9. Tina wants to estimate the heights of two. a) Tina s shadow is 2.4 m and the first tree s. b) Tina s shadow is 0.

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1 b) J 1 15 G F 9. Tina wants to estimate the heights of two trees. For each tree, she stands so that one end of her shadow coincides with one end of the shadow of the tree. Tina s friend measures the lengths of her shadow and the tree s shadow. Tina is 1.7 m tall. c) E Jaquie is 1.6 m tall. When her shadow is. m long, the shadow of the school s flagpole is 16 m long. How tall is the flagpole, to the nearest tenth of a metre? Sun s ras 8. ssessment Focus Work with a partner. Use the method described in question 7. hoose an object whose height ou cannot measure directl. a) raw a labelled diagram. b) Indicate which triangles are similar. c) etermine the height of the object. Show our work. Sun s ras 1.6 m 16 m. m 1.7 m a) Tina s shadow is. m and the first tree s shadow is 1.8 m. What is the height of the tree? b) Tina s shadow is.8 m and the second tree s shadow is 1.8 m. What is the height of the tree? 1. When the shadow of a building is 16 m long, a -m fence post casts a shadow 3 m long. a) Sketch a diagram. b) How tall is the building? 11. This scale diagram shows the measurements a surveor made to determine the length of Lac Lalune. What is this length? How do ou know? Lac Lalune Sun s ras 3 m m 1 m 35 UNIT 7: Similarit and Transformations

2 1. To help calculate the distance PQ across a river, Emil drew the diagram below based on measurements he made. What is the distance across the river? P Q 1. The foot of a ladder is 3 m from the base of a wall. The ladder just touches the top of a 1.-m fence that is. m from the wall. How high up the wall does the ladder reach? How do ou know? 11 m River R m T S 15 m 1. m Take It Further 13. Phillipe places a mirror M on the ground 6. m from a tree. When he is 1.7 m from the mirror, he can see the top of the tree in the mirror. His ees are 1.5 m above the ground. The diagram below shows the equal angles. How can ou use similar triangles to determine the height of the tree to the nearest tenth of a metre? 15. In the diagram below, how high are the two supports and for the conveor belt? 1 m. m 3. m 15 m 18 m 16 m 1.5 m 1.7 m M 6. m Reflect How do the properties of similar triangles help ou to determine distances that cannot be measured directl? Include an eample in our eplanation. 7. Similar Triangles 351

3 Mid-Unit Review photo of a gmnast is to be enlarged. The dimensions of the photo are 15 cm b 1 cm. What are the dimensions of the 7 enlargement with a scale factor of? 5. computer chip has dimensions 15 mm b 8 mm. Here is a scale drawing of the chip These quadrilaterals have corresponding angles equal. F E H.8 G J 6.75 K 7. a) etermine the scale factor of the diagram. b) raw a scale diagram of the chip with a scale factor of a) op this polgon on 1-cm grid paper. b) raw a scale diagram of the polgon 3 with a scale factor of. Show an 5 calculations ou made.. This top view of a swimming pool is drawn on.5-cm grid paper. The dimensions of the pool are 6 m b m. etermine the scale factor of the reduction as a fraction or a decimal N a) re an of these quadrilaterals similar? Justif our answer. b) hoose one quadrilateral. raw a similar quadrilateral. How do ou know the quadrilaterals are similar? 6. window has the shape of a heagon. raw a heagon that is similar to this heagon. Eplain how ou know the heagons are similar. M tree casts a shadow 8 m long. t the same time a -m wall casts a shadow 1.6 m long. a) Sketch a diagram. b) What is the height of the tree? 35 UNIT 7: Similarit and Transformations

4 7.5 Reflections and Line Smmetr FOUS raw and classif shapes with line smmetr. How can ou use this photograph to show what ou know about line smmetr? Investigate Your teacher will give ou a large cop of the shapes below. E F G H I J K L Which shapes have the same number of lines of smmetr? Sort the shapes according to the number of lines of smmetr the have. Which shapes do not have line smmetr? How can ou tell? Reflect Share & Share our sorting with another pair of students. ompare strategies for identifing the lines of smmetr. 7.5 Reflections and Line Smmetr 353

