CHAPTER. Symmetry and Surface Area GET READY 2. Math Link Warm Up Line Symmetry Warm Up 15

Transcription

1 CHAPTER 1 Symmetry and Surface Area GET READY Math Link Warm Up Line Symmetry 6 1. Warm Up Rotation Symmetry and Transformations Warm Up Surface Area 6 Graphic Organizer 36 Chapter 1 Review 37 Key Word Builder 40 Chapter 1 Practice Test 41 Math Link: Wrap It Up! 43 Challenge 44 Answers 46

2 Get Ready Using Translations Transformations include translations, reflections, and rotations. A translation is a slide along a straight line. It can be horizontal ( ), vertical ( ), or oblique (slanted or ). A transformation moves a shape to a different position. B A - y C 0 A B C x This is a translation 3 units horizontally to the right, R3. E y D F E' D' x F' This is a translation 3 units horizontally to the right and units vertically down, R3 and D. 1. Describe each translation. a) B' C' B y A' A x units up b) E' G' y E F G H x C F' H' Drawing Reflections A reflection is a mirror image in a line of reflection. A point and its reflection are the same perpendicular ( ) distance from the line of reflection. The line of reflection is a horizontal line at y = 1. Both N and N are 1 unit from the line of reflection. y N line of reflection N 0 4 x. Draw the reflection image for each figure. a) y b) y r line of reflection line of reflection r x x MHR Chapter 1: Symmetry and Surface Area

3 Drawing Rotations A rotation is a turn about a point or centre of rotation. It can be clockwise ( ) or counter-clockwise ( ). The centre of rotation is at A. The rotation is 180 counter-clockwise ( ) about A. y centre of rotation 0 - A x 180 is halfway around a circle a) Rotate figure DEFG 90 clockwise about its centre of rotation, F. Draw the rotation image. Label it D E F G. b) Write the coordinates of D, E, F, and G. D = (, ) E = F = G = D E - y G 4 F x Using Surface Area Surface area is the sum of the areas of all the faces of a 3-D object. Find the surface area of the right rectangular prism. Front or back Top or bottom Ends A = l w A = l w A = l w = 6 8 = 6 = 8 = 48 cm = 1 cm = 16 cm Area of front and back: Area of top and bottom: Area of ends: A = 48 A = 1 A = 16 = 96 cm = 4 cm = 3 cm Total surface area = = 15 cm 8 cm 6 cm cm 4. Calculate the surface area of the right rectangular prism. Front or back Top or bottom Ends Get Ready MHR 3

4 Math Link Reflections on Our World symmetry an image or object has symmetry if a line can divide it into identical parts an image has symmetry if it can fit onto itself by reflection or rotation 1. a) This flower has several lines of symmetry. Place a Mira or mirror from the top left corner to the bottom right corner. Does this show a line of reflection? Circle YES or NO. b) Use the Mira or a mirror to draw other lines of reflections. How many can you draw?. a) Use a ruler to draw a vertical line ( ) through the centre of the dragonfly. Is this a line of reflection? Circle YES or NO. b) Are there any other lines of reflection? Circle YES or NO. 3. The dashed line shows the line of reflection, r. C a) Use a Mira to help you draw the reflected image. b) In the original image, the highest point is. D E r c) In the new image, the lowest point is. d) The new image has been reflected in the line of reflection. (horizontally or vertically) 4. a) Figure ABC has been translated to make figure A B C. Fill in the blanks to describe a rule for this translation. A to A : move spaces right and spaces up. B to B : C to C : y A A C B C B x b) Describe a translation that would place the image for ABC in a different quadrant. IV III I II 4 MHR Chapter 1: Symmetry and Surface Area

5 1.1 Warm Up 1. Line is the line of reflection. Draw the reflection image of each figure. Name the coordinates of the reflection image. a) y b) 4 B C A E D F l y 0 l D O G x 0 4 x A (, ) D B (, ) E C F D O G. Circle the diagrams that show perpendicular line segments. a) C A B b) K J Line segments are perpendicular ( ) if they meet at right angles. D I L c) E G F d) O H M N 3. Use a ruler to draw a perpendicular line segment. P a) 4 cm b) 4. List examples of perpendicular line segments in your classroom. E H G F 3 cm 1.1 Warm Up MHR 5

6 1.1 Line Symmetry line of symmetry a line that divides an object or design in half each side of the line is a mirror image of the other can be vertical ( ), horizontal ( ), or oblique (slanted ), also called line of reflection a figure may have more than 1 line of symmetry vertical line of symmetry horizontal line of symmetry oblique line of symmetry Link the Ideas line symmetry a type of symmetry where a line of symmetry divides an image or object into identical, reflected halves identical halves can be reflected in a vertical ( ), horizontal ( ), or oblique (slanted ), line of symmetry Working Example 1: Find Lines of Symmetry Each diagram shows line symmetry. For each part, use a different method to find the line(s) of symmetry. State the number of lines of symmetry and describe each one. a) Solution Use a Mira. There is horizontal line of symmetry. number of description of line lines of symmetry 6 MHR Chapter 1: Symmetry and Surface Area

