Answers. Chapter Line Symmetry, pages Answers MHR a) 4. a) b) c)

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1 nswers hapter 1 9. a) 1.1 Line Smmetr, pages a) 5 b) c) a) b) b) (-3, 6), (, ), (-5, 3) c) Yes, the original image and the reflected image show line smmetr. However, each individual figure does not show line smmetr within itself. 10. ample: a) 5 c) , D, and. The can be folded in two different was so that the overlap eactl. ach of the other figures has more than two lines of smmetr. 7. a) b) b) (7, 6), (6, ), (9, 3) c) No. The image was not reflected and does not contain line smmetr within itself. d) No. See eplanation in 10c) a) b) 11. ample: I agree because these shapes show horizontal and/or vertical line smmetr within themselves, a horizontal or vertical translation of the shape results in line smmetr between the original and new images. figure with onl vertical line smmetr within itself will show line smmetr after a horizontal translation onl. nswers MHR 53

2 1. a) Yes, the flag has horizontal line smmetr. If folded from bottom to top through the middle of the horizontal blue stripe, the upper and lower halves will overlap. b) Moving the vertical blue and white stripes to the centre of the flag would give it two lines of smmetr. 13. a) one horizontal b) one vertical c) two: one horizontal and one vertical d) four: one horizontal, one vertical, and two oblique 1. ample: b) The digital clock can show horizontal line smmetr at 8:08, 8:38, 3:03 (not 1:01 or 10:10, etc., because of the shape of the number 1); the analogue clock can show line smmetr at an time when the line of smmetr bisects the angle between the hands and bisects the squares representing the numbers or when the time is 6:00 or 1:00.. (,-3) (,3) 0 lines 1 line lines lines 15. a),, D,, H, I, K, O, X b), H, I, M, O, T, U, V, W, X, Y c) H, I, O, X 16. a) ample: WOW W OW HH H H b) ample: HI H I c) ample: WHO W HO (-6,) (6,) 17. a) 0, I, 3, 8 b) ample: I00I c) ample: 80I a) 70 square units 3. 6 (5, 5) b) The number of equal, interior angles equals the number of lines of smmetr. c) Yes. s the number of interior angles increases, ou approach a circle shape, which is smmetrical from all angles a) b) Different colours mean that figure becomes a non-smmetric figure. c) Figure has five lines of smmetr =10 Triangle is the result of a horizontal translation 0 units to the right. 1. a) If the two-dot separator in the digital clock is ignored, then both clocks show line smmetr at some time. Triangle is the image created b rotating the original triangle 90 about the origin.. ample: Yes, a three-dimensional object such as a cube is smmetric because all faces and edges are of equal size. plane cutting the cube parallel to a face and through the centre will create two identical rectangular prisms. 1. Rotation Smmetr and Transformations, pages 1 5. a) order of rotation =, angle of rotation = 90, fraction of a turn = 1, centre of rotation is at centre of figure. b) order of rotation =, angle of rotation = 180, fraction of a turn = 1, centre of rotation is at centre of figure. c) order of rotation =, angle of rotation = 180, fraction of a turn = 1, centre of rotation is between 9 and a) Yes; angle of rotation = 90 b) Yes; angle of rotation = 10 c) Yes; angle of rotation = a) number of lines of smmetr = 6, order of rotation = 6 b) number of lines of smmetr =, order of rotation = c) number of lines of smmetr =, order of rotation = 5 MHR nswers

3 7. a) number of lines of smmetr = 3, angle of rotation = 10 b) number of lines of smmetr = 5, angle of rotation = 7 8. a) b) ach image is oriented with the longer dimension along the horizontal and the order of labelling switches between clockwise and counter-clockwise. b) Rotate the original figure 180º and join the two figures. Translate the new figure to the right so it does not overlap. Join the two figures. Now join this new figure with the original one on the right. D D 9. a) 3 b) 0 6 D D c) No, because the image does not show line smmetr. 10. a) ample: line smmetr: pu, pa; rotation smmetr: ki, ku; 11. a) oth. Lines of smmetr: the vertical black line with three red squares to its left and to its right in each row; the horizontal black line with two red squares above it and below it in each column. The centre of rotation is located where the two lines of smmetr intersect. b) Neither. There would be a vertical line of smmetr through the noses of the centre column of faces if the face colours on each side of the line matched each other. c) Neither. There would be 180º rotation smmetr if the pink and blue dolphins were the same colour. d) Rotation smmetr onl of order. The centre of rotation is at the centre of the figure. 1. a) The vertices of the images are: (, 1), (, 1), (, 5), D (, 5); (-1, ), (-1, ), (-5, ), D (-5, ); (, -1), (, -1), (, -5), D (, -5). D D 13. a) order of rotation =, angle of rotation = 90 b) No; the rotation of the design makes line smmetr impossible. 1. a) There are eight lines of smmetr; the angle between the lines is.5 b) order of rotation = 8, angle of rotation = ample: 16. a) b) 0 6 D D nswers MHR 55

