What Should I Recall?

Transcription

1 What Should I Recall? Suppose I have to solve this problem: etermine the unknown measures of the angles and sides in. The given measures are rounded to the nearest whole number. I think of what I alread know about triangles. I see that and have the same hatch marks; this means the sides are equal. So, 5 cm I know that a triangle with equal sides is an isosceles triangle. So, is isosceles. n isosceles triangle has equal angles that are formed where the equal sides intersect the third side. I use 3 letters to describe an angle. So, 37 Since is isosceles, the height is the perpendicular bisector of the base. So, and 90 I can use the thagorean Theorem in to calculate the length of. 5 cm 3 cm 37 5 cm 5 cm 3 cm cm 5 cm 3 cm UNIT 7: Similarit and Transformations

2 cm So, cm 8 cm I know that the sum of the angles in a triangle is 180. So, I can calculate the measure of cm 5 cm 3 cm cm cm 8 cm M friend Janelle showed me a different wa to calculate. She recalled that the line is a line of smmetr for an isosceles triangle. So, is congruent to. This means that Janelle calculated the measure of in Then, cm 5 cm cm heck 1. alculate the measure of each angle. a) b) and c) JK and KJ 8 K J 76 Start Where You re 317

3 7.1 Scale iagrams and nlargements OUS raw and interpret scale diagrams that represent enlargements. ow are these photos alike? ow are the different? Investigate You will need 0.5-cm grid paper. ere is an actual size drawing of a memor card for a digital camera and an enlargement of the drawing. ctual Size nlargement op the drawings on grid paper. Measure the lengths of pairs of matching sides on the drawings. Label each drawing with these measurements. or each measurement, write the fraction: Write each fraction as a decimal. What do ou notice about these numbers? 318 UNIT 7: Similarit and Transformations Length on enlargement Length on actual size drawing

4 ractice heck. etermine the scale factor for each scale diagram. a) b) Original diagram Scale diagram ppl 6. photo of a surfboard has dimensions 17.5 cm b 1.5 cm. nlargements are to be made with each scale factor below. etermine the dimensions of each enlargement. Round the answers to the nearest centimetre. a) scale factor 1 b) scale factor 0 7 c) scale factor d) scale factor 7. ere is a scale diagram of a salmon fr. The actual length of the salmon fr is 30 mm. Measure the length on the diagram to the nearest millimetre. etermine the scale factor for the scale diagram. 17 Original diagram Scale diagram 5. Scale diagrams of different squares are to be drawn. The side length of each original square and the scale factor are given. etermine the side length of each scale diagram. length 8. The head of a pin has diameter mm. etermine the scale factor of this photo of the pinhead. diameter Side length of Scale factor original square a) 1 cm 3 b) 8 mm 5 9. This view of the head of a bolt has the shape of a regular heagon. ach angle is 10. Use a protractor and ruler to draw a scale diagram of the bolt with scale factor.5. c) 1.55 cm. d) 5 mm 3.8 e) 0.8 cm Scale iagrams and nlargements 33

5 10. raw our initials on 0.5-cm grid paper. Use different-sized grid paper to draw two different scale diagrams of our initials. or each scale diagram, state the scale factor. 11. ssessment ocus or each set of diagrams below, identif which of diagrams,,, and are scale diagrams of the shaded shape. or each scale diagram ou identif: i) State the scale factor. ii) plain how it is a scale diagram. a) 13. Look in a newspaper, magazine, or on the Internet. ind an eample of a scale diagram that is an enlargement and has its scale factor given. What does the scale factor indicate about the original diagram or object? 1. raw a scale diagram of the shape below with scale factor.5. b) 1. One frame of a film in a projector is about 50 mm high. The film is projected onto a giant screen. The image of the film frame is 16 m high. a) What is the scale factor of this enlargement? b) penguin is 35 mm high on the film. ow high is the penguin on the screen? 15. On a grid, draw O with vertices O(0, 0), (0, 3), and (, 0). a) raw a scale diagram of O with scale factor 3 and one verte at (3, 3). Write the coordinates of the new vertices. b) Is there more than one answer for part a? If our answer is no, eplain wh no other diagrams are possible. If our answer is es, draw other possible scale diagrams. Take It urther 16. One micron is one-millionth of a metre, or 1 m 10 6 microns. a) human hair is about 00 microns wide. ow wide is a scale drawing of a human hair with scale factor 00? ive our answer in as man different units as ou can. b) computer chip is about microns wide. scale diagram of a computer chip is 5 cm wide. What is the scale factor? Reflect Suppose ou are given a scale diagram. Wh is it important to know the scale factor? 3 UNIT 7: Similarit and Transformations