7 b) Solution vertical line of symmetry Count squares on the grid to find the lines of symmetry. The same number of squares are above and below the horizontal line of symmetry. The same number of squares are to the left and right of the vertical line of symmetry. horizontal line of symmetry Number of lines of symmetry: Description: One line is ( ), and one line is ( ). c) Solution Use folding. If the shape on each side of the fold is the same, the fold is a line of symmetry. You can sketch the complete figures to prove that the fold lines are lines of symmetry. You can fold this figure along 4 different lines to make mirror images: There is 1 ( ) line of symmetry, 1 ( ) line of symmetry, and (, ) lines of symmetry. 1.1 Line Symmetry MHR 7

8 How many lines of symmetry does each figure have? Describe each line of symmetry as vertical, horizontal, or oblique. a) Number of lines of symmetry: Description: b) Number of lines of symmetry: Description: c) Number of lines of symmetry: Description: d) Number of lines of symmetry: Description: 8 MHR Chapter 1: Symmetry and Surface Area

9 Working Example : Complete Drawings Using Symmetry Each drawing shows half of a figure. The dashed grey line shows a line of symmetry. Draw the whole image for each figure. a) Solution Method 1: Use Paper Folding Fold a piece of paper in half. Draw the figure so the line of symmetry is along the folded edge. Cut out the figure. Unfold the paper to show the whole figure. Method : Use Measurement or Counting B C B C B C A D A X D A D B C B C Draw the half figure on the grid. Label the vertices A, B, C, and D. Mark the line of symmetry. Reflect the points that are not on the line of symmetry to the opposite side of the line. These are points B and. Draw the reflected points the same distance from the fold line. This distance must be perpendicular ( ) to the fold line. Literacy Link Use B and C to label the new positions of B and C after a transformation. B is read as B prime. So, BX = B X and CD =. Join A to B, B to C, and C to D to draw the figure. 1.1 Line Symmetry MHR 9

10 b) Solution From each vertex, mark the perpendicular distance on the opposite side of the line of symmetry. Connect the lines to draw the figure. Complete the figure. C B B C A A F The whole figure is the letter H. The line of symmetry is not part of the final figure. D E Complete each shape. Use the line of symmetry. a) b) 10 MHR Chapter 1: Symmetry and Surface Area

11 Check Your Understanding Communicate the Ideas 1. Any rectangle has only lines of symmetry. Do you agree? Circle YES or NO. Give 1 reason for your answer.. Sasha says a parallelogram has lines of symmetry. Basil says it has 4 lines of symmetry. Kendra says it has no lines of symmetry. Who is correct? Circle SASHA or BASIL or KENDRA. Give 1 reason for your answer. Practise 3. Draw all the lines of symmetry on each figure. a) b) c) d) 1.1 Line Symmetry MHR 11

12 4. Draw all the lines of symmetry on each diagram. a) b) 5. Circle the figures that have only lines of symmetry. 6. The dashed line is the line of symmetry. Complete each diagram. a) b) c) d) 1 MHR Chapter 1: Symmetry and Surface Area

13 Apply 7. a) The y-axis is the line of reflection. Draw the reflection image. Label the reflected vertices A, C, and E. y 5 A b) What are the coordinates of the reflected image? E A (, ) C E -5 C 0 5 x c) Do the original figure and the reflection image show line symmetry? Circle YES or NO. Give 1 reason for your answer. 8. How many lines of symmetry does each flag have? a) b) c) d) 9. Look at the upper-case letters of the English alphabet. a) Which letters have a horizontal line of symmetry? b) Which letters have a vertical line of symmetry? c) Which letters have both horizontal and vertical lines of symmetry? 1.1 Line Symmetry MHR 13

14 10. Using upper-case letters, you can write the word MOM vertically or horizontally. The word has 1 vertical line of symmetry in each position. Write another word that has 1 vertical line of symmetry when you print it vertically or horizontally. M O M MOM 11. Write a -digit number that has lines of symmetry. Show the lines of symmetry. Math Link Imagine you work for a company that designs playing cards. Your job is to make an interesting design for a deck of cards. Your design must have at least 1 line of symmetry. Draw your design in the box or on the grid. 14 MHR Chapter 1: Symmetry and Surface Area

15 1. Warm Up 1. a) Translate the figure 4 units to the right. Label the new coordinates A, B, and C. A y 8 b) What are the coordinates of A, B, and C? 6 A (, ) 4 B C B C - - x c) Reflect A B C over the x-axis. Label the new coordinates A, B, and C a) Draw the lines of symmetry for each figure. Then, write the number of lines of symmetry. -8 Figure Draw Lines of Symmetry Number of Lines of Symmetry Equilateral triangle Square Regular pentagon Regular hexagon b) What pattern do you notice? 3. How many lines of symmetry does each figure have? a) b) 1. Warm Up MHR 15

