirrors together and set them up as shown. Draw a straight line on a piece of paper. Set the mirror on top of the paper so that the line etends from the bottom edge of one mirror to the bottom edge of the other. Look in. Eplain wh the shape ou see has rotation smmetr. NEL 8.2 Rotation Smmetr 381

21 8.3 Smmetr in rt YOU WILL NEED art from a number of sources GOL Identif smmetr in art. EXPLORE the Math Nola, Viktor, and Luc showed their friends eamples of art forms from their cultures that use smmetr. Inuit art: pendant carving Ukrainian art: egg decoration French-Canadian art: ceinture fléchée (a woven sash worn around the waist to keep a jacket closed) Eplore different tpes of art such as paintings, jeweller, quilts, tiles, murals, and cultural artwork.? How do artists use smmetr in their art? 382 Chapter 8 Smmetr NEL

22 Curious MTH Kaleidoscopes Kaleidoscopes were invented in With a kaleidoscope, ou can look at things in a new wa. These beautiful images are all from kaleidoscopes. YOU WILL NEED two mirrors a protractor adhesive tape a small, flat object such as a heartshaped sticker Make our own kaleidoscope images. 1. Tape two mirrors together so that the stand up on the table at an angle of Put a small flat object on the table between them. 3. Look at the image. How man images, including the object, do ou see? 4. Change the angle of the mirrors to 72, 60, 4, and 30. How man images do ou see, including the object, at each angle?. Suppose ou could draw a line to connect the images in questions 3 and 4. What geometric shapes would ou create? 6. Identif the order of rotation smmetr and the angle of rotation in those shapes. 7. Eplain how changing the angle in the kaleidoscope changes the resulting rotation smmetr. NEL Curious Math 383

23 CHPTER 8 Mid-Chapter Review Stud See Lesson 8.1, Eamples 1, 2, and 3. Tr Mid-Chapter Review questions 1, 2, and 3. Stud id id See Lesson 8.2, Eamples 1, 2, and 3. Tr Mid-Chapter Review questions 4,, 6, and 7. FREQUENTLY SKED Questions Q: How can ou tell that a 2-D object has line smmetr? : 2-D object has line smmetr if ou can draw a line through it that creates two halves, so that ever point in one half is the same perpendicular distance from the line as the mirror image of the point. Q: What is the order of rotation smmetr of a shape? : This is the number of times that ou can turn a shape about its centre to fit into its outline. ll shapes have rotation smmetr of order at least 1. ll regular polgons have rotation smmetr of order greater than 1. ll shapes with two or more lines of smmetr, such as this pentagon, have rotation smmetr. Shapes can have rotation smmetr, but no line of smmetr, such as this parallelogram. Practice Lesson Sort these pictures of instruments into those that have more than one line of smmetr and those that do not. Eplain our answer... C. 384 Chapter 8 Smmetr NEL

24 2. shape has more lines of smmetr than another, but fewer sides. What could the two shapes be? 3. Complete each shape using the red line of smmetr. a) b) Lesson a) Determine the order of rotation smmetr and the angle of rotation of this snowflake b tracing and rotating the shape. b) How can ou fold a square piece of paper to make a shape with the same order of rotation smmetr? Check b making a paper snowflake of the same order of rotation smmetr.. Louise sas the kings, queens, and jacks in a pack of cards are an eample of designs with rotation smmetr. Do ou agree or disagree? Eplain. 6. Draw two shapes or designs that do have rotation smmetr but do not have lines of smmetr. Do not draw parallelograms. 7. Show that a shape with si sides can have a greater order of rotation smmetr than a shape with eight sides. NEL Mid-Chapter Review 38

25 8.4 Smmetr on the Coordinate Plane YOU WILL NEED grid paper a protractor a ruler a transparent mirror GOL Recognize how transformations relate to line smmetr and rotation smmetr. LERN OUT the Math Rani sas that ou can translate, reflect, or rotate ^C to create a design with smmetr. Nola sas that ou can t. (1, 1) 0 (2,3) C(, 1)? Do ou agree with Rani or Nola? EXMPLE 1 Determining how transformations result in shapes or designs with smmetr Use ^C to create a shape with smmetr. Rani s Solution: Reflecting, rotating, and translating to create smmetr C 0 I reflected ^C along side C. I ended up with a shape with four sides. The vertices are at (1, 1), (2, 3), (, 1), and (2, 21). The shape has a line of smmetr along side C. It makes sense that reflections result in shapes with lines of smmetr since the two halves match, but are opposite, when there is a line of smmetr. The two shapes are related b line smmetr. 386 Chapter 8 Smmetr NEL

