8Similarity ONLINE PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8.

Transcription

1 8.1 Kick off with S 8. Similr ojects 8. Liner scle fctors 8Similrity 8.4 re nd volume scle fctors 8. Review

2 Plese refer to the Resources t in the Prelims section of your eookplus for comprehensive step-y-step guide on how to use your S technology. 8.1 Kick off with S Liner, re nd volume scle fctors The re of circle is given y the formul = πr, where r is the length of the rdius. 1 Using S, define nd sve the formul to clculte the re of circle. Use the formul defined in prt to clculte the re of circle with rdius of 7 cm. If the rdius in 1 incresed y scle fctor of (i.e. ws multiplied y ), use your formul from 1 to determine the scle fctor the re of the circle hs incresed y. The re of tringle is given y the formul = 1 h, where is the length of the se nd h is the perpendiculr height. Using S, define nd sve the formul to clculte the re of tringle. Use the formul defined in prt to clculte the re of tringle with se length of 6 cm nd perpendiculr height of 9 cm. 4 If the se length nd perpendiculr height in oth incresed y scle fctor of (i.e. were multiplied y ), use your formul from to determine the scle fctor the re of the tringle hs incresed y. The volume of rectngulr prism is given y the formul V = lwh, where l is the length, w is the width nd h is the height. Using S, define nd sve the formul to volume of rectngulr prism. Use the formul defined in prt to clculte the volume of rectngulr prism with length of cm, width of cm nd height of 7 cm. 6 If the length, width nd height in ll incresed y scle fctor of (i.e. were multiplied y ), use your formul from to determine the scle fctor the volume of the rectngulr prism hs incresed y. 7 omment on your nswers to questions, 4 nd 6.

3 8. Units 1 & OS 4 Topic oncept 1 Similr figures oncept summry Prctice questions WORKeD exmple 1 Similr ojects Ojects re clled similr when they re ectly the sme shpe ut hve different sizes. Ojects tht re ectly the sme size nd shpe re clled congruent. Similrity is n importnt mthemticl concept tht is often used for plnning purposes in res such s engineering, rchitecture nd design. Scled-down versions of much lrger ojects llow designs to e trilled nd tested efore their construction. onditions for similrity Two-dimensionl ojects re similr when their internl ngles re the sme nd their side lengths re proportionl. This mens tht the rtios of corresponding side lengths re lwys equl for similr ojects. We use the symols ~ or to indicte tht ojects re similr. Show tht these two ojects re similr. 1. think 1 onfirm tht the internl ngles for the ojects re the sme. lculte the rtio of the corresponding side lengths nd simplify. 1. WritE The digrms indicte tht ll ngles in oth ojects re equl. Rtios of corresponding sides: 1. =. nd =. Stte the nswer. The two ojects re similr s their ngles re equl nd the rtios of the corresponding side lengths re equl. 00 Mths Quest 11 GeNeRL MtheMtIs Ve units 1 nd

4 Interctivity Similr tringles int-67 similr tringles The conditions for similrity pply to ll ojects, ut not ll of them need to e known in order to demonstrte similrity in tringles. If pirs of tringles hve ny of the following conditions in common, they re similr. 1. ngle ngle ngle () D WORKeD exmple If two different-sized tringles hve ll three ngles identified s eing equl, they will e similr. = D, = E, = F. Side side side (SSS) If two different-sized tringles hve ll three sides identified s eing in proportion, they will e similr. DE = DF = EF. Side ngle side (SS) If two different-sized tringles hve two pirs of sides identified s eing in proportion nd their included ngles re equl, they will e similr. DE = DF nd = D Show tht these two tringles re similr. think WritE 1 Identify ll possile ngles nd side lengths. The ngles in the lue tringle re: 8, 7 nd 180 (8 + 7) = 70. The ngles in the red tringle re: 70, 7 nd 180 (70 + 7) = 8 Use one of, SSS or SS to check for similrity. ll three ngles in the two tringles re equl. Stte the nswer. The two tringles re similr s they stisfy the condition E E E 7 D D F F F topic 8 similrity 01