4 17. a) order of rotation = 5, angle of rotation = 7 b) order of rotation = 7, angle of rotation = 51. c) order of rotation = 6, angle of rotation = 60 d) order of rotation = 1, angle of rotation = a) ample: Some parts of the diagram appear to be rotated and projected five times, whereas others (such as the interior bolts) appear four times. Depending on which part is chosen, the order of rotation ma seem different. b) dding another bolt so that the total on the inside rim is five would give this diagram rotation smmetr. 19. a) The top half of the card, along with the K smbol, is rotated 180 (rotation order ). b) ards are designed so the can be read while being held from either end. c) No; attempting to fold the card along an line does not result in an overlap. 0. Rachelle is correct. lthough there are 0 wedges on the board, the alternating red and green colours must be grouped together and then rotated to reproduce the image. 1. a) H, I, N, O, S, X, Z b) 0, I, 8 c) ample: X08OI. ample: heagon-shaped sign, a si-sided snowflake, or other object. 3. a) : no smmetr because of the variation in overlap of green and blue circles in pairs of opposite ras; : rotational smmetr of order and line smmetr b) ample: The logo of Sun Microsstems shows rotational smmetr, and UNSO shows vertical line smmetr.. a) m = 1. The letter m represents the number of teeth in the gear. b) n = 16 c).5 turns d) 6 turns e) ()(m) = number of turns 5. a) ll of the objects have at least one eample of line smmetr. ll of Group show multiple lines of smmetr. Onl the left object in Group does not show rotation smmetr. b) ample: cube has man lines of smmetr because the edges are all of equal length. 6. r 7. a) 17 cm b 9 cm b 11 cm b) The surface area with the cutout is the same as the surface area without the cutout (750 cm ). 8. a) For one bo: width = cm, height = 1 cm, depth = 3 cm b) 96 cm c) one bo: cm ; si boes: 13 cm. The ratio of the surface area for the si combined boes to the surface area for si separate boes is 8 : a) 30 cm b) 5 70 cm c) Onl three surface areas need to be calculated (shelf, side, back). 10. a) 36 cm b) ample: 1 cm cm 5 cm rectangular prism has a surface area of 3 cm, while a 1 cm 1 cm 10 cm rectangular prism has a surface area of cm. 11. ample: Surface area is important to consider when painting a building, icing a cake, and packaging items. 1. a) 0.06 m. This allows rainwater to flow awa from the house, off the roof to the ground below. b) m ; You must assume that there is no bottom included in this calculation, all angles are 90, the garage door is to be included, and an average height of. m. 13. a) left mug: 86.5 cm ; right mug: 98.6 cm b) The left mug has better heat-retaining properties. It has less surface area eposed to the air resulting in lower heat loss m 15. a) The object s top and bottom faces, left and right faces, and front and back faces are smmetrical. b) 3.57 cm 16. a) You must use the Pthagorean theorem three times. b) 51.7 m c) 33 bundles of shingles for a cost of $ a) The two flues are each cm wide and 0 cm tall. b) 67 cm cm ; To calculate the surface area of the inside of the bo, ou must assume the metal has zero thickness. 19. a) 5 cm 6 cm 6 cm 5 cm 5 cm b) The shape made would be a heagon. The order of rotation for the shape is Surface rea, pages a) stimate the total surface area of a solid, 5 rectangular prism. The total surface area of the object is 7 cm. b) stimate the total surface area of a solid, 6 prism. The total surface area of the object is 11 cm. 5. a) 16 cm b) 56 cm 6. a) 1 cm b) 1 cm c) square cake: 1850 cm ; 8 square-cake slices: cm ; round cake: 15.3 cm ; 8 round-cake slices: cm ; surface area increase of square cake: 18.6 cm ; surface area increase of round cake: 100 cm ; For both cakes, the eight equal slices increase the total surface area b almost double compared to the unsliced cakes. 56 MHR nswers

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6 7. a) greater than 1 b) equal to 1 c) less than 1 8. Sierra Leone 9. a) enlargement b) 100. The lens makes all dimensions of the original image appear to be enlarged b 100 times. 10. amine the font used in both posters. Mia s font is 0.5 cm high, and Hassan s is 0.5 cm high. Mia s font is twice the height of Hassan s, so the scale factor is. 11. ample: Measure the width of the sunglasses in both images. Determine the scale factor. Then, see if the scale factor applies to another pair of corresponding parts (e.g. the width of the mouth). 1. a) width = 7 cm, length = 5 cm b) width =.5 cm, length = 9 cm 13. waist front back waist waist front back lower edge front back scale factor lower edge scale factor lower edge scale factor 3. Scale Diagrams, pages a) Divide 1 b 3. b) Divide 117 b a) 13 b) a) 110 cm or 1.1 m b) 16 mm 7. a) 38.9 m b) 1 m 8. a) 0.15 b) a) 0.0 b) a) 6.3 cm b) Yes, an actual egg could have a length of approimatel 6.3 cm. 1. a) 1 b) The length of the footprint image is 15 approimatel 3. cm. With the scale factor of 1 15, the actual length of the footprint is approimatel 51 cm. c) ample: The span of a human hand to the footprint could be approimatel 1 :.1. The footprint is approimatel times as large as a human hand span Yes, it will fit. The model will measure.5 m in height, giving it a 0.5 m clearance. 17. length = 17. m, height =.35 m 18. a) b) 3 c) 1.5 d) 1 3 e) a) b) 700 cm or 7 m 0. a) ample: You could reduce the image with a scale factor of 1. Then, width = 10.5 cm, height = 7.5 cm, depth =.5 cm 17. a) -6 0 D 6 F -6 b) b) Yes, the sides of the larger triangle are 3 times the length of the sides of the smaller triangle. c) 1 3 d) 3 e) rea of is.5 square units; area of DF is 0.5 square units. f) 1: 0.1; 1: 9 g) The scale factor of the area is 3 times larger than the scale factor of the sides (when comparing DF : ) 1. a).6 m b) 5. m. a).5, b) ample: Scale factors between a 5 smaller object and a larger one are often easier to use. nswers MHR 63