6 ractice heck. Write each fraction in simplest form, then epress it as a decimal. 5 5 a) b) c) d) ppl 7. ere are two drawings of a dog. etermine the scale factor of the reduction as a fraction and as a decimal. 5. etermine the scale factor for each reduction as a fraction or a decimal. a) Reduction b) Original 8. Which of rectangles,, and is a reduction of the large rectangle? Justif our answer. cm 3 cm Original Reduction 6. or each pair of circles, the original diameter and the diameter of the reduction are given. etermine each scale factor as a fraction or a decimal. iameter of ctual ircle iameter of Reduction a) 50 cm 30 cm b) 30 cm 0 cm c) 16 cm 3 cm d) 5 m cm e) km 300 m 9. Which two polgons have pairs of corresponding lengths that are proportional? Identif the scale factor for the reduction. 7. Scale iagrams and Reductions 39

7 op the table below. Use our results from the first tables to complete this table. Write the ratios of the lengths of the sides as decimals to the nearest hundredth. What do ou notice about the measures of the matching angles? What do ou notice about the ratios of matching sides? Reflect Share & ompare our results with those of another pair of students. Work together to draw two other quadrilaterals that have sides and angles related the same wa as our quadrilaterals. ow does this work relate to scale drawings that show enlargements and reductions? onnect When one polgon is an enlargement or a reduction of another polgon, we sa the polgons are similar.similar polgons have the same shape,but not necessaril the same size. ere are two similar pentagons. R cm R 1.5 cm 96 Q 15 cm.5 cm 110 S.5 cm.5 cm Q 15 3 cm 110 S 3.75 cm cm T.5 cm T Matching angles are corresponding angles. Matching sides are corresponding sides. We list the corresponding angles and the pairs of corresponding sides. 7.3 Similar olgons 335

8 b) raw a reduction. hoose a scale factor that is less than 1, such as. Let the similar pentagon be. Multipl each side length of b to get the corresponding side lengths of cm.8 cm.0 cm cm 1. cm.0 cm Since, Since, then then 1.0 cm 1. cm The corresponding angles are equal. So: Use a ruler and protractor to draw pentagon. The pentagons are similar because corresponding angles are equal and corresponding sides are proportional. That is, the length of each side of the reduction is the corresponding side of the original pentagon. 1 the length of 1. cm 1. cm cm.0 cm 1.0 cm ample 3 Solving roblems Using the roperties of Similar olgons These two octagonal garden plots are similar. a) alculate the length of. b) alculate the length of N. 5. m 8.1 m I J M N 7.0 m K L Q 3. m Solution a) To calculate, consider polgon as a reduction of polgon IJKLMNQ. 7.3 Similar olgons 339

9 ractice heck. alculate the side length, in units, in each proportion. 8 a) b) c) d) 5. alculate the value of the variable in each proportion..5 z a) b) 1. a 0.7 c) d) Identif similar quadrilaterals. List their corresponding sides and corresponding angles Use isometric dot paper. onstruct a heagon similar to heagon. ppl 9. re an of these rectangles similar? Justif our answer. 5. m 1.9 m I J 7.5 m.5 m X N U V W L K M I M.8 m J 1.6 m K Q T 7. Use grid paper. onstruct a quadrilateral similar to quadrilateral MNQ. R S Q 10. or each polgon below: i) raw a similar larger polgon. ii) raw a similar smaller polgon. plain how ou know the polgons are similar. a) b) M Q N 7.3 Similar olgons 31