5 onnect The pentagon E has one line of smmetr G, because G divides the pentagon E into two congruent parts: polgon G is congruent to polgon EG. Line of smmetr lso, each point on one side of the line of smmetr has a corresponding point on the other side of the line. These two points are the same distance, or equidistant from the line of smmetr: points and E correspond, F FE, and E G. F G E line of smmetr is also called a line of reflection. If a mirror is placed along one side of a shape, the reflection image and the original shape together form one larger shape. The line of reflection is a line of smmetr of this larger shape. Original shape Original shape and its reflection image Line of smmetr Eample 1 Identifing Lines of Smmetr in Tessellations Identif the lines of smmetr in each tessellation. a) b) 35 UNIT 7: Similarit and Transformations

6 Solution a) The red line is the line of smmetr b) This tessellation has lines of for this tessellation. Each point on one smmetr. For each line, a point side of the line has a corresponding point on one side of the line has a on the other side. The pattern on one side matching point on the other side. of the line of smmetr is a mirror image nd, the pattern on one side of the of the pattern on the other side. line is a mirror image of the pattern on the other side. Two shapes ma be related b a line of reflection. Eample Identifing Shapes Related b a Line of Reflection Identif the triangles that are related to the red triangle b a line of reflection. escribe the position of each line of smmetr Reflections and Line Smmetr 355

7 Solution Triangle is the reflection image of the red triangle in the blue line through 5 on the -ais. Triangle is the reflection image of the red triangle in the red line through 3 on the -ais. Triangle is not a reflection image of the red triangle. Triangle is the reflection image of the red triangle in the green line through the points (9, 1) and (1, 9) We can use a coordinate grid to draw shapes and their reflection images. Eample 3 ompleting a Shape Given its Line of Smmetr Quadrilateral is part of a larger shape. raw the image of after each reflection below. Write the coordinates of the larger shape formed b and its image. escribe the larger shape and its smmetr. a) a reflection in the horizontal line through on the -ais b) a reflection in the vertical line through 6 on the -ais c) a reflection in an oblique line through (, ) and (6, 6) 6 Solution The red line is the line of reflection. Each image point is the same distance from this line as the corresponding original point. a) Point Image (, ) (, ) (, ) (, ) (6, ) (6, ) (6, ) (6, ) The larger shape has coordinates: (, ), (, ), (6, ), (6, ), (, ) This shape is a pentagon with line smmetr. The line of smmetr is the red line. 356 UNIT 7: Similarit and Transformations

8 b) Point Image (, ) (1, ) 8 1 (, ) (8, ) (6, ) (6, ) (6, ) (6, ) The larger shape has coordinates: (, ), (, ), (8, ), (1, ) This shape is an isosceles trapezoid with line smmetr. The line of smmetr is the red line. c) 6 Point Image (, ) (, ) (, ) (, ) (6, ) (, 6) (6, ) (, 6) 6 The larger shape has coordinates: (, ), (, 6), (, 6), (, ), (6, ), (6, ) This shape is a concave heagon with line smmetr. The line of smmetr is the red line. iscuss the ideas 1. How do ou identif whether a shape has a line of smmetr?. How are a line of reflection and a line of smmetr related? Practice heck 3. You ma have seen these hazardous substance warning smbols in the science lab. Which smbols have line smmetr? How man lines of smmetr? a) b) c) d) e) f) 7.5 Reflections and Line Smmetr 357

9 ppl. Identif the lines of smmetr in each tessellation. a) b) 6. State the number of lines of smmetr in each design. a) a tessellation created b M.. Escher 5. op each polgon on grid paper. It is one-half of a shape. Use the red line as a line of smmetr to complete the shape b drawing its other half. Label the shape with the coordinates of its vertices. a) P Q b) a Haida button blanket b) c) S R 6 6 E F U T V 6 7. ssessment Focus a) raw a triangle on a grid. b) hoose one side of the triangle as a line of reflection. i) raw the reflection image. ii) Label the vertices of the shape formed b the original triangle and its image. iii) Write the coordinates of each verte. iv) How man lines of smmetr does the shape have? c) Repeat part b for each of the other two sides of the triangle. o ou alwas get the same shape? Eplain. d) Repeat parts a to c for different tpes of triangles. e) Which tpes of triangle alwas produce a shape that is a quadrilateral with line smmetr? Justif our answer. 358 UNIT 7: Similarit and Transformations

