National 5 Mathematics Course Materials Topic 7: Similar shapes

Transcription

1 SCHOLAR Study Guide National 5 Mathematics Course Materials Topic 7: Similar shapes Authored by: Margaret Ferguson Reviewed by: Jillian Hornby Previously authored by: Eddie Mullan Heriot-Watt University Edinburgh EH14 4AS, United Kingdom.

2 First published 2014 by Heriot-Watt University. This edition published in 2016 by Heriot-Watt University SCHOLAR. Copyright 2016 SCHOLAR Forum. Members of the SCHOLAR Forum may reproduce this publication in whole or in part for educational purposes within their establishment providing that no profit accrues at any stage, Any other use of the materials is governed by the general copyright statement that follows. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, without written permission from the publisher. Heriot-Watt University accepts no responsibility or liability whatsoever with regard to the information contained in this study guide. Distributed by the SCHOLAR Forum. SCHOLAR Study Guide Course Materials Topic 7: National 5 Mathematics 1. National 5 Mathematics Course Code: C747 75

3 Acknowledgements Thanks are due to the members of Heriot-Watt University's SCHOLAR team who planned and created these materials, and to the many colleagues who reviewed the content. We would like to acknowledge the assistance of the education authorities, colleges, teachers and students who contributed to the SCHOLAR programme and who evaluated these materials. Grateful acknowledgement is made for permission to use the following material in the SCHOLAR programme: The Scottish Qualifications Authority for permission to use Past Papers assessments. The Scottish Government for financial support. The content of this Study Guide is aligned to the Scottish Qualifications Authority (SQA) curriculum. All brand names, product names, logos and related devices are used for identification purposes only and are trademarks, registered trademarks or service marks of their respective holders.

4

5 1 Topic 1 Similar shapes Contents 7.1 Length scale factors Similar triangles Area scale factors Volume scale factors Learning points End of topic test

6 2 TOPIC 1. SIMILAR SHAPES Learning objectives By the end of this topic, you should be able to: identify similar shapes; identify enlargement and reduction scale factors; use a scale factor to find an unknown length; work with similar triangles; identify an area scale factor; use an area scale factor to find an unknown area; identify a volume scale factor; use a volume scale factor to calculate an unknown volume.

7 TOPIC 1. SIMILAR SHAPES Length scale factors Similar shapes Shapes are said to be congruent if they are identical in every way. These prints of the painting by Claude Monet are congruent because one is an exact copy of the other. Shapes are said to be similar if one is an enlargement or reduction of the other. These prints of the Mona Lisa by Leonardo da Vinci are similar because the one on the right is a reduction of the one on the left. One is a scaled version of the other. Shapes are also said to be similar if one is a mirror image of the other. These dogs are similar because the one on the right is a reflection and enlargement of the one the left.

8 4 TOPIC 1. SIMILAR SHAPES Scale factors Go online These rectangles are congruent because pairs of corresponding sides are equal and corresponding angles are equal. To identify the scale factor choose a pair of corresponding sides and write down the ratio. 5 5 =1 5 5 =1 3 3 = =1 Congruent shapes have a length scale factor of To identify the scale factor choose a pair of corresponding sides and write down the ratio = = 1 2 A length scale factor less than 1 is a reduction scale factor. These rectangles are similar. The rectangle on the right is a reduction of the one on the left. To identify the scale factor choose a pair of corresponding sides and write down the ratio. 8 4 =2 6 3 = scale factor. =2 A length scale factor greater than 1 is an enlargement These triangles are similar. The triangle on the right is an enlargement of the one on the left.

9 TOPIC 1. SIMILAR SHAPES 5 Examples 1. Using scale factors Problem: These shapes are similar. Calculate the breadth x. Solution: The side that we are looking for is on the larger shape. We need an enlargement scale factor. We know the height of each heart so we have one pair of corresponding sides. 7 There are two possible ways to display the ratios, 28 = or 7 =4. An enlargement scale factor is greater than 1. The enlargement scale factor = 28 7 =4. All we have to do is scale the corresponding side of x. x = 4 6 = 24 cm 2. Using scale factors Problem: These shapes are similar. Calculate the height y. Solution: The side that we are looking for is on the smaller shape. We need a reduction scale factor.