10 onnect When two polgons are similar: the measures of corresponding angles must be equal and the ratios of the lengths of corresponding sides must be equal. triangle is a special polgon. When we check whether two triangles are similar: the measures of corresponding angles must be equal;or the ratios of the lengths of corresponding sides must be equal roperties of Similar Triangles To identif that QR and STU are similar, we onl need to know that: S and Q T and R U; or Q QR R ST TU SU R Q S U T These triangles are similar because: Q 75 R 6 3 When we name two similar triangles, we order the letters to match the corresponding angles. We write: QR Then we can identif corresponding sides: corresponds to QR. corresponds to R. corresponds to Q R 6 75 Q 3 Your World Satellite imager consists of photographs of arth taken from space. The images are reductions of regions on arth. The qualit of an image depends upon the instrument used to obtain it and on the altitude of the satellite. The Landsat 7 satellite can create images of objects as small as 10 cm long. to come <atch: wncp9_u07_le0_p18 photo of Quickird satellite or Landsat 7 satellite> 3 UNIT 7: Similarit and Transformations

11 ample 3 Using Overlapping Similar Triangles to etermine a Length surveor wants to determine the width of a lake at two points on opposite sides of the lake. She measures distances and angles on land, then sketches this diagram. ow can the surveor determine the length N to the nearest metre? Lake N 305 m 30 m Q 10 m J Solution Identif the two triangles, then draw them separatel. onsider NJ and QJ. rom the diagram: NJ QJ NJ QJ J is the common angle to both triangles. Since 3 pairs of corresponding angles are equal, NJ QJ Two corresponding sides are: J 305 m 10 m and J 10 m 515 m So, NJ is an enlargement of QJ with a scale factor of. Write a proportion that includes the unknown length N. N Q N 30 N 30 Substitute Q 30. To solve for N, multipl each side b N m J 10 m 30 m J Q N The width of the lake, N, is about 56 m. 7. Similar Triangles 37

12 ractice heck. Which triangles in each pair are similar? ow do ou know? a) b) R M Q T 8 3 S 10 5 J U 6 30 N 5. In each diagram, identif two similar triangles. plain wh the are similar. a) b) c) M N J K c) Q R ppl R Q d) 5 S T etermine the length of in each pair of similar triangles. a) V 7. Similar Triangles 39

13 Mid-Unit Review photo of a gmnast is to be enlarged. The dimensions of the photo are 15 cm b 10 cm. What are the dimensions of the 7 enlargement with a scale factor of?. computer chip has dimensions 15 mm b 8 mm. ere is a scale drawing of the chip These quadrilaterals have corresponding angles equal 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 0.5-cm grid paper. The dimensions of the pool are 60 m b 0 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. ow do ou know the quadrilaterals are similar? 6. window has the shape of a heagon. raw a heagon that is similar to this heagon. plain 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

14 Solution a) The red line is the line of smmetr b) This tessellation has lines of for this tessellation. ach point on one smmetr. or 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. ample 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

15 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. ample 3 ompleting a Shape iven 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 (0, 0) and (6, 6) 0 6 Solution The red line is the line of reflection. ach image point is the same distance from this line as the corresponding original point. a) oint Image (, ) (, ) (, ) (, 0) 0 (6, ) (6, 0) (6, ) (6, ) The larger shape has coordinates: (, ), (, ), (6, ), (6, 0), (, 0) This shape is a pentagon with line smmetr. The line of smmetr is the red line. 356 UNIT 7: Similarit and Transformations

16 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.. scher 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) Q b) a aida button blanket b) c) S R U T V 6 7. ssessment ocus 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) ow 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? plain. 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

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) 90 clockwise b) 180 about M about M b) c) 70 counterclockwise about Y Y 10. op each shape on isometric dot paper. raw the rotation image after each given rotation. a) 60 clockwise about 1. This octagon is part of a larger shape that is to be completed b a rotation of 180 about the origin. 0 b) 10 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. plain wh ou think the smmetr occurred. 366 UNIT 7: Similarit and Transformations

18 onnect On this grid, rectangle has been rotated 180 about ( 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. ach point on rectangle has a corresponding point on rectangle. These points are equidistant from the line of smmetr 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 ample 1 etermining whether Shapes re Related b Smmetr or each pair of rectangles and, determine whether the are related b smmetr. a) b) c) Identifing Tpes of Smmetr on the artesian lane 369

19 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 180 about the point S(0, 3), coincides with. So, the rectangles are related b rotational smmetr of order about S(0, 3). 180 S 0 b) ach point on has a corresponding point on. 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 180 about the point (.5, 0), coincides with. So, the two rectangles are related b rotational smmetr c) When is rotated 90 clockwise about point J( 5, ), coincides with. 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 J ample 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 180 about the origin b) a reflection in the -ais c) a translation units right and 1 unit down UNIT 7: Similarit and Transformations