10 8. Quadrilateral PQRS is part of a larger shape. S P fter each reflection below: raw the image of PQRS. Write the coordinates of the larger shape formed b PQRS and its image. escribe the larger shape and its smmetr. a) a reflection in the horizontal line through on the -ais b) a reflection in the vertical line through 8 on the -ais c) a reflection in the oblique line through (1, 1) and (, ) 9. a) Graph these points on grid paper: ( 3, ), ( 1, 1), (, 3), (1, 1), E(3, ). Join the points to form polgon E. b) Reflect the polgon in the -ais. raw and label its image. c) Write the coordinates of the shape formed b the polgon and its image. d) How man lines of smmetr does this shape have? How do ou know? Q R Identif the pentagons that are related to the blue pentagon b a line of reflection. escribe the position of each line of smmetr Take It Further a) On a grid, plot the points P(, ), Q(6, ), and R(, ). Join the points to form PQR. b) Reflect PQR in the line through the points (, ) and (, ). raw the reflection image. c) Reflect PQR in the line through the points (, ) and (, ). raw the reflection image. d) Reflect PQR in the -ais. raw the reflection image. e) Look at the shape formed b the triangle and all its images. How man lines of smmetr does this shape have? Reflect When ou see two shapes on a grid, how can ou tell if the are related b a line of reflection? Include eamples of shapes that are related and are not related this wa. 7.5 Reflections and Line Smmetr 359

11 Make Your Own Kaleidoscope The kaleidoscope was invented in It uses mirrors placed at different angles to produce patterns with smmetr. You will need small rectangular mirrors masking tape To make a simple kaleidoscope, use masking tape to join two mirrors so the stand at an angle. Place our mirrors on the arms of each angle below. Sketch and describe what ou see. Include an lines of smmetr in our sketch UNIT 7: Similarit and Transformations

12 7.6 Rotations and Rotational Smmetr FOUS raw and classif shapes with rotational smmetr. Look at these photographs. How are the windmills the same? How are the different? Investigate 3 You will need a protractor, a sharp pencil, tracing paper, and grid paper or isometric dot paper. Each of ou chooses one of these shapes and copies it on grid paper or dot paper. Trace our shape and place the tracing to coincide with the shape. Place a pencil point on the red dot. Rotate the tracing, counting the number of times the tracing coincides with the original shape, until ou make a complete turn. Repeat the rotation. This time, measure and record the angle ou turned the tracing through each time. Work together to draw a shape that coincides with itself times as ou rotate it. Reflect Share & Share our results with another group. What is the relationship between the number of times the shape coincided with itself and the angle ou turned it through each time? 7.6 Rotations and Rotational Smmetr 361

13 onnect tracing of this shape is rotated about its centre. We draw a line segment to help identif the angle the shape turned through before it coincided with itself. Rotation 1 Rotation Rotation 3 Rotation Original ack at the start The shape coincided with itself times in one complete turn; that is, during a rotation of 36. shape has rotational smmetr when it coincides with itself after a rotation of less than 36 about its centre. The number of times the shape coincides with itself, during a rotation of 36, is the order of rotation. The shape above has rotational smmetr of order. For each match, the shape rotated through We sa the angle of rotation smmetr is 9. This is. In general, for rotational smmetr: 36 the angle of rotation smmetr the order of rotation shape that requires a rotation of 36 to return to its original position does not have rotational smmetr. shape cannot have rotational smmetr of order 1. Eample 1 Identifing Shapes with Rotational Smmetr etermine which heagons below have rotational smmetr. State the order of rotation and the angle of rotation smmetr. a) b) c) 36 UNIT 7: Similarit and Transformations