7 .3 Similar Triangles, pages orresponding angles: P and T, Q and U, R and V. orresponding sides: PQ and TU, PR and TV, QR and UV. 5. orresponding angles: and Y, and W, and X. orresponding sides: and YW, and WX, and YX. 6. Yes, the triangles are similar because the sides are the proportional; the sides are related b a scale factor of No, the triangles are not similar because the sides are not proportional. 8., FG, and L KLM are similar F. 3 G 9 K = = m m m 15. = 76.5 cm 16. Peter is taller. Michael is 19.3 cm tall. Peter 168 cm Michael 9 M 18. a) = 0.8 m b) The shadow ma not reach the street level due to surrounding buildings. 19. a) No, the corresponding angles are not equal. The angle measures of one triangle are: 50, 60, and 70. The angle measures of the other triangle are: 50, 50, and 80. b) Yes, the triangles are similar because the both have angle measures of 5, 60, and a) 13.3 cm and 16.0 cm b) 1 : First, measure our height, and the length of the building s shadow. fter measuring our own shadow, find the ratio of our shadow to the building s shadow. Then, divide our height b that value to find the height of the building.. ZY =.9 cm 3. The area is 150 cm.. Similar Polgons, pages a) Similar b) Not similar. 5. = 6. =.8 m 7. No. The corresponding angles must be the same. 8. a) Michael s shadow 0 cm Peter s shadow 5 cm 17. ample: Two buildings, and, stand side b side. uilding casts a shadow of 10 m and is 60 m tall. uilding has a shadow of 60 m. Using the diagram, find the height of uilding. 60 m = 60 uilding is 30 m tall. s shadow 60 m s shadow 10 m similar dissimilar b) The two similar heagons are similar to the photo because the interior angles are the same and the side lengths are related b a scale factor. The two dissimilar heagons do not have these properties. 9. The side length of the game board will be 15.0 cm. 10. a) 7.5 m b) 1080º. ample: n octagon can be divided into si non-overlapping triangles. 11. a) The final enlargement should be 6 times the size of the original diagram. b) The corresponding angles are equal, and the dimensions are all enlarged b the same proportion cm cm 6 MHR nswers

8 1. a) 1 0 b) 7.5 cm 7.5 cm 7.5 cm 1 cm 7.5 cm 15. = The polgons are not similar cm 18. = 7.; = a) c) length model = 15 cm, width model = 1 cm L 16. The ratio of areas to the ratio of corresponding side lengths in similar polgons is equal to the scale factor comparing side lengths squared. 17. The volume ratio is the same as the side ratio cubed. 18. a) The similar polgons have 7 sides, so the are heptagons. b) ample: ach heptagon is a reduction of the centre heptagon, with the scale factor decreasing with distance from the centre. hapter Review, pages POLYGON. SIMILR 3. SL FTOR. PROPORTION 5. a) b) 6. The vertical height of the drawing is 3 cm. The enlarged egg will have a vertical height of 9 cm. 7. The vertical height of the drawing is 3 cm. The reduced drawing will have a vertical height of 1.5 cm. 8. a) b) c) a) 1 cm b) 13.9 cm cm No. The corresponding sides are not proportional. 1. = 10

9 nswers hapter 1 Practice Test 1. D. D 3.. D 5. a) 8 b) 5 6. a) Horizontal lines of smmetr:,, D,, H, I, K, O, X; Vertical lines of smmetr:, H, I, M, O, T, U, V, W, X, Y; Oblique lines of smmetr: O b) H, I, N, O, S, X, Z 7. The two surface areas are identical cm 9. a) The dimensions are width = 1 cm, height = 1 cm, depth = 36 cm. b) The dimensions are width = 3 cm, height = 3 cm, depth = cm. c) ample: ver long side can often increase surface area more than having sides of equal or near-equal value. 10. There are vertical, horizontal, and two oblique lines of smmetr. There is also rotation smmetr, of order 8 with an angle of rotation of 5. hapter Practice Test 1.. D cm 6. scale factor 7. Look for a reduction using a scale factor of 0.5. ample: 13. The triangles are not similar. The corresponding sides are not proportional in length. 1. ample: mm 11. Yes, the are similar. ll corresponding angles are equal in measure and the corresponding sides are proportional using a scale factor of m 15. = ; =.88