16 1. Rotation Symmetry and Transformations centre of rotation the point about which an object or a design is turned or rotated rotation symmetry when a shape or design can be turned about its centre of rotation so it fits onto itself more than once in 1 complete turn Link the Ideas Working Example 1: Find Order and Angle of Rotation order of rotation the number of times a shape fits onto itself in 1 complete turn angle of rotation the smallest measure of the angle needed to turn a shape onto itself measured in degrees ( ) or fractions of a turn angle of rotation = 360 order of rotation For each shape, state the order of rotation and the angle of rotation. Write the angle of rotation in degrees and as a fraction of a revolution. centre of rotation 6 4 A revolution is 1 complete turn. a) Solution Trace the shape onto a piece of tracing paper. Place your copy over the original. Rotate it to see the order and angle of rotation. Order of rotation =, because when you rotate the flag 1 complete turn it fits on itself times. Angle of rotation in degrees = 360 = Angle of rotation as a fraction of a turn order of rotation 1 turn = 1 = turn 16 MHR Chapter 1: Symmetry and Surface Area

17 b) Solution Trace the shape onto a piece of tracing paper. Place your copy over the original. Rotate it to see the order and angle of rotation. Order of rotation = 5. When you rotate the pentagon 1 complete turn, it fits on itself times. Angle of rotation in degrees = = Angle of rotation as a fraction of a turn = 1 turn order of rotation = 1 turn 1. Rotation Symmetry and Transformations MHR 17

18 c) Solution Trace the shape onto a piece of tracing paper. Place your copy over the original. Rotate it to see the order and angle of rotation. Order of rotation = 1. When you rotate the kite 1 complete turn, it fits on itself time. Angle of rotation in degrees = 360 When you turn it about its centre of rotation it does not fit onto itself more than once in 1 complete turn. Angle of rotation as a fraction of a turn = 1 turn This figure does not have rotational symmetry. Complete the chart. a) Figure Order of Rotation Angle of Rotation (Degrees) Angle of Rotation (Fraction of a Turn) 360 = 1 turn 1 = turn order of rotation b) c) 18 MHR Chapter 1: Symmetry and Surface Area

19 Working Example : Relating Symmetry to Transformations Look at the figures. Figure 1 Figure Figure 3 a) Complete the chart for each figure. Solution Figure 1 Figure Figure 3 Type of Symmetry rotation line rotation and line Number of lines Number of lines Number and Direction of Lines of Symmetry No lines of symmetry of symmetry: Direction: ( ) of symmetry: Direction: 1 1 Order of Rotation 3 1 Angle of Rotation 360 = = 1. Rotation Symmetry and Transformations MHR 19

20 b) What shape is used to make each design? How is each design made? Solution Use translation, reflection, rotation. Figure 1 is made from 1 arrow. Rotate the arrow 1 3 of a turn about the centre of. Figure is made from 1. Translate the circle times. Figure 1 Figure 3 is made from 1. Reflect the hexagon both vertically ( ) and ( ). or Figure Translate the hexagon times. Figure 3 Figure 3 Look at each figure. Figure A Figure B a) Circle the type of symmetry that each figure shows. Figure A: LINE SYMMETRY or ROTATION SYMMETRY or BOTH Figure B: LINE SYMMETRY or ROTATION SYMMETRY or BOTH b) For Figure A, draw the line(s) of symmetry. c) What is the order of rotation? Figure A: Figure B: d) How can you make each design using part of the figure and translations, reflections, or rotations? Figure A: Figure B: 0 MHR Chapter 1: Symmetry and Surface Area

21 Check Your Understanding Communicate the Ideas 1. Describe rotation symmetry. Use the design to help you. A A A A. Maurice says the design shows rotation symmetry. Claudette says the design shows line symmetry. Who is correct? Circle MAURICE or CLAUDETTE or BOTH. Give 1 reason for your answer. Practise 3. Each shape has rotation symmetry. Figure 1 Figure a) Mark the centre of rotation in each figure with a black dot. b) Complete the chart. Figure 1 Order of Rotation Angle of Rotation (Degrees) Angle of Rotation (Fraction of a Turn) turn 1 = order of rotation = 4 turn Figure 1. Rotation Symmetry and Transformations MHR 1

22 4. Complete the chart. Use tracing paper to help you. Figure Rotation Symmetry (Yes or No) Order of Rotation Angle of Rotation (Degrees) a) b) c) XOX 5. Each design has line and rotation symmetry. Find the number of lines of symmetry and the order of rotation. a) b) Number of lines of symmetry = Order of rotation = Number of lines of symmetry = Order of rotation = Apply 6. Complete the drawing of the rotation. Use the centre of rotation. The order of rotation is. E MHR Chapter 1: Symmetry and Surface Area