14 Solution For each heagon: Join one verte to the red dot. Trace the heagon. Rotate the tracing about the red dot and record the order of rotation. alculate the angle of rotation smmetr. a) The order of rotation is The angle of rotation smmetr is: b) The order of rotation is. 36 The angle of rotation smmetr is: c) This heagon is rotated one complete turn before it coincides with itself. It does not have rotational smmetr. rotation is another tpe of transformation. We use a square grid to draw rotation images after a rotation of 9, or an multiple of 9, such as 18 and 7. We use isometric dot paper to draw rotation images after a rotation of 6, or an multiple of 6, such as 1 and Rotations and Rotational Smmetr 363

15 Eample rawing Rotation Images a) Rotate pentagon E b) Rotate trapezoid FGHJ 9 clockwise about verte E. 1 counterclockwise raw the rotation image. about verte F. raw the rotation image. E F G J H Solution Trace each shape and label the vertices on the tracing. a) Rotate pentagon E 9 b) Rotate trapezoid FGHJ clockwise about E. Side E 1 counterclockwise moves from being vertical to about F. The angle between being horizontal. FG and FG is E F G G H J J H Eample 3 Identifing Smmetr after Rotations a) Rotate rectangle : i) 9 clockwise about verte ii) 18 clockwise about verte iii) 7 clockwise about verte raw and label each rotation image. b) Look at the shape formed b the rectangle and all its images. Identif an rotational smmetr in this shape. 36 UNIT 7: Similarit and Transformations

16 Solution a) Trace rectangle and label the vertices. i) Rotate 9 clockwise about. Vertical side becomes horizontal side G. E F The rotation image is EFG. ii) Rotate 18 clockwise about. P G Vertical side becomes vertical side K. H The rotation image is HJK. 7 iii) Rotate 7 clockwise about. N M Vertical side becomes horizontal side P. K J The rotation image is MNP. b) The resulting shape EFGHJKMNP has rotational smmetr of order about point. iscuss the ideas 1. How do ou determine whether a shape has rotational smmetr?. How can ou determine: a) the order of rotational smmetr? b) the angle of rotation smmetr? 3. How is rotational smmetr related to rotation images? Practice heck. What is the angle of rotation smmetr for a shape with each order of rotational smmetr? a) 3 b) 5 c) 9 d) 1 5. What is the order of rotational smmetr for each angle of rotation smmetr? a) 6 b) c) 5 d) What is the order of rotational smmetr and angle of rotation smmetr for each regular polgon? a) an equilateral b) a regular pentagon triangle c) a square d) a regular octagon ppl 7. oes each picture have rotational smmetr? If it does, state the order and the angle of rotation smmetr. a) b) 7.6 Rotations and Rotational Smmetr 365

17 8. oes each shape have rotational smmetr about the red dot? If it does, state the order and the angle of rotation smmetr. a) b) 11. Identif and describe an rotational smmetr in each design. a) 9. op each shape on grid paper. raw the rotation image after each given rotation. a) 9 clockwise b) 18 about M about E M b) E c) 7 counterclockwise about Y Y 1. op each shape on isometric dot paper. raw the rotation image after each given rotation. a) 6 clockwise about G 1. This octagon is part of a larger shape that is to be completed b a rotation of 18 about the origin. G b) 1 counterclockwise about a) On a coordinate grid, draw the octagon and its image. b) Outline the shape formed b the octagon and its image. escribe an rotational smmetr in this shape. Eplain wh ou think the smmetr occurred. 366 UNIT 7: Similarit and Transformations

18 13. ssessment Focus Rotate each shape. a) rectangle i) 18 about verte ii) 18 about centre E b) square FGHJ counterclockwise through i) 9 about verte F ii) 9 about centre K F G J E c) equilateral triangle MNP clockwise through i) 1 about verte M ii) 1 about centre Q N P d) How are the images in each of parts a, b, and c the same? How are the different? Eplain what ou notice. 1. a) Rotate square PQRS clockwise about verte P through: i) 9 ii) 18 iii) 7 raw and label each rotation image. P K H M Q Q b) Outline the shape formed b the square and all its images. Identif an rotational smmetr. Eplain what ou notice. 15. Triangle is part of a larger shape that is to be completed b three rotations. a) Rotate clockwise about verte through: i) 9 ii) 18 iii) 7 raw and label each rotation image. 3 3 b) List the coordinates of the vertices of the larger shape formed b the triangle and its images. escribe an rotational smmetr. Take It Further 16. a) raw a polgon on a coordinate grid. hoose an angle of rotation and a centre of rotation to complete a larger polgon with order of rotation: i) ii) List the coordinates of the centre of rotation, and the vertices of the larger polgon. b) raw a polgon on isometric dot paper. hoose an angle of rotation and a centre of rotation to complete a larger polgon with order of rotation: i) 3 ii) 6 S R Reflect How do ou decide if a given shape has rotational smmetr? If it does, how do ou determine the order of rotation and the angle of rotation smmetr? Include an eample in our eplanation. 7.6 Rotations and Rotational Smmetr 367