10 6 TOPIC 1. SIMILAR SHAPES We have one pair of corresponding sides and two possible ratios, = or 1 5 = 4 3. A reduction scale factor is less than 1. The reduction scale factor = y = = 9 m Key point If two shapes are similar then: pairs of corresponding angles are equal; and the ratios of pairs of corresponding sides are equal. Using scale factors exercise Go online Q1: Are the rectangles below similar, Yes or No? Q2: These rhombi are similar. What is the reduction scale factor?

11 TOPIC 1. SIMILAR SHAPES 7 Q3: These regular octagons are similar. What is the enlargement scale factor? Q4: The shapes below are similar. Do you need an enlargement scale factor or reduction scale factor to find the unknown length? a) enlargement b) reduction Q5: What is the scale factor needed to find the length of the big sun? Q6: What is the length of the big sun?

12 8 TOPIC 1. SIMILAR SHAPES Q7: The mountain bikes below are similar. Do you need an enlargement scale factor or reduction scale factor to find the height of the small bike? a) enlargement b) reduction Q8: What is the scale factor needed to find the height of the small bike? Q9: What is the height of the small bike? Q10: These prints of Leonardo da Vinci s "The Last Supper" are similar. Calculate the height of the small print.

13 TOPIC 1. SIMILAR SHAPES 9 Q11: The smiley faces are similar. The length of the mouth on the small smiley face is 0 8 cm. Calculate the length of the mouth on the big smiley face Similar triangles Key point Triangles are special. If two triangles are similar then: pairs of corresponding angles are equal; or the ratios of pairs of corresponding sides are equal. When all pairs of corresponding angles in two triangles are equal we say that the triangles are equiangular. Similar triangles Similar triangles are special. Go online There are two triangles in the above diagram, ABC and ADE. From our knowledge of angle properties we can identify that corresponding angles are equal.

14 10 TOPIC 1. SIMILAR SHAPES Both triangles have a common angle at A. Angles ABC and AED are corresponding or "F" angles and are equal. Angles ACB and ADE are corresponding or "F" angles and are equal. We say that these triangles are equiangular and as such are similar. Triangle ABC is similar to triangle ADE. There are three pairs of corresponding sides; AD and AC; DE and CB; AE and AB. In some triangles corresponding sides are harder to spot. There are two triangles in this diagram ABC and CDE. From our knowledge of angle properties we can identify that alternate angles and vertically opposite angles are also equal.

15 TOPIC 1. SIMILAR SHAPES 11 A B C D E Angles ACB and DCE are vertically opposite or "X" angles and are equal. Angles ABC and CDE are alternate or "Z" angles and are equal. Angles CAB and DEC are alternate or "Z" angles and are equal. These triangles are also equiangular and similar. Triangle ABC is similar to triangle CDE. There are three pairs of corresponding sides but they are harder to spot. AC is opposite the green star so its corresponding side is CE because it is also opposite the green star. DE and AB are corresponding sides because they are both opposite the red star. CD and CB are corresponding sides because they are both opposite the blue star.

16 12 TOPIC 1. SIMILAR SHAPES Example Problem: Calculate the length of the side marked x. Solution: These triangles are equiangular and therefore similar. It is often easier to pull the similar triangles apart in a sketch. Notice 40 cm = 24 cm +16cm The side that we are looking for is on the smaller triangle. We need a reduction scale factor. We know one pair of corresponding sides. For a reduction scale factor the ratio must be less than 1 so the smaller of the corresponding sides goes on the numerator. The reduction scale factor = or 3 5. All we have to do is scale the corresponding side of x. x = 24 / = 27 cm When identifying the scale factor in a non-calculator question it is best to simplify the fraction if you can.

17 TOPIC 1. SIMILAR SHAPES 13 Example Problem: Calculate the length of the side marked x. Solution: These triangles are equiangular and therefore similar. We have one pair of corresponding sides. It is easier to mark alternate angles in this question to help identify the corresponding sides. The corresponding sides are opposite the green alternate angles so we want the 20 mm and 32 mm sides. (Notice that the 19 mm side is not required.) The side that we are looking for is on the larger triangle. We need an enlargement scale factor. For an enlargement scale factor the ratio of corresponding sides must be greater than 1 so the larger of the sides goes on the numerator. The enlargement scale factor = or 8 5. All we have to do is scale the corresponding side of x. x = 32 / = 40 mm