23 7. Look at the design. a) Draw the shape used to make the design. b) Describe how to make this design using 1 or more of the following: translations reflections rotations 8. a) Describe the symmetry shown on the playing card. b) Why do you think the card is designed like this? c) Does this card have line symmetry? Circle YES or NO. Give 1 reason for your answer. 9. Part of each object has rotation symmetry of order 6. Draw other objects that have rotation symmetry of order Rotation Symmetry and Transformations MHR 3

24 Math Link Your company asks you to create a design using rotation symmetry. a) Follow the steps to create this design with a rotation symmetry of 45. Use tracing paper to copy the diamonds at the top and bottom of the design. Mark the centre. Draw a line of symmetry through the diamonds and the centre. This will be your design template. Copy your tracing onto another piece of paper. Set a protractor on the centre of your diagram and mark points at 45, 90, and 135. Set your template behind the second piece of paper along the line of symmetry Rotate it to 45. Copy the design. Rotate it to 90. Copy the design. Rotate it to 135. Copy the design. On a separate sheet of paper, create a design of your own. Your design must have rotational symmetry. You will use this design for the cover of a box to hold your playing cards. 4 MHR Chapter 1: Symmetry and Surface Area

25 1.3 Warm Up 1. Use a formula to find the surface area of each right prism. What are the shapes of the faces? a) 6 m 10 m 4 m Area of front or back : A = Area of top or bottom : A = Area of end : A = SA = ( area of front or back) + ( area of top or bottom) + ( area of end) = ( ) + ( ) + ( ) = + + = m b) 5 cm 10 cm SA = (π r ) + (π d h). Find the area of the shaded region. Shaded area = area of large rectangle area of small rectangle = ( ) ( ) 13 cm 40 cm 5 cm 10 cm = = cm 1.3 Warm Up MHR 5

26 1.3 Surface Area Link the Ideas surface area the sum of the areas of all the faces of an object The formula for the surface area of a rectangular prism and a cylinder is the same: SA = (area of base) + (perimeter of base) (height) Surface area of rectangular prism: SA = (area of base) + (perimeter of base) (height) = (4 ) + ( ) 1 cm Perimeter = distance around a figure 4 cm 1 cm = = = 8 cm Use the order of operations. Surface area of cylinder: SA = (area of base) + (perimeter of base) (height) = (π r ) + ( π r) h = πr + πrh r h Working Example 1: Calculating Surface Area of a Solid All the angles in the solid are right angles. front top 8 cm 16 cm 0 cm 4 cm 15 cm front 8 cm 8 cm 0 cm side (right) 16 cm 4 cm a) What are the dimensions of the cutout piece? Solution The cutout piece is a right rectangular prism. 8 cm 8 cm 16 cm Dimensions are the length, width, and height. The dimensions of the piece are 8 cm by 8 cm by. 6 MHR Chapter 1: Symmetry and Surface Area

27 b) What is the total surface area of the solid? Solution Method 1: Find the Surface Area of Each Face Find the area of the 9 faces. Number the faces to help keep track of them. 7 (back) (bottom) 8 (right side) 4 6 Face View Surface Area Calculation 1 15 cm 4 cm 8 cm 8 cm 0 cm 8 cm 16 cm Shaded area = area of large rectangle area of square = (15 0) (8 8) = = 36 cm Shaded area = area of large rectangle area of small rectangle = (0 4) (8 16) = 3 and 5 8 cm 0 cm 16 cm = cm Area = 8 16 = cm cm 8 cm 15 cm 16 cm 4 cm 15 cm 8 cm Area = 8 8 = cm Shaded area = area of large rectangle area of small rectangle = (15 4) (8 16) = = cm Area = 15 4 = cm 4 cm Area = cm 0 cm = cm Area = cm = cm 4 cm 1.3 Surface Area MHR 7

28 Total SA = = cm The total surface area is cm. Method : Use Symmetry By symmetry, opposite faces match. face + face 5 = face 9 (bottom) face 1 + face 4 = face 8 (right side) 7 (back) (bottom) 8 (right side) 4 6 face 6 + face 3 = face (back) So, calculate the area of faces 9, 8, and 7. face 9 (bottom): 0 4 = face 8 (right side): 15 0 = face 7 (back): 15 4 = Total of 3 faces = + + = cm To find the total surface area, double the area for face 9 + face 8 + face = The total surface area is cm. Did You Know? If you cut a right rectangular piece out of 1 corner of a rectangular prism, the surface area does not change. The surface area does change if the cutout goes across the solid. Figure 1 Figure 3 Figure 8 MHR Chapter 1: Symmetry and Surface Area