19 7.7 Identifing Tpes of Smmetr on the artesian Plane What smmetr do ou see in each picture? FOUS Identif and classif line and rotational smmetr. Investigate 3 You will need grid paper and tracing paper. Plot these points on a coordinate grid: (1, 3), (3, 1), and (5, 5) Join the points to form. Each of ou chooses one of these transformations: a translation units right and units down a rotation of 18 about verte a reflection in a line through raw the image for the transformation ou chose. Record the coordinates of each verte on the image. On a separate piece of paper, record an smmetr in the triangle and its image. Trade grids with a member of our group. Identif an smmetr in the triangle and its image. Reflect Share & ompare the tpes of smmetr ou found. id an grid show both rotational smmetr and line smmetr? Eplain wh both tpes of smmetr occurred. Which grid showed onl one tpe of smmetr? 368 UNIT 7: Similarit and Transformations

20 onnect On this grid, rectangle has been rotated 18 about E( 1, ) to produce its image, rectangle. We can etend our meaning of line smmetr to relate the two rectangles. The line through 1 on the -ais is a line of smmetr for the two rectangles. Each point on rectangle has a corresponding point on rectangle. These points are equidistant from the line of smmetr E 6 When a shape and its transformation image are drawn, the resulting diagram ma show: no smmetr line smmetr rotational smmetr both line smmetr and rotational smmetr Eample 1 etermining whether Shapes re Related b Smmetr For each pair of rectangles and EFGH, determine whether the are related b smmetr. a) b) E F H G 3 H E c) E H F G 6 G F Identifing Tpes of Smmetr on the artesian Plane 369

21 Solution a) There is no line on which a mirror can be placed so that one rectangle is the reflection image of the other. So, the rectangles are not related b line smmetr. Trace the rectangles. Use guess and check to determine if a centre of rotation eists. When is rotated 18 about the point S(, 3), coincides with GHEF. So, the rectangles are related b rotational smmetr of order about S(, 3). 18 S E H F G b) Each point on has a corresponding point on EFGH. These points are equidistant from the -ais. So, the two rectangles are related b line smmetr; the -ais is the line of smmetr. Trace the rectangles. Use guess and check to determine if a centre of rotation eists. When a tracing of is rotated 18 about the point P(.5, ), coincides with GHEF. So, the two rectangles are related b rotational smmetr. P H G 18 E F c) When is rotated 9 clockwise about point J( 5, ), coincides with FGHE. Then, the polgon formed b both rectangles together has rotational smmetr of order about point J. So, the two rectangles are related b rotational smmetr. 6 E 9 F J H G 8 6 Eample Identifing Smmetr in a Shape and Its Transformation Image raw the image of rectangle after each transformation. Write the coordinates of each verte and its image. Identif and describe the tpe of smmetr that results. a) a rotation of 18 about the origin b) a reflection in the -ais c) a translation units right and 1 unit down 37 UNIT 7: Similarit and Transformations