20 Solution a) Use tracing paper to rotate 180 about the origin. oint Image ( 1, 1) (1, 1) (3, 1) ( 3, 1) (3, 0) ( 3, 0) ( 1, 0) (1, 0) The octagon, formed b both rectangles together, has rotational smmetr of order about the origin, and no line smmetr. b) Reflect in the -ais. oint Image ( 1, 1) ( 1, 1) (3, 1) (3, 1) (3, 0) (3, 0) ( 1, 0) ( 1, 0) 180 Rotational smmetr Line smmetr The rectangle,formed b both rectangles,has rotational smmetr oforder about the point (1, 0). 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. oint Image ( 1, 1) (3, 0) (3, 1) (7, 0) (3, 0) (7, 1) ( 1, 0) (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, 0). There is no line of smmetr relating the rectangles. In ample, 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 lane 371

21 iscuss the ideas 1. ow can ou tell if two shapes are related b line smmetr?. ow can ou tell if two shapes are related b rotational smmetr? ractice 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) 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 lane 373

22 Stud uide Scale iagrams or an enlargement or reduction, the scale factor is: Length on scale diagram Length on original diagram n enlargement has a scale factor 1. reduction has a scale factor 1. Similar olgons Similar polgons are related b an enlargement or a reduction. When two polgons are similar: their corresponding angles are equal: ; ; ; and their corresponding sides are proportional: n of the ratios,,, and is the scale factor * * Similar Triangles When we check whether two triangles are similar: their corresponding angles must be equal: S and Q T and R U or their corresponding sides must be proportional: Q ST QR TU R SU Q QR R SU n of the ratios,, and is the scale factor. ST TU 3.0 R S 1.5 U Q.5 T 3.0 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 360 about its centre. The number of times the shape coincides with itself is the order of rotation. The angle of rotation smmetr 360 the order of rotation 376 UNIT 7: Similarit and Transformations

23 Review This photo of participants in the rctic Winter ames is to be enlarged. 5. ina plans to build a triangular dog run against one side of a dog house. ere 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? a) 3 b).5 c) d). 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 70 cm b 138 cm. model of a pool table has dimensions 180 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?. ere 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 5.0 cm 3.0 cm 3.5 cm Y 3.0 cm 5.5 cm X 7.0 cm 1.0 cm 5.5 cm 1.50 cm 6.0 cm 1.35 cm.5 cm * Z 1.5 cm 3.15 cm.70 cm 0.90 cm Review 377

24 7. These two courtards are similar. 1 m 8 m 11. ow can ou use similar triangles to calculate the distance in this scale diagram?.5 m 1.5 m 5 m 7. etermine each length. a) b) c) 8. These two quadrilaterals are similar. V 3.6 cm S Q M. cm 3.5 cm T 3.0 cm 1.6 cm N U alculate the length of: a) N b) TS 9. To determine the distance, d,across a pond, ri uses this diagram. What is the distance across the pond? ond d 3 m T 8 m 35 m S Which of these traffic signs have line smmetr? ow man lines of smmetr in each case? a) b) c) d) 13. eagon is a part of a larger shape. op the heagon on a grid. Q 10. 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 0 m 18 m 16 m River 378 UNIT 7: Similarit and Transformations R 3 0 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.

25 What is the order of rotational smmetr of each shape? ow do ou know? a) b) 17. or each diagram, determine whether the two pentagons are related b an smmetr. escribe each tpe of smmetr. a) b) 0 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) a) Rotate rectangle as indicated, then draw and label each image. i) 90 counterclockwise about the point (, ) ii) 180 about verte iii) 70 counterclockwise about the point (, ) b) Which diagrams in part a have rotational smmetr? ow do ou know? 16. Look at the diagrams in question 15. Which diagrams have line smmetr? ow do ou know? 19. a) Translate quadrilateral as indicated, then draw and label each image. 0 6 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. Review 379

26 ractice 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.. Scott wants to calculate the height of a tree. is friend measures Scott s shadow as 3.15 m. t the same time, the shadow of the tree is 6.30 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) ow 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) or 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.. lot these points on a grid: (, 1), (1, ), (1, ), (, 5), (3, ), (3, ) or 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 90 clockwise about the point (, 3) b) a translation R c) a reflection in the line UNIT 7: Similarit and Transformations