29 A set of concrete steps has these dimensions. Face 6 Face 4 Face 7 a) Calculate the area of the faces that are not against the ground. Draw each face and label the measurements: Face 1: Face : Face 3: Face 1 80 cm Face 3 Face 0 cm 60 cm Face 5 90 cm 40 cm 80 cm 0 cm Face 4: Face 5: Face 6: Face 7: 60 cm 0 cm 90 cm 40 cm Find the area of faces 1 to 7: Find the total area: Sentence: b) What is the area of the face of the step against the ground? Sentence: 1.3 Surface Area MHR 9

30 Working Example : Painting a Bookcase Raubyn wants to paint the shelves and the outside of the bookcase. He will not paint the back. The back is very thin plywood. It does not add any extra depth. 115 cm cm a) What assumptions could you make about how he calculates the area to be painted? Solution 140 cm Assumptions could include: He paints the underside of shelves. The measurement of the shelves is the same as the top. He does not paint the area of the base on which the bookcase stands. He paints the bookcase after it is put together. Literacy Link An assumption is something you have not been told, but you believe to be true. b) Find the surface area that Raubyn needs to paint. Solution Group similar size surfaces together. Draw and label the dimensions of each group. Group 1: Both sides of top + top and bottom of each shelf = 8 rectangular areas Surface area = 8 area of top or bottom of shelf = 8 l w = = cm 115 cm cm Group : Both sides of the ends of the bookcase cm Surface area = 4 area of side = 4 l w = 4 = cm Total surface area = = cm The surface area Raubyn needs to paint is cm. Side 140 cm 30 MHR Chapter 1: Symmetry and Surface Area

31 Calculate the outside surface area of the building. Do not include the bottom of the building. 5 m Label the dimensions of similar surfaces. 4 m 5 m 10 m triangles: rectangles (front and back): 6 m 4 m 4 m 6 m 6 m rectangles (sides): rectangles (roof ): Find the area of each group. Total surface area: Sentence: 1.3 Surface Area MHR 31

32 Check Your Understanding Communicate the Ideas 1. Use 4 interlocking blocks to build different solid objects. a) Explain how symmetry helps you find the surface area of 1 of your objects. Group matching surfaces. b) Slide the objects together. What is the area of overlap between the objects? area where objects touch Sentence: c) How does the overlap affect the total surface area of your composite object?. Nick makes a -layer cake. He puts strawberry jam between the layers. He wants to cover the outside of the cake with chocolate icing. Describe how he can find the area that needs icing. Literacy Link A composite object is an object made from or more separate objects. 3 MHR Chapter 1: Symmetry and Surface Area

33 Practise 3. Each object is made with centimetre cubes. Calculate the surface area. Include the bottom area. a) b) Number the faces. Area of each face: Count the cubes on each face. Face 1: Face : Face 3: Face 4: Face 5: Face 6: Face 7: Face 8: Total surface area: 4. a) If you build these cubes and slide them together, what is the area of the overlap? Assume the dots are 1 cm apart. b) What is the surface area of the composite solid? Number the sides. 1.3 Surface Area MHR 33

34 5. In the solid, all angles are right angles. top front right side 9 cm 5 cm 18 cm 5 cm 35 cm 14 cm What are the dimensions of the cutout piece? Write them on the diagram. length: = width: height: 5 = Apply 6. Use centimetre cubes to build this object. Look at the diagram of the top. Look at the diagram of the front. a) What is the surface area of the object? 1 cm Sentence: b) Using the same 10 cubes, build a rectangular prism. Find the surface area of your rectangular prism. Sentence: c) Are the surface areas the same? Circle YES or NO. Give 1 reason for your answer. 7. List 3 places or situations where surface area is important. 34 MHR Chapter 1: Symmetry and Surface Area

35 8. a) How can you use symmetry to find the surface area of this object? 8 cm 4 cm b) Find the surface area of the outside of the object. 1 cm 3 cm Area of front or back: A = (l w) (π r ) Area of rectangle area of circle Area of inside of hole: A = π r length of hole Area of top or bottom: Area of end of rectangular prism: Total surface area: Sentence: Math Link You are making a box to store the deck of playing cards you designed in Math Link 1.1. a) How many cards are in your deck? b) What are the dimensions of your deck of cards? length: width: height: c) What is the surface area of your deck of cards? Sentence: 1.3 Math Link MHR 35

36 Graphic Organizer Fill in the boxes. Rotation Symmetry Symmetry and Surface Area Line Symmetry Definition: Definition: Surface Area Example: centre of rotation Definition: Example: Define order of rotation: Ways to find the surface area of an object Give 1 example of each of the lines of symmetry: vertical Example: For the fan above, the order of rotation is Define angle of rotation: horizontal oblique Example: For the fan above, the angle of rotation is Example: For the fan above, the fraction of a turn is You can complete a symmetrical drawing by or 36 MHR Chapter 1: Symmetry and Surface Area