22 Solution a) Use tracing paper to rotate 18 about the origin. Point Image ( 1, 1) (1, 1) (3, 1) ( 3, 1) (3, ) ( 3, ) ( 1, ) (1, ) 18 The octagon, formed b both rectangles together, has rotational smmetr of order about the origin, and no line smmetr. b) Reflect in the -ais. Point Image ( 1, 1) ( 1, 1) (3, 1) (3, 1) (3, ) (3, ) ( 1, ) ( 1, ) 18 Rotational smmetr Line smmetr The rectangle, formed b both rectangles, has rotational smmetr of order about the point (1, ). It also has lines of smmetr: the -ais and the vertical line through 1 on the -ais. c) Translate units right and 1 unit down. Point Image ( 1, 1) (3, ) (3, 1) (7, ) (3, ) (7, 1) 18 ( 1, ) (3, 1) The two rectangles do not form a shape; but the have a common verte at (or ). The two rectangles are related b rotational smmetr of order about the point (3, ). There is no line of smmetr relating the rectangles. In Eample, we could write the translation units right and 1 unit down in a shorter form as R, 1. In this shorter form, a translation of 7 units left and units up would be written as L7, U. 7.7 Identifing Tpes of Smmetr on the artesian Plane 371

23 Eample 3 Identifing Smmetr in Shapes and their Translation Images raw the image of pentagon PQRST after each translation below. Label the vertices of the pentagon and its image, and list their coordinates. If each diagram has smmetr, describe it. If each diagram does not have smmetr, eplain how ou know. a) a translation L b) a translation L, 3 T S P Q R Solution a) Translate each verte of pentagon PQRST units left. Point Image P( 3, ) Q(, 3) R(, 5) S(, 5) P ( 5, ) T(, 3) S(, 5) S ( 6, 5) T(, 3) T ( 6, 3) S S The diagram has line smmetr because the vertical line through ST is a line of reflection. The diagram does not have rotational smmetr because there is no point about which it can be rotated so that it coincides with itself. 6 T P T P Q R b) Translate each verte of pentagon PQRST units left and 3 units down. Point P( 3, ) Q(, 3) R(, 5) Image P ( 5, 5) Q (, 6) R (, 8) 6 P S(, 5) T(, 3) S ( 6, 8) T ( 6, 6) The diagram does not have line smmetr because there is no line on which a mirror can be placed so that one pentagon is the reflection image of the other. The diagram does not have rotational smmetr because there is no point about which it can be rotated so that it coincides with itself. T S T P S Q R Q R UNIT 7: Similarit and Transformations

24 iscuss the ideas 1. How can ou tell if two shapes are related b line smmetr?. How can ou tell if two shapes are related b rotational smmetr? Practice heck c) d) 3. escribe the rotational smmetr and line smmetr of each shape. a) a parallelogram b) a rhombus c) an isosceles d) a kite trapezoid 5. escribe the smmetr of each face of a die. op each face. Mark the centre of rotation and the lines of smmetr. ppl 6. Look at the squares below.. escribe the rotational smmetr and line smmetr of each wheel cover. On a cop of the wheel covers, mark the centre of rotation and the line of reflection. a) b) 6 8 Which of squares,,, and are related to the red square: a) b rotational smmetr about the origin? b) b line smmetr? 7.7 Identifing Tpes of Smmetr on the artesian Plane 373

25 7. For each diagram, determine whether the two polgons are related b line smmetr, b rotational smmetr about the origin, or b both. a) b) 1. Identif and describe the tpes of smmetr in each piece of artwork. a) c) d) b) 8. For each diagram, determine whether the two octagons are related b line smmetr, b rotational smmetr, b both tpes of smmetr, or b neither. a) b) 9. Triangle F G H is the image of FGH after a rotation about the origin. Identif an smmetr. F H G G H F 11. op each shape on grid paper. raw the image after the translation given. Label each verte with its coordinates. oes each diagram have line and rotational smmetr? If our answer is es, describe the smmetr. If our answer is no, describe how ou know. a) 6 units up b) units right G E F 37 UNIT 7: Similarit and Transformations