18 14 TOPIC 1. SIMILAR SHAPES Similar triangles exercise Go online Q12: Looking at the image above, do you need an enlargement or reduction scale factor to find y? a) enlargement b) reduction Q13: What is the scale factor? Q14: What is the length of y? Q15: What is the length of z? Q16: Looking at the image above, do you need an enlargement or reduction scale factor to find x? a) enlargement b) reduction

19 TOPIC 1. SIMILAR SHAPES 15 Q17: What is the scale factor? Q18: What is the length of x? Q19: Looking at the image again, do you need an enlargement or reduction scale factor to find y? a) enlargement b) reduction Q20: What is the length of y? A piece of wire is bent to form the shape below. The verticals are parallel. Q21: What is the height h? Q22: What is the length d?

20 16 TOPIC 1. SIMILAR SHAPES Q23: Some pieces of wood have been nailed together. The pieces labelled BC and KL are parallel. AK = 51 cm, KL = 45 cm and BC = 60 cm as shown. Calculate the length of CK. 1.2 Area scale factors Area scale factors Go online The length scale factor for enlargement is 30 / 10 = 3. The area scale factor for enlargement is 180 / 20 = 9.

21 TOPIC 1. SIMILAR SHAPES 17 Key point 9 = 3 2 so Area scale factor = (Length scale factor) 2 Example Problem: The bolts of lightening are similar. Calculate the area of the large lightening bolt. Solution: Since we want the area of the large shape we need the scale factor for enlargement. The length scale factor for enlargement = The area scale factor for enlargement = ( 12 5 The area of the large lightening bolt = ( ) 2 ) 2 36 = 100 cm 2 Area scale factors exercise Q24: If the length scale factor for enlargement is 5, what is the area scale factor for enlargement? Go online Q25: If the length scale factor for reduction is , what is the area scale factor for reduction?

22 18 TOPIC 1. SIMILAR SHAPES The mirrors above come in 2 sizes and are similar. The area of glass in the small mirror is 540 cm 2. Q26: What is the length scale factor for enlargement? Q27: What is the area scale factor? Q28: What is the area of glass in the large mirror? The regular seven pointed stars are similar. Q29: What is the length scale factor for reduction? Q30: What is the area scale factor? Q31: What is the area of the small star?

23 TOPIC 1. SIMILAR SHAPES 19 The rectangles below are similar. Q32: What is the length scale factor for reduction? Q33: What is the area scale factor? Q34: What is the area of the large rectangle? Q35: What is the area of the small rectangle? 1.3 Volume scale factors Volume scale factors Go online The length scale factor for enlargement is 6 / 2 = 3. The volume scale factor for enlargement is 1080 / 40 = 27.

24 20 TOPIC 1. SIMILAR SHAPES Key point 27 = 3 3 so Volume scale factor = (Length scale factor) 3 Examples 1. Problem: The cylinders below are similar. Calculate the volume of the small cylinder. Solution: For the small cylinder we need a reduction scale factor. The length scale factor for reduction = The volume scale factor for reduction = ( The volume of the small cylinder = ( ) = 2389 cm 3 2. Problem: The bottles of perfume are similar. Calculate the cost of the large bottle of perfume. ) 3

25 TOPIC 1. SIMILAR SHAPES 21 Solution: For the large bottle of perfume we need the enlargement scale factor. The length scale factor for enlargement = Cost is dependent on Volume. You are not told this in the question but you will be expected to know it. Since cost is dependent on volume we need the volume scale factor. The volume scale factor = ( ) Cost of the large bottle = ( ) = Volume scale factors exercise Q36: If the length scale factor for enlargement is 4, what is the volume scale factor for enlargement? Go online Q37: If the length scale factor for reduction is , what is the volume scale factor for reduction? The bottles below are similar. Q38: What is the length scale factor for reduction? Q39: What is the volume scale factor for reduction? Q40: What is the volume of the smaller bottle?

26 22 TOPIC 1. SIMILAR SHAPES The tubes of toothpaste are similar. Q41: What is the length scale factor for enlargement? Q42: What is the volume scale factor for enlargement? Give your answer as a fraction to a power ( x / y ) z. Q43: What is the volume of the large tube of toothpaste? These cans of beans are similar. The large can is 15 cm tall and costs 65 pence. The small can is 12 cm tall. Q44: What is the length scale factor for reduction? Q45: What is the volume scale factor for reduction? Q46: Calculate the cost of the small can of beans to the nearest penny.