37 Chapter 1 Review For #1 to #6, choose the number that best matches the description. 1. line symmetry. rotation symmetry 3. angle of rotation 4. surface area 5. line of symmetry 6. order of rotation another name for a reflection line type of symmetry where the shape is divided into reflected halves the total area of all the faces of an object type of symmetry where a shape is turned onto itself number of times a shape fits onto itself in 1 turn the size of turn for a shape to rotate onto itself 1.1 Line Symmetry, pages Draw the lines of symmetry. Write the number of lines of symmetry for each design. Then, describe each line of symmetry using the terms vertical, horizontal, and oblique. a) b) Number of lines of symmetry Description 8. Half of a figure is drawn. The dashed line is the line of symmetry. Finish drawing each figure. a) b) Chapter 1 Review MHR 37

38 9. a) Draw a reflection of the shape in the y-axis. Label the image A, B, C, D, E, and F. A E F y D 4 b) Write the coordinates of the reflection image. B C A (, ) B C D E F x c) Using a different colour, translate the original shape R6, D3. Label the image A, B, C, D, E, and F. d) Which transformation shows symmetry? Circle REFLECTION or TRANSLATION. Describe the symmetry using the terms vertical, horizontal, and oblique. 1. Rotation Symmetry and Transformations, pages Complete the chart for each diagram. Diagram Order of Rotation Angle of Rotation (Degrees) Angle of Rotation (Fraction of a Turn) a) 360 = 1 b) 11. What type of symmetry does the design have? Circle ROTATION SYMMETRY or LINE SYMMETRY or BOTH. Give 1 reason for your answer. 38 MHR Chapter 1: Symmetry and Surface Area

39 1.3 Surface Area, pages The triangular prism has 1 triangular end fastened to the wall. All the other faces are showing. What is the surface area of the faces that are showing? 7.8 cm 18 cm 9 cm Sentence: 13. Two blocks are placed 1 on top of the other. a) If the blocks are separated, what is the surface area of each block? Small Block: Area of front or back: Area of top or bottom: Area of side: 3 cm 4 cm 18 cm 16 cm 8 cm 4 cm Total surface area: Large Block: Area of front or back: Area of top or bottom: Area of side: Total surface area: b) What is the total surface area of the blocks when separated? Sentence: c) What is the surface area of the stacked blocks? (answer to part b) ( area of shape where blocks touch) Sentence: Chapter 1 Review MHR 39

40 Key Word Builder Use the clues to find the key words in Chapter 1. Then, write them in the crossword puzzle. Across 6. the sum of the areas of all the faces of an object ( words) 8. the measure of an angle to turn a shape onto itself (3 words) Down 1. a type of symmetry where an object is divided into identical halves by a line ( words). occurs when a shape can be turned about its centre of rotation and fit onto itself more than once in a complete turn ( words) 3. a line running through the centre of an object so that the halves on each side are mirror images (3 words) 4. the number of times a shape or design fits onto itself in 1 complete turn (3 words) 5. the point about which the rotation of an object turns (3 words) 7. when a shape or design can fit onto itself by reflection or rotation, it has (1 word) 40 MHR Chapter 1: Symmetry and Surface Area

41 Chapter 1 Practice Test For #1 to #4, choose the best answer. 1. Which design has rotation symmetry of order? A B C D How many lines of symmetry does the design have? A 0 B 1 C D 4 3. Which figure does NOT have rotation symmetry? A B OHO C D Complete the statements in #4. 4. The design has rotation symmetry. a) The order of rotation is b) The angle of rotation is Chapter 1 Practice Test MHR 41

42 Short Answer 5. Use these upper-case letters: a) List 5 letters that have horizontal line symmetry. b) List 5 letters that have vertical line symmetry. c) List 1 letter that has oblique line symmetry. d) List 5 letters that have rotation symmetry where the angle of rotation is a) What is the surface area of the rectangular prism? 7 cm 1 cm 5 cm b) The rectangular prism has a 1-cm cube cut out of 4 of its corners. How many faces of each cube are visible before they are removed? 7 cm 1 cm How many faces are visible after they are removed? 5 cm What is the surface area of this rectangular prism with 4 of its corners removed? 4 MHR Chapter 1: Symmetry and Surface Area

43 Math Link: Wrap It Up! You employer wants to see the playing card box that you designed in the Math Links from this chapter. Include the design from Math Link 1.1, page 14. Describe the type of line symmetry your card design shows. Use the terms horizontal, vertical, and oblique. Include the design from Math Link 1., page 4. This will be the design for the cover of your box. Write the dimensions and the surface area of the box needed to hold 1 set of your playing cards. Use your answers from Math Link 1.3, page 35. Write the total surface area of 6 boxes of cards. SA of 1 box: Total SA of 6 boxes: Your company wants to sell a large package containing 6 decks of your playing cards. Draw how you would arrange the decks for packaging. Using your dimensions for 1 box of cards, calculate the measurements for your large package. Write them on your design. Calculate the surface area of your large package. Math Link: Wrap It Up! MHR 43