5 Eercise 8. PRctise Similr ojects 1 WE1 Show tht the two ojects in ech of the following pirs re similr Show tht rectngle with side lengths of 4. cm nd 18. cm will e similr to one with side lengths of cm nd 4.6 cm. WE Show tht the two tringles in ech of the following pirs re similr Which of the following pirs of tringles re similr? Mths Quest 11 GENERL MTHEMTIS VE Units 1 nd

6 onsolidte Which of the following pirs of rectngles re similr? c d Which of the following pirs of polygons re similr? Topic 8 Similrity 0

7 c 6. d In ech of the following groups, which two tringles re similr? c d Eplin why ech of the following pirs of ojects must e similr. 04 Mths Quest 11 GENERL MTHEMTIS VE Units 1 nd

8 c 9 lculte the rtios of the corresponding sides for the following pirs of ojects..44 c Evlute the rtios of the corresponding side lengths in the following pirs of similr ojects Evlute the unknown side lengths in the following pirs of similr ojects Topic 8 Similrity 0

9 Mster 1 Verify tht the following re similr. squre of side length 8. cm nd squre of side length 0.84 cm n equilterl tringle of side length 1.6 cm nd n equilterl tringle of side length 14.4 cm 1 lculte the vlue of required to mke the pirs of ojects similr in ech of the following digrms lculte the vlue of for the following similr shpes. Give your nswer correct to 1 deciml plce lculte the vlues of nd y in the following digrm. y Mths Quest 11 GENERL MTHEMTIS VE Units 1 nd

10 16 For the polygon shown, drw nd lel similr polygon where: the corresponding sides re 4 the size of those shown the corresponding sides re 4 the size of those shown. Liner scle fctors onsider the pir of similr tringles shown in the digrm: The rtios of the corresponding side lengths re: D 18 DE : = 18 : 6 = : 1 EF : = 16. :.4 = : 1 DF : = 0 : 10 = : 1 Note: In this topic we will put the imge first when clculting rtios of corresponding lengths. In fct, the side lengths of tringle DEF re ll three times the lengths of tringle. In this cse we would sy tht the liner scle fctor is. The liner scle fctor for similr ojects cn e evluted using the rtio of the corresponding side lengths. Δ ~ ΔDEF Liner scle fctor: DE = EF = DF = k F 0 E 16. F D E length of imge Liner scle fctor = length of oject liner scle fctor greter thn 1 indictes enlrgement, nd liner scle fctor less thn 1 indictes reduction. Topic 8 Similrity 07

11 WORKeD exmple lculte the liner scle fctor for the pir of tringles shown in the digrm think 1 lculte the rtio of the corresponding side lengths nd simplify. WritE 1 7 = = = Stte the nswer. The liner scle fctor is. EErcisE 8. PrctisE Liner scle fctors 1 WE lculte the liner scle fctor for the pirs of tringles shown c d lculte the liner scle fctors for the pirs of similr ojects shown Mths Quest 11 GeNeRL MtheMtIs Ve units 1 nd

12 c d lculte the unknown side lengths in the pirs of similr shpes shown. onsolidte y c d z y Topic 8 Similrity 09

13 4 lculte the unknown side lengths in the digrms shown.. y y lculte the liner scle fctors for the following rtios of corresponding side lengths. : 1 : c : 4 d 8 : 68 6 lculte the missing vlues for the following. = = 6 = = c 7 = 81 9 = d = = 0.6 e 16. = 4 =. 7 lculte the length of. 8 lculte the vlue of. 4 E The side of house csts shdow tht is 8.4 m long on horizontl ground. t the sme time, n 800-mm verticl grden stke hs shdow tht is 1.4 m long. Wht is the height of the house? When the house hs shdow tht is 10 m long, how long is the grden stke s shdow? D 10 Mths Quest 11 GENERL MTHEMTIS VE Units 1 nd

14 10 lculte the vlue of in the digrm. 10 Mster 11 section of ridge is shown in the digrm. How high is point ove the rodwy of the ridge?.6 m. m 7.4 m. m To clculte the distnce cross rvine, surveyor took direct line of sight from the point to fied point on the other side nd then mesured out perpendiculr distnce of 18 m. From tht point the surveyor mesured out smller similr tringle s shown in the digrm. lculte the distnce cross the rvine long the line. 1 Over horizontl distnce of 6. m, n escltor rises 1.7 m. If you trvel on the escltor for horizontl distnce of 4. m, wht verticl distnce hve you risen? 1 m.. 14 m 8 m 18 m Topic 8 Similrity 11