26 1. ssessment Focus a) On a grid, draw E with vertices (, 3), (, 1), and E(3, ). b) raw the translation image E after the translation R1, U3. c) Label all the vertices with their ordered pairs. d) Eplain wh the translation does not result in line or rotational smmetr. e) Find a translation that does result in one tpe of smmetr. raw the image. How do ou know the diagram has smmetr? Show our work. 13. a) raw the image of parallelogram EF after each transformation below. b) The parallelogram and its image form a diagram. If each diagram has smmetr, describe it. If each diagram does not have smmetr, describe how ou know. i) a rotation of 9 clockwise about (, ) ii) a reflection in the horizontal line through 1 on the -ais iii) a translation R 1. The digits to 9 on a digital clock are made up from horizontal and vertical segments. F E 6 a) Sketch each digit on dot paper. Identif an smmetr it has. b) For each digit with line smmetr, plot a part of the digit on grid paper and draw a line of smmetr so that the digit can be completed b a reflection. c) For each digit with rotational smmetr, plot a part of the digit on grid paper. Locate the point about which the digit can be completed b a rotation. d) Is there a pair of digits that are related b line or rotational smmetr? Justif our answer b plotting the digits on a artesian plane. 15. This heagon is part of a larger shape that N is completed b rotating the heagon 18 about the origin. M a) raw the rotation image. b) List the coordinates of the vertices of the larger shape. c) escribe the smmetr in the larger shape. Take It Further 16. The -hour clock represents midnight as : and three-thirt.m. as 3:3. The time 3:3 has line smmetr with a horizontal line of reflection. List as man times from midnight onward that have line smmetr, rotational smmetr, or both. escribe the smmetr for each time ou find. G H K J 3 Reflect When ou see a shape and its transformation image on a grid, how do ou identif line smmetr and rotational smmetr? Include eamples in our eplanation. 7.7 Identifing Tpes of Smmetr on the artesian Plane 375

27 Stud Guide Scale iagrams Length on scale diagram For an enlargement or reduction, the scale factor is: Length on original diagram n enlargement has a scale factor 1. reduction has a scale factor 1. Similar Polgons Similar polgons are related b an enlargement or a reduction. When two polgons are similar: their corresponding angles are equal: E; F; G; H and their corresponding sides are proportional: EF FG GH HE n of the ratios,,, and is the EF FG GH HE scale factor.. E 3. F * * H 3.75 G Similar Triangles When we check whether two triangles are similar: their corresponding angles must be equal: P S and Q T and R U or their corresponding sides must be proportional: PQ QR PR ST TU SU PQ QR PR n of the ratios,, and is the scale factor. ST TU SU 3. P R S 1.5 U Q.5 T 3. Line Smmetr shape has line smmetr when a line divides the shape into two congruent parts so that one part is the image of the other part after a reflection in the line of smmetr. Rotational Smmetr shape has rotational smmetr when it coincides with itself after a rotation of less than 36 about its centre. The number of times the shape coincides with itself is the order of rotation. 36 The angle of rotation smmetr the order of rotation 376 UNIT 7: Similarit and Transformations

28 Review This photo of participants in the rctic Winter Games is to be enlarged. 5. Gina plans to build a triangular dog run against one side of a dog house. Here is a scale diagram of the run. The wall of the dog house is m long. alculate the lengths of the other two sides of the dog run. Wall of dog house Measure the photo. What are the dimensions of the enlargement for each scale factor? 3 1 a) 3 b).5 c) d) 5. raw this pentagon on 1-cm grid paper. Then draw an enlargement of the shape with a scale factor of full-size pool table has dimensions approimatel 7 cm b 138 cm. model of a pool table has dimensions 18 cm b 9 cm. a) What is the scale factor for this reduction? b) standard-size pool cue is about 1 cm long. What is the length of a model of this pool cue with the scale factor from part a?. Here is a scale diagram of a ramp. The height of the ramp is 1.8 m. Measure the lengths on the scale diagram. What is the length of the ramp? Which pentagon is similar to the red pentagon? Justif our answer. 1.5 cm *.5 cm.5 cm 3.75 cm 5. cm 3. cm 3.5 cm Y 3. cm 5.5 cm X 7. cm 1. cm 5.5 cm 1.5 cm 6. cm 1.35 cm.5 cm * Z 1.5 cm 3.15 cm.7 cm.9 cm Review 377