27 TOPIC 1. SIMILAR SHAPES Learning points Similar shapes Congruent shapes are an exact copy of each other. Congruent shapes have a length scale factor of 1. Similar shapes are an enlargement or reduction of each other. If two shapes are similar then: pairs of corresponding angles are equal; and the ratios of pairs of corresponding sides are equal. An enlargement scale factor is greater than 1 e.g. 5, 22 / 7. A reduction scale factor is less than 1 e.g. 1 /2, 12 / 25. If two triangles have equal angles they are equiangular. Similar triangles If two triangles are similar then: pairs of corresponding angles are equal; or the ratios of pairs of corresponding sides are equal. Triangles which are equiangular are similar. In similar triangles: corresponding sides are opposite corresponding angles; there are corresponding, alternate and/or vertically opposite angles. Area scale factor Area scale factor = (length scale factor) 2 Volume scale factor Volume scale factor = (length scale factor) 3

28 24 TOPIC 1. SIMILAR SHAPES 1.5 End of topic test End of topic 7 test Go online Length scale factor Q47: These guitars are similar and come in adult and child sizes. The adult size guitar is 39 cm wide. What is the width of the child size guitar? Q48: The lamps below are similar and come in two sizes, small and medium. What is the height of the medium size lamp?

29 TOPIC 1. SIMILAR SHAPES 25 Similar triangles Q49: The diagram shows a 30,60,90 set square. Calculate the length of x. EAC and CBD are equal. Q50: Calculate the length of AC. Q51: Calculate the length of BD.

30 26 TOPIC 1. SIMILAR SHAPES Area scale factor Q52: The two Christmas trees are similar. Calculate the area coloured green on the large tree. Q53: The sheepskin rugs below are similar. Calculate the area of the small sheepskin rug.

31 TOPIC 1. SIMILAR SHAPES Volume scale factor Q54: The glasses of milk below are similar. Calculate the volume of the small glass of milk. Q55: The two children s balls below are similar. Calculate the cost of the large ball to the nearest penny. H ERIOT-WATT U NIVERSITY 27

32 28 ANSWERS: TOPIC 7 Answers to questions and activities 7 Similar shapes Using scale factors exercise (page 6) Q1: Hint: Although corresponding angles in rectangles are all right angles, the ratios of pairs of corresponding sides are not the same e.g or Answer: No 2 Q2: 6 5 Q3: 3 Q4: a) enlargement Q5: Hint: An enlargement scale factor is greater than 1. Answer: 4 Q6: Hint: scale factor length of little sun = length of big sun So, 4 7 =? Answer: 28 cm Q7: b) reduction Q8: Hint: A reduction scale factor is less than 1. So, 25 / 150 =? Answer: 1 6 Q9: Hint: scale factor height of big bike = height of small bike So, 1 / 6 72 =? Answer: 12 cm

33 ANSWERS: TOPIC 7 29 Q10: Steps: Do we need an enlargement or reduction scale factor? reduction What is the scale factor? 30/80 Use the scale factor to calculate the height of the small print. Answer: Height = 18 cm Q11: Steps: Do we need an enlargement or reduction scale factor? enlargement What is the scale factor? 15/5 Use the scale factor to calculate the length of the mouth on the large face. Answer: Length = 2 4 cm Similar triangles exercise (page 14) Q12: b) reduction Q13: Hint: A reduction scale factor is less than 1. Answer: 10/18 or simplified to 5/9 Q14: Hint: scale factor 36 = y So, 10 / =? Answer: 20 Q15: Hint: 36 y = z So, =? Answer: 16 Q16: b) reduction Q17: Hint:

34 30 ANSWERS: TOPIC 7 A reduction scale factor is less than 1. Answer: 8/12, or simplified to 4/6 or 2/3 Q18: Hint: scale factor 9-y=x So, 8 / 12 9 =? Answer: 6 Q19: b) reduction Q20: 10 Q21: Steps: Do you need an enlargement or reduction scale factor to find h? reduction Identify the pair of corresponding sides. 20, 32 What is the scale factor? 20/32 Use the scale factor to find the height h. Answer: Height h = mm Q22: Steps: Do you need an enlargement or reduction scale factor to find d? enlargement What is the scale factor? 32/20 Use the scale factor to find the length d. Answer: Length d = 33 6 mm Q23: Steps: Do we need an enlargement or reduction scale factor to find AC? enlargement What is the scale factor? 60/45 What is the length of AC? AC = 68 cm Use this answer to calculate CK. Answer: Length CK = 19 cm