44 Challenge Making a Paper Airplane Materials paper ruler scissors 1. Make a paper airplane by following the folding instructions. In Step 6, make 4 small cuts as shown to create tabs. Fold the tabs up to make flaps. Step 1 Step Step 3 Step 4 Step 5 Step 6. a) Using your airplane from #1, find the total area of the top view of the wings. Include the flaps. Measurement of wings of airplane: Area of triangle ( wings): b) Fly the airplane 5 times. Record the distance travelled in metres. Record the direction travelled in each flight. Flight Distance Travelled (m) Direction of Flight 44 MHR Chapter 1: Symmetry and Surface Area

45 3. Design and create a non-symmetrical airplane. a) Find the new surface area of the wings. Literacy Link Non-symmetrical means it is not the same on both sides. Drawing of wings (include measurements): Area of wings: Sentence: b) Fly the airplane 5 times. Record the distance travelled, in metres. Record the direction travelled in each flight. Flight Distance Travelled (m) Direction of Flight 4. Which airplane flew the best? Circle SYMMETRICAL or NON-SYMMETRICAL. Give reasons why. Include the words surface area and symmetry in your explanation: Reason 1 (surface area): Reason (symmetry): Challenge MHR 45

46 Answers Get Ready, pages 3 1. a) 4, vertically up b) 3 units horizontally left, 4 units vertically down. a) y b) y line of reflection r r x 1.1 Line Symmetry, pages 6 14 Working Example 1: Show You Know a) 1; horizontal b) 1; vertical c) 4; 1 horizontal, 1 vertical, oblique d) 1; vertical Working Example : Show You Know a) b) x 3. a) y b) D = (, 3) E = ( 1, 3) D G F = ( 1, ) G = (, ) E D E F G x cm Math Link 1. a) YES b) 3. a) YES b) NO 3. a) C b) C c) C D E d) vertically r D' E' Communicate the Ideas 1. NO. A rectangle that is a square has 4 lines of symmetry.. KENDRA. It does not overlap itself if it s cut out and folded. Practise 3. a) b) c) d) C' 4. a) A to A : 5, 3; B to B : 5 spaces right and 3 spaces up; C to C : 5 spaces right and 3 spaces up b) Answers will vary. Example: 4 spaces left and 4 spaces down would put the translation in quadrant III. 1.1 Warm Up, page 5 1. a) y 4 A 0 A E B C 4 D A (0, 1); B (1, ); C (3, ); D (3, 1); E (, 1); F (1, 0) b) y D (0, 0); O (, ); G (, 1) O G B F F 0 E D D C D. a, b, d 3. a) b) O G 4 cm x x 3 cm 4. a) b) a) b) c) d) Apply 7. a) y b) A ( 3, 6); C (, ); E ( 5, 3) c) YES; The line of reflection A divides the original image 5 and the reflection image into identical parts. E C 4. Answers will vary x 46 MHR Chapter 1: Symmetry and Surface Area

47 8. a) 1 b) 1 c) d) 4 9. a) B, C, D, E, H, I, K, O, X b) A, H, I, M, O, T, U, V, W, X, Y c) H, I, O, X 10. Answers will vary: Examples: WOW W WHO W HAH H OW HO AH HI H I Working Example : Show You Know a) Figure A: BOTH; Figure B: ROTATION SYMMETRY b) 11. Answers will vary. Example: 88 Math Link Answers will vary. 1. Warm Up, page a) and c) B y B b) A (1, 1); B (1, 9); C (4, 1) c) Figure A: ; Figure B: 4 d) Figure A: reflection along a horizontal or vertical line; Figure B: rotation about the centre Communicate the Ideas 1. Answers will vary. Example: Rotation symmetry is when a section of a design can be rotated around the centre and shows symmetry with other parts.. BOTH; The design can be folded over a horizontal or vertical line of symmetry to overlap itself, or a corner piece can be rotated 4 times to create the design. Practise 3. a) A C A A x C C. a) Figure Lines of Symmetry Equilateral triangle B Number of Lines of Symmetry 3 b) Order of Rotation Figure 1 4 Angle of Rotation (Degrees) 360 = 90 4 Figure 180 Angle of Rotation (Fraction of a Turn) 1 turn 1 = turn order of rotation 4 1 turn Square 4 4. Figure Rotation Symmetry (Yes or No) Order of Rotation Angle of Rotation (Degrees) a) Yes 4 90 Regular pentagon 5 Regular hexagon 6 b) The number of sides is equal to the number of lines of symmetry. 3. a) 5 b) 0 1. Rotation Symmetry and Transformations, pages 16 4 Working Example 1: Show You Know Order of Rotation a) Angle of Rotation (Degrees) 360 Angle of Rotation (Fraction of a Turn) 1 turn 1 = 180 = turn order of rotation b) turn c) turn 1 = turn order of rotation 3 b) Yes 3 10 XOX c) Yes a) Number of lines of symmetry = 6, Order of rotation = 6 b) Number of lines of symmetry =, Order of rotation = Answers MHR 47