29 7. These two courtards are similar. 1 m 8 m 11. How can ou use similar triangles to calculate the distance in this scale diagram?.5 m 1.5 m 5 m 7. F etermine each length. a) b) c) 8. These two quadrilaterals are similar. V 3.6 cm S Q M. cm 3.5 cm T 3. cm 1.6 cm N U P alculate the length of: a) PN b) TS 9. To determine the distance, d, across a pond, ri uses this diagram. What is the distance across the pond? P 3 m Pond d T 8 m E F 35 m S E Which of these traffic signs have line smmetr? How man lines of smmetr in each case? a) b) c) d) 13. Heagon EF is a part of a larger shape. op the heagon on a grid. Q 1. This scale diagram shows a surveor s measurements taken to determine the distance across a river. What is the approimate distance across the river? d m 18 m 16 m River 378 UNIT 7: Similarit and Transformations R 3 F E a) omplete the shape b reflecting the heagon: i) in the -ais ii) in the -ais iii) in the line through (, 1) and (, 3) b) omplete the shape with a translation R. c) List the ordered pairs of the vertices of each completed shape. d) State whether each completed shape has line smmetr.

30 What is the order of rotational smmetr of each shape? How do ou know? a) b) 17. For each diagram, determine whether the two pentagons are related b an smmetr. escribe each tpe of smmetr. a) b) 3 c) d) 18. Identif and describe the tpes of smmetr in each piece of artwork. a) 15. Rectangle is part of a larger shape that is to be completed b a transformation image. b) 7.7 a) Rotate rectangle as indicated, then draw and label each image. i) 9 counterclockwise about the point (, ) ii) 18 about verte iii) 7 counterclockwise about the point (, ) b) Which diagrams in part a have rotational smmetr? How do ou know? 16. Look at the diagrams in question 15. Which diagrams have line smmetr? How do ou know? 19. a) Translate quadrilateral EFG as indicated, then draw and label each image. G 6 F i) L, ii) R1, U b) oes each translation result in line smmetr or rotational smmetr? If our answer is es, describe the smmetr. If our answer is no, eplain wh there is no smmetr. E Review 379

31 Practice Test 1. These two quadrilaterals are similar. 75 X 5. m m. m Y.5 m m W Z 7.8 m a) alculate the length of. b) alculate the length of WZ. c) raw an enlargement of quadrilateral WXYZ with scale factor. 1 d) raw a reduction of quadrilateral with scale factor. 3. Scott wants to calculate the height of a tree. His friend measures Scott s shadow as 3.15 m. t the same time, the shadow of the tree is 6.3 m. Scott knows that he is 1.7 m tall. a) Sketch two triangles Scott could use to calculate the height of the tree. b) How do ou know the triangles are similar? c) What is the height of the tree? 3. Use isometric dot paper or grid paper. a) raw these shapes: equilateral triangle, square, rectangle, parallelogram, trapezoid, kite, and regular heagon b) For each shape in part a: i) raw its lines of smmetr. ii) State the order and angle of rotation smmetr. c) raw a shape that has line smmetr but not rotational smmetr. d) raw a shape that has rotational smmetr but not line smmetr.. Plot these points on a grid: (, 1), (1, ), (1, ), (, 5), E(3, ), F(3, ) For each transformation below: i) raw the transformation image. ii) Record the coordinates of its vertices. iii) escribe the smmetr of the diagram formed b the original shape and its image. a) a rotation of 9 clockwise about the point G(, 3) b) a translation R c) a reflection in the line 38 UNIT 7: Similarit and Transformations

32 Unit Problem esigning a Flag Part 1 t sea, flags are used to displa messages or warnings. Here are some nautical flags. escribe the smmetries of each flag in as much detail as possible. lassif the flags according to the numbers of lines of smmetr. Part esign our own flag. The flag ma be for a countr, an organization, or it ma be a flag with a message. It must have line smmetr and rotational smmetr. escribe the smmetries in our flag. The actual flag must be at least 3 m b m. raw a scale diagram of our flag, including the scale factor ou used. escribe what our flag will be used for. Your work should show: a description and classification of the smmetries of the nautical flags a scale diagram of our flag, in colour, including the scale factor a description of the smmetries in our flag a description of what our flag will be used for Reflect on Your Learning How does knowledge of enlargements and reductions in scale diagrams help ou understand similar polgons? How are line smmetr and rotational smmetr related to transformations on a grid? Unit Problem 381