26 C 0 1 I noticed that ^C looked like 4 of a pinwheel. If I rotate the triangle 90 over and over, with point as the centre of rotation, I knew I would end up with a design with rotation smmetr. The resulting shape had 9 vertices: (1, 1), (2, 3), (, 1), (1, ), (21, 2), (23, 1), (0, 21), (1, 23), and (3, 0). C 0 I reflected to make a shape with line smmetr. I rotated and translated to make a shape with rotation smmetr of order 4. These shapes are related b rotation smmetr. Nola s Solution: Translating to create smmetr I translated the shape so its centre was at the origin. The shape still had rotation smmetr. The new vertices were (0, 0), (1, 2), (4, 0), (0, 4),(22, 1), (24, 0), (21, 22), (0, 24), and (2, 21). Communication Tip Two shapes are said to be related b line smmetr if the form a shape with line smmetr when the are combined. Two shapes are said to be related b rotation smmetr if one shape can be rotated over the other. C 0 I saw what Rani did, but wondered if ou could do just a translation to make a shape that had smmetr. I tried a vertical translation (D2). This design was made up of two triangles, with coordinates (1, 1), (2, 3), (, 1), (1,21), (2, 1), and (, 21). It did not have an reflection or rotation smmetr. Communication Tip Translations are often described b abbreviations like D2, which means down 2, and L4, which means left 4. The letters U and R can also be used, to describe up and right. NEL 8.4 Smmetr on the Coordinate Plane 387

27 C 0 I tried a horizontal translation (L4). This design was also made up of two triangles, with coordinates (1, 1), (2, 3), (, 1), (23, 1), and (22, 3). It did not have an reflection or rotation smmetr either. C 0 I tried a diagonal translation (L1, D2). This design was made up of two triangles, with coordinates (1, 1), (2, 3), (, 1), (0, 21), and (4, 21). Like m other designs, it did not have reflection or rotation smmetr. I think that ou won t get a shape with smmetr if ou just translate the triangle. I think we are both right, but in different was. There was no line of smmetr or rotation point since each point on the plane moves when it is translated. One point has to sta put when ou rotate and a whole line of points has to sta put when ou reflect. Reflecting. Wh does a reflection result in a design with line smmetr?. Rani rotated the triangle 90 each time. Wh did she need to rotate the triangle four times to end up with a design with rotational smmetr? C. How might the coordinates of the design after ou rotated or reflected help ou decide if the design has smmetr? 388 Chapter 8 Smmetr NEL

28 WORK WITH the Math EXMPLE 2 Eamining coordinate changes for specific transformations Determine what happens to the coordinates of a shape as a result of each transformation: reflection across the -ais reflection across the -ais reflection across the line 180 rotation about (0, 0) Zachar s Solution C C 10 I drew a right triangle on a grid to see what happened with each motion. When ou reflect across the -ais, points above the ais move below the ais, and vice versa. When ou reflect across the -ais, the -coordinates sta the same and the -coordinates become opposites. 10 C 10 0 C 10 (4, 6) becomes r(4, 26) (9, 2) becomes r(9, 22) C(4, 2) becomes C r(4, 22) When ou reflect across the -ais, points to the right of the ais move to the left, and vice versa. 10 When ou reflect across the -ais, the -coordinates sta the same and the -coordinates become opposites. (4, 6) becomes r(24, 6) (9, 2) becomes r(29, 2) C(4, 2) becomes C r(24, 2) NEL 8.4 Smmetr on the Coordinate Plane 389

29 10 10 C 0 C 10 When ou reflect across the line, both coordinates change. 10 When ou reflect across the line, the -coordinates and -coordinates switch C 0 10 C 10 When ou rotate 180 about (0, 0), the image coordinates are both opposites of the original coordinates. (4, 6) becomes r(6, 4) (9, 2) becomes r(2, 9) C(4, 2) becomes Cr(2, 4) When ou rotate 180 about (0, 0), the shape ends up in the diagonall opposite quadrant. (4, 6) becomes r(24, 26) (9, 2) becomes r(29, 22) C(4, 2) becomes Cr(24, 22) EXMPLE 3 Determining a transformation using coordinates Francis transformed a shape, and the combined shape that resulted was this square. What was the original shape and what was the transformation? How could ou use the coordinates to determine the transformation? (1, 1) (4, 1) (1, 2) (4, 2) 390 Chapter 8 Smmetr NEL

30 Luc s Solution (1, 1) (4, 1) (1, 2) (4, 2) 2. Length of square Erin s Solution The original shape could be a rectangle with vertices at (1, 1), (2., 1), (1, 22), and (2., 22) and the line of reflection was the line 2.. I imagined a vertical line of smmetr through the centre of the shape. I saw that one wa to think of the shape was as the result of reflecting the left half of it across a vertical line down the centre and looking at the combined shape. I subtracted the -coordinates of (1, 1) and (4, 1) to determine the side length of the square. I know that the line of smmetr goes through the centre of the side. I had to add half the side length to 1 or subtract it from 4 to get the -coordinate for the reflection line (1, 1) (4, 1) (1, 2) (4, 2) I know that a square has rotation smmetr of order 4 about its centre. Length of square I determined the centre b dividing the side length in half. I added half the side length to the -coordinate of (1, 1) I subtracted that half from the -coordinate of (1, 1). The square could be the result of a rotation of itself about the centre, (2., 20.). NEL 8.4 Smmetr on the Coordinate Plane 391

31 In Summar Ke Ideas The combined figure created when a figure is reflected will have line smmetr. It ma or ma not have rotation smmetr. The combined figure created when a figure is rotated ma have rotation smmetr, but not alwas. It ma or ma not have line smmetr. The combined shape or design created when a figure is translated will not usuall have smmetr. Need to Know If a shape is reflected across the -ais, its -coordinates remain the same, and its -coordinates become the opposite. P(a, b) S P r(a, 2b) If a shape is reflected across the -ais, its -coordinates remain the same, and its -coordinates become the opposite. P(a, b) S P r(2a, b) If a shape is reflected across the line, its -coordinates and -coordinates switch. P(a, b) S P r(b, a) If a shape is rotated 180 about the origin, its -coordinates and -coordinates are the opposites of the original. P(a, b) S P r(2a, 2b). Checking 1. a) Reflect ^C across the -ais and the C(4, ) -ais. b) Rotate ^C 180 about the origin. c) Write the coordinates of the three images. d) What tpe of smmetr will the completed (1, 2) (, 1) design have? Eplain a) Reflect figure CDE to create a design with smmetr. Write the coordinates of r, r, C r, D r, and E r. b) Rotate CDE about the verte E to C create a design with smmetr. Write the E D coordinates of r, r, C r, D r, and E r. 0 c) Translate CDE using the rule (L1, D1). Write the coordinates of r, r, C r, D r, and E r. Does the combined design have smmetr? 392 Chapter 8 Smmetr NEL

32 Practising 3. a) What tpe(s) of smmetr does the combined shape have? Describe the line of reflection or the centre of rotation. b) Eplain how figure CD was transformed to create r rc rd r. 4. Jonathan reflected figure EFGH to create this design. He sas that it has line smmetr. Do ou agree or disagree? Eplain.. Multiple choice. a) Which two shapes are not related b line smmetr?. P and Q C. Q and R. P and R D. Q and S b) Which two shapes are related b rotation smmetr about the origin?. R and S C. Q and R. P and R D. Q and S D C 0 C D F E E 0 H F G H G P Q S 0 R 6. Multiple choice. Choose the correct description of this design.. It has no line smmetr and no rotation smmetr.. It has line smmetr, but no rotation smmetr. C. It has no line smmetr, but does have rotation smmetr. D. It has line smmetr and rotation smmetr. 0 NEL 8.4 Smmetr on the Coordinate Plane 393

33 7. In each case, figure CD has been transformed. Write the coordinates of each image. Describe the transformations that ma have been used and how ou know. Describe the smmetr of the design. a) b) D D C C C C C D D 4 0 D C 4 D C C D D 8. a) Draw a polgon that has line smmetr on a coordinate grid. b) Show that ou can reflect that shape on its line of smmetr and end up with the same shape b describing the image of each original verte. c) Show that ou can reflect the shape across a different line and end up with a different design with line smmetr. Identif the new line of smmetr. d) Show that ou can rotate the original shape 180 about a verte to get a shape with rotation smmetr of order 2. e) Show that sometimes ou can rotate a shape and end up with a combined shape with line smmetr. E F 9. This shape is a quarter of a tile pattern. The entire pattern has four lines of smmetr and a rotational smmetr of order 4 with an angle of rotation of 90. Use transformations to complete the pattern. 0 D C 10. Show that an rectangle on a coordinate grid can be created b reflecting a different rectangle. Closing 11. Show that it is possible to translate a shape and end up with a combined shape with both line smmetr and rotation smmetr. 394 Chapter 8 Smmetr Etending 12. transformation moves the verte of a shape from (1, 2) to the point r(21, 2). a) What kind of transformation might it be? b) Could the resulting design have smmetr? If so, what kind? 13. shape is rotated 90 ccw about the point (0, 0). What happens to the coordinates of the original vertices? NEL

34 YOU WILL NEED Word Search Sheet for each plaer a timer What s the Message? Number of plaers: 2 to 4 How to pla 1. There are ten words in the Word Search Sheet. The words have been reflected or rotated. Take turns looking for the words. llow one minute for each turn. 2. Score one point for each letter in the word that has line smmetr and one point for each letter in the word that has rotation smmetr of order 2 or more. 3. When all ten words have been found, take turns tring to form a sentence. llow one minute for each turn. Score 10 points for forming a sentence. 4. The plaer with the higher score wins. Luc s Turn I found SEE. It has two letters with line smmetr (E), and one letter with rotation smmetr of order 2 (S). I score 3 points. Math GME NEL 39 Math Game G E W W K Q J U X N I U S L D M I N W N L D P I L H I N T Z Q H P O W O X N G F F O U G Y D S I U D T H I R J J G M O O C C L Q X T U U T E X R G T L D Y M R M L F S Q K I M D Y U V D D F X K K E C P N P E O W F Y C C U E V V Y R O I Q G K C U E I X U N S R U M E Z Z M J J C S E V K Y P S D V G U Y H T N T N C T T S Z F K R R L I O I E E R S Q Y W L O Z X R T E S M S N T U R E Z Z E W O U F U I H U F J G Y V P G N Z N O X L U W N O U V D P

35 8. Solve Problems Using Diagrams YOU WILL NEED grid paper a ruler GOL Solve problems involving smmetr b using diagrams. LERN OUT the Math Francis wrote these instructions for a design, but some of the paper was torn. Viktor has to complete the design. Francis s Design I drew rectangle CD with one verte in each quadrant of the Cartesian plane. I rotated CD to form its image, ' ' C' D'. The combined shape has rotation smmetr of order 4. The vertices of the original rectangle are (4, ),? How can Viktor complete the design? EXMPLE 1 Rotating rectangles to create a shape I decided to draw a diagram to solve the problem. Viktor s Solution 1. Understand the Problem I need to rotate a rectangle with one verte in each quadrant to form a shape with an order of rotation smmetr of Make a Plan I ll plot point (4, ) and draw CD using the -ais and the -ais as lines of smmetr. I think that if I rotate CD 90 cw, the rectangle will fit over itself 4 times in a full turn, so the shape I create will have rotation smmetr of order 4. That s because Chapter 8 Smmetr NEL

36 3. Carr Out the Plan D( 4, ) (4, ) 0 With the -ais as a line of smmetr, must be at (4, 2). With the -ais as a line of smmetr, D must be at (24, ). Verte C is at (24, 2). I joined the vertices. C( 4, ) (4, ) D( 4, ) (4, ) C (, 4) D (, 4) (0, 0) (, 4) (, 4) I rotated CD 90 cw about the origin to create rrcrdr. The combined shape has rotation smmetr of order 4. I figured it out b rotating the shape in m head with the centre of rotation at (0, 0). C( 4, ) (4, ) This design has rotation smmetr of order 4. I think it is Francis s design. Reflecting. Can Viktor be sure he drew Francis s design? Eplain.. How did Viktor s diagrams help him solve the problem? WORK WITH the Math EXMPLE 2 Solving a problem using rotation smmetr Rotate a triangle once to create a star with si lines of smmetr and rotation smmetr of order 6. Rani s Solution 1. Understand the Problem I need to determine which kind of triangle to choose and which point to rotate it about, so that the original triangle and the image will form a star. NEL 8. Solve Problems Using Diagrams 397

37 2. Make a Plan I figured that I would be more likel to end up with rotation smmetr of order 6 if I started with rotation smmetr of order 3. So, I decided to use an equilateral triangle and not a scalene or isosceles one; the don t have an rotation smmetr. 3. Carr Out the Plan I drew an equilateral triangle using a ruler and a protractor. I drew the outline of the triangle and tried different points of rotation. I used a verte as the point of rotation. That didn t make a star. I decided to rotate the triangle about its centre. To locate the centre, I drew the lines of smmetr. The centre is where those lines intersect. I rotated the triangle 180. The triangle and its image made a star with si points with onl one rotation. This star has si lines of smmetr and rotation smmetr of order Chapter 8 Smmetr NEL

38 EXMPLE 3 Locating coordinates using rotation smmetr Part of a design has the vertices (0, 8), (4, 6), C(3, ), D(, 2), and E(0, 0). The entire design has rotation smmetr of order 4 about the origin. What are the other vertices? Nola s Solution 1. Understand the Problem The entire design has rotation smmetr of order 4. So, to see the entire design, I need to rotate the part that I know three times, b 90 each time. 2. Make a Plan I will trace the part I know and rotate the tracing 90 cw. Then, I will draw the figure and locate its coordinates. I will do the same for rotations of 180 cw and 90 ccw. 3. Carr Out the Plan E C D I plotted the points, drew the pattern, and traced it. 8 4 C E D 8 4 E C D 8 0 I rotated m tracing 90 cw about the origin. The new coordinates are r(8, 0), r(6, 24), Cr(, 23), Dr(2, 2), and E r(0, 0). NEL 8. Solve Problems Using Diagrams 399

39 8 4 C E D 8 4 E 4 8 D E 4 C C D 8 0 I rotated m original tracing 180 cw about the origin. The new coordinates are s(0, 28), s(24, 26), Cs(23, 2), Ds(2, 22), and Es(0, 0). 8 D C 4 C D E E 8 4 E 4 8 D E 4 C C D 8 0 I rotated m original tracing 90 ccw. The new coordinates are t(28, 0), t(26, 4), Ct(2, 3), Dt(22, ), and Et(0, 0). This is the combined shape when the images rotated and the original drawing are all drawn. This shape has rotation smmetr of order 4 when rotated 90 about the origin. In Summar Ke Idea Man combined shapes and designs can be analzed b drawing a diagram on the Cartesian plane. The diagram can suggest how to solve a problem regarding the tpe of transformations required to create that shape. Checking 1. polgon has these features: Four of its vertices are (0, 4), (4, 4), (6, 2), and (22, 2). It has four lines of smmetr. The point at (2, 4) is on one line of smmetr. The point at (22, 0) is on another line of smmetr. Determine the coordinates of the other vertices. 400 Chapter 8 Smmetr NEL

40 Practising 2. Cop this design and colour the grid so that the design has two lines of smmetr and rotation smmetr of order a) ^C has vertices (2, 1), (, 4), and C (, 1). Transform ^C to create a design with four lines of smmetr and rotation smmetr of order 4. b) Transform ^C to create a design with no line of smmetr but rotation smmetr of order Complete this design. The red and blue lines are both lines of smmetr.. a) Draw a shape on a coordinate grid that has rotation smmetr of order 8. b) Reflect it so that the resulting combined shape also has rotation smmetr of order Matthew is creating a stained glass window that will have rotation smmetr of order 6. This green and white triangle is one part of the window. Complete the pattern for the window. 7. Matthew is creating another stained glass window. It will have the same shape as the window in question 6, but will have two lines of smmetr and rotation smmetr of order 2. Using the same piece as in question 6, complete the pattern for the window. 8. a) Draw a triangle with the vertices (, 1), (, 22), and (1, 0). b) Reflect the shape in three was so that each design has line smmetr. Describe each line of reflection. c) Rotate the shape in two was so that each design has rotation smmetr. Describe the order of rotation and the centre of rotation. 9. You translate a heagon and the combined shape has a line of smmetr. Show what shape it could have been and what the translation was. Closing 10. Create and solve a problem about transformations and smmetr that would be more easil solved b using a diagram. Etending 11. You draw two perpendicular lines on a coordinate grid. Show that, if ou reflect a shape first across one line, then the other, then the first again, ou get a shape with rotation smmetr of order 2. Reading Strateg Questioning What clarifing questions can ou ask ourself as ou work through the solution to questions 6 and 7? NEL 8. Solve Problems Using Diagrams 401

41 CHPTER 8 Chapter Self-Test 1. Sort these tiles according to the number of lines of smmetr their designs have. 2. Complete the other side of this puzzle piece using the red line of smmetr. 3. Determine whether each shape has rotation smmetr, and if it does, state the order and angle of rotation. a) b) c) 4. Which of these shapes have rotation smmetr of order about the centre?.. C. D. 402 Chapter 8 Smmetr NEL

42 . Determine whether each tessellation has line smmetr or rotation smmetr. If it has rotation smmetr, identif the order of rotation smmetr and the angle of rotation. a) b) c) 6. Dimitri sas CD and rrc rdr together make a design that has line smmetr and rotation smmetr. Milos sas it onl has rotation smmetr. Who is right? Eplain. C D a) Create a design with rotation smmetr b rotating quadrilateral CD about. D 0 D C C 0 b) Record the coordinates of r, r, C r, and Dr. 8. a) Create a design with line smmetr b reflecting quadrilateral CD from question 7 across the -ais. b) Record the coordinates of r, r, C r, and Dr. WHT DO You Think Now? Revisit What Do You Think? on page 36. How have our answers and eplanations changed? NEL Chapter Self-Test 403

43 CHPTER 8 Stud See Lesson 8.4, Eamples 1 and 2. Tr Chapter Review questions 4 and 9. Stud id See Lesson 8.4, Eamples 1 and 2. Tr Chapter Review questions and 9. Stud id id See Lesson 8.4, Eamples 1 and 2. Tr Chapter Review question 6. Chapter Review FREQUENTLY SKED Questions Q: What smmetr will a design have if a shape is reflected? : The design will have line smmetr. The line across which the shape is reflected will be the line of smmetr. D 0 D C C 0 C D C 0 D D Q: What smmetr will a design have if a shape is rotated? C : The design ma or ma not have rotation smmetr. This design has rotation smmetr of order 2 about (0, 0), since it was rotated 180 about that point. D C 0 C D Q: What smmetr will a design have if a shape is translated? : The smmetr depends on the translation and also on the original shape s smmetr. The design on the left has neither line nor rotation smmetr, but the design on the right has line smmetr. C 0 C 404 Chapter 8 Smmetr NEL

44 Practice Lesson Complete each shape using the red line of smmetr. a) b) Lesson Sketch each wheel. Determine if each sketch has rotation smmetr, and if it does, state the order and angle of rotation. a) b) c) d) Lesson Identif and describe the tpes of smmetr in each rug. a) b) NEL Chapter Review 40

45 Lesson D C a) Reflect quadrilateral CD across the -ais to create a new design. b) Write the coordinates of r, r, C r, and Dr. c) Describe what smmetr the new design has, if an.. a) Rotate quadrilateral CD 180 ccw about D to create a new design. b) Write the coordinates of r, r, C r, and Dr. c) Describe what smmetr the new design has, if an. 6. a) Translate quadrilateral CD b the translation rule (L6, D3) to create a new design. b) Write the coordinates of r, r, C r, and Dr. c) Tell wh the new shape does not have smmetr. Lesson Colour the squares of this mosaic to have four lines of smmetr and rotation smmetr of order Determine whether each tessellation has line smmetr or rotation smmetr. If it does have rotation smmetr, identif the order of rotation smmetr and the angle of rotation. a) b) C 9. a) Which two shapes shown at left are related b line smmetr? b) Which two shapes shown at left are related b rotation smmetr? 406 Chapter 8 Smmetr NEL

46 CHPTER 8 Chapter Task Smmetrical T-shirt Designs You will make three designs to put on T-shirts. The theme is smmetr. YOU WILL NEED grid paper? How can ou make smmetrical designs?. Draw three different designs on grid paper.. Each design must include the coordinates of vertices a combination of line smmetr and rotation smmetr a different order of rotation smmetr than our other two designs a different number of lines of smmetr than our other two designs smmetr of colour as well as shape C. For each design, eplain how our design meets the conditions given in part wh our design represents the theme of smmetr well Task Checklist Did ou include all of the drawings needed? Did ou include all of the tpes of smmetries needed? Did ou use the appropriate math language? NEL Chapter Task 407