15 8.4 Interctivity re scle fctor int In gme of illirds, ll trvels in stright line from point onethird of the distnce from the ottom 1. m of the right side nd reounds from point three-eighths of the distnce long the ottom side. The ngles.4 m etween the ottom side nd the ll s pth efore nd fter it reounds re equl. lculte the perpendiculr distnce, correct to deciml plces, from the ottom side fter the ll hs trvelled distnce of 0.8 m prllel with the ottom side fter reounding. If the ll hs een struck with sufficient force, t wht point on n edge of the tle will it net touch? Give your nswer correct to deciml plces. re nd volume scle fctors re scle fctor onsider three squres with side lengths of 1, nd cm. Their res will e 1 cm, 4 cm nd 9 cm respectively. 1 cm cm cm 1 cm cm cm re = 1 cm re = 4 cm re = 9 cm The liner scle fctor etween squre nd squre will e, nd the liner scle fctor etween squre nd squre is. When we look t the rtio of the res of the squres, we get 4 : 1 for squres nd nd 9 : 1 for squres nd. In oth cses, the re scle fctor is equl to the liner scle fctor rised to the power of two. Liner scle fctor re scle fctor : 4 : 9 ompring squres nd, the rtio of the side lengths is :, resulting in liner scle fctor of or 1.. From the rtio of the res we get 4 : 9, which once gin indictes n re scle fctor 9 =.; tht is, the liner scle fctor to the 4 power of two. In generl, if the liner scle fctor for two similr ojects is, the re scle fctor will e. 1 Mths Quest 11 GENERL MTHEMTIS VE Units 1 nd

16 WORKeD exmple 4 lculte the re scle fctor for the pir of tringles shown in the digrm think 1 lculte the rtio of the corresponding side lengths. Squre the liner scle fctor to otin the re scle fctor. WritE 1. = =. 1.6 = = 4 Stte the nswer. The re scle fctor is 4. Units 1 & OS 4 Topic oncept Similrity of solids oncept summry Prctice questions Interctivity Volume scle fctor int-6479 WORKeD exmple Volume scle fctor Three-dimensionl ojects of the sme shpe re similr when the rtios of their corresponding dimensions re equl. When we compre the volumes of three similr cues, we cn see tht if the liner scle fctor is, the volume scle fctor will e. ue : ue Scle fctor Liner : 1 re 4 : 1 = 4 Volume 8 : 1 = If the liner scle fctor for two similr ojects is, the volume scle fctor will e. lculte the volume scle fctor for the pir of spheres shown in the digrm. 9 7 think 1 lculte the rtio of the corresponding dimensions. WritE 7 9 = ue the result to otin the volume scle fctor. = 9 Stte the nswer. The volume scle fctor is 9. topic 8 similrity 1

17 Eercise 8.4 PRctise re nd volume scle fctors 1 WE4 lculte the re scle fctor for ech of the pirs of tringles shown. c lculte the re scle fctor for ech of the pirs of similr ojects shown WE lculte the volume scle fctor for the pir of spheres shown lculte the volume scle fctor for ech of the pirs of similr ojects shown c Mths Quest 11 GENERL MTHEMTIS VE Units 1 nd

18 onsolidte lculte the res of the two similr tringles shown in the digrm. How mny times lrger in re is the iggest tringle? 1 c lculte the liner scle fctor. d lculte the re scle fctor. 8 6 hegon is mde up of si equilterl tringles of side length cm. If similr hegon hs n re of 4 liner scle fctor. cm, clculte the 7 rectngulr swimming pool is shown on the plns for uilding development with length of 6 cm nd width of. cm. If the scle on the plns is shown s 1 : 0: clculte the re scle fctor clculte the surfce re of the swimming pool. 8 The re of the tringle DE in the digrm is 100 cm, nd the rtio of DE : is : 1. lculte the re of tringle. 9 The floor of squre room hs n re of 1 m. lculte the re tht the room tkes up in digrm with scle of