48 Apply 6. E E 7. a) ; square b) Rotate the figure 180 and join the figures. Translate the new figure to the right so it does not overlap. Join the two figures. Now, join the new figure with the original one on the right. 8. a) The top half is rotated 180. b) Cards are designed so they can be read while being held from either end. c) NO; If you fold it in half, the design does not overlap. 9. Math Link Answers will vary. 1.3 Warm Up, page 5 1. a) 48 m b) 471 cm. 470 cm 1.3 Surface Area, pages 6 35 Working Example 1: Show You Know a) The total area of the faces that are not against the ground is cm. b) The area of the step that is against the ground is 7 00 cm. Working Example : Show You Know The surface area is 84 m. Communicate the Ideas 1. Answers will vary. Example: a) Apply 6. a) The surface area is 36 cm. b) The surface area of a 1 5 rectangular prism is 34 cm. c) NO. There are more surfaces that overlap in the 1 5 rectangular prism. 7. Answers will vary. Examples: when icing a cake, painting a building, or wrapping a present. 8. a) The top and bottom, left and right, and front and back are symmetrical so they have the same surface area. b) The surface area is 34.6 cm. Math Link Answers will vary. Examples: a) 5 b) length: 10 cm; width: 7 cm; height: cm c) The surface area is 08 cm. Graphic Organizer, page 36 Rotation Symmetry Definition: when a shape or design can be turned about its centre of rotation so it fits onto itself more than once in 1 complete turn Define order of rotation: the number of times a shape fits onto itself in 1 complete turn Example: 8 Define angle of rotation: the minimum measure of the angle needed to turn a shape or design onto itself Example: 45º Example: 1 8 turn Surface Area Definition: the number of square units needed to cover an object Ways to find the surface area of an object: find the surface area of each face and add them together; use symmetry; for a composite shape, find the surface area of each part and subtract the area of the overlapping parts Line Symmetry Definition: a type of symmetry where a line of symmetry divides an image or object into identical, reflected halves Example: the Canadian flag has line symmetry Examples: vertical line of symmetry: horizontal line of symmetry: oblique line of symmetry: You can use symmetry to find the surface area of 1 side, and then you can multiply it by to get the surface area of the opposite side. b) The area of overlap is 1 cm. c) The overlap reduces the total surface area by twice the amount of the surface area that is touching the other shape.. Nick should find the area of each shape and add them together. Then he should subtract times the area of the overlap (the bottom of the cylinder). He should also subtract the area of the bottom. Practise 3. a) 7 cm b) 11 cm 4. a) The area of overlap is 4 cm. b) The surface area is 70 cm. 5. length = 17 cm; width = 9 cm; height = 11 cm You can complete a symmetrical drawing by: paper folding; measuring; counting Chapter 1 Review, pages type of symmetry in which the shape can be divided into reflected halves. type of symmetry in which a shape is turned onto itself 3. the size of turn for a shape to rotate onto itself 4. the total area of all the faces of an object 5. another name for a reflection line 6. number of times a shape fits onto itself in 1 complete turn 48 MHR Chapter 1: Symmetry and Surface Area

49 7. a) Number of lines of symmetry: 4 Description: 1 horizontal, 1 vertical, oblique 10. Diagram Order of Rotation Angle of Rotation (Degrees) Angle of Rotation (Fraction of a Turn) a) = b) b) a) b) 9. a) and c) A B -4 Number of lines of symmetry: 6 Description:1 horizontal, 1 vertical, 4 oblique y E D 4 D E F F A - C - C B E D A F 4 x B C b) A (5, 3) B (5, ) C (1, ) D (1, 4) E (3, 4) F (3, 3) d) REFLECTION; The reflection shows a vertical line of symmetry. 11. ROTATION SYMMETRY; There is no line symmetry because of the shading and overlapping. 1. The total surface of the faces that are showing is 51.1 cm. 13. a) small block: 444 cm ; large block: 1088 cm b) The surface area of the blocks when separated is 153 cm. c) The surface area of the stacked blocks is 144 cm. Key Word Builder, page 40 Across: 6. surface area 8. angle of rotation Down: 1. line symmetry. rotation symmetry 3. line of symmetry 4. order of rotation 5. centre of rotation 7. symmetry Chapter 1 Practice Test, pages D. D 3. D 4. a) 8 b) Answers will vary. Examples: a) B, C, D, E, H, I, K, O, X b) A, H, I, M, O, T, U, V, W, X, Y c) O d) H, I, N, O, S, X, Z 6. a) 358 cm b) 3; 3; 358 cm Math Link: Wrap It Up!, page 43 Answers will vary. Challenge, page Answers will vary. Answers MHR 49

50