35 ANSWERS: TOPIC 7 31 Area scale factors exercise (page 17) Q24: 25 Q25: ( 1.5 / 2.5 ) 2, ( 3 / 5 ) 2 and ( 6 / 10 ) 2 are also correct Q26: 45 / 20, or simplified to 9 / 4 Q27: ( 45 / 20 ) 2, or simplified to ( 9 / 4 ) 2 Q28: cm 2 Q29: 4 5 6, other simplified scale factors are 45 / 60, 9 / 12 and 3 / 4 Q30: ( ) 4 5 2, 6 other simplified scale factors are ( 45 / 60 ) 2, ( 9 / 12 ) 2 and ( 3 / 4 ) 2 Q31: cm 2 Q32: 18 / 30, or simplified to 9 / 15, 6 / 10, 3 / 5 or 18 / 30 Q33: ( 18 / 30 ) 2, or simplified to ( 9 / 15 ) 2, ( 6 / 10 ) 2, ( 3 / 5 ) 2 or ( 18 / 30 ) 2 Q34: 1890 mm 2 Q35: mm 2 Volume scale factors exercise (page 21) Q36: 64 Q37: ( ) 1 3 3, ( 2 4 other simplified answers accepted would be 13 ) 3 24 Q38: 18 / 24, or simplified to 9 / 12, 6 / 8 or 3 / 4 Q39: ( 18 / 24 ) 3, or simplified to ( 9 / 12 ) 3, ( 6 / 8 ) 3 or ( 3 / 4 ) 3 Q40: 81 cm 3 Q41: 22 / 10, or simplified to 11 / 5 Q42: ( 22 / 10 ) 3, or simplified to ( 11 / 5 ) 3 Q43: 1331 cm 3 Q44: 12 / 15, or simplified to 4 / 5 Q45: ( 12 / 15 ) 3, or simplified to ( 4 / 5 ) 3 Q46: 33 pence

36 32 ANSWERS: TOPIC 7 End of topic 7 test (page 24) Q47: Steps: Do we need an enlargement or reduction scale factor? reduction What is the scale factor? 50 / 75 Answer: Width = 26 cm Q48: Steps: Do we need an enlargement or reduction scale factor? enlargement What is the scale factor? 42 / 28 Use the scale factor to calculate the height of the medium size lamp. Answer: Height = 84 cm Q49: Steps: Do we need an enlargement or reduction scale factor? reduction What is the scale factor? 9 / 23 Use the scale factor to calculate the length x. Answer: Length x = 10 6 cm Q50: Steps: Do we need an enlargement or reduction scale factor to find AC? enlargement What is the scale factor? Use the scale factor to calculate the length of AC. Answer: AC = 1 6 m Q51: Steps: Do we need an enlargement or reduction scale factor to find BD? reduction What is the scale factor? Use the scale factor to calculate the length of BD. Answer: BD = 2 45 m Q52: Steps:

37 ANSWERS: TOPIC 7 33 Do we need an enlargement or reduction scale factor? enlargement What is the length scale factor? 76 5 What is the area scale factor? ( ) 2 Use the area scale factor to calculate the area coloured green on the large Christmas tree. Answer: Area = 2023 cm 2 Q53: Steps: Do we need an enlargement or reduction scale factor? reduction What is the length scale factor? ) 2 What is the area scale factor? ( Use the area scale factor to calculate the area of the small sheepskin rug. Answer: Area = 0 8 m 2 Q54: Steps: Do we need an enlargement or reduction scale factor? reduction What is the length scale factor? 18 / 28 What is the area scale factor? ( 18 / 28 ) 3 Use the volume scale factor to calculate the volume of milk in the small glass. Answer: Volume = 66 4 ml Q55: Steps: Do we need an enlargement or reduction scale factor? enlargement What is the length scale factor? What is the volume scale factor? Use the volume scale factor to find the cost of the large ball. Give your answer to the nearest penny. Answer: Cost = 9 84 (to the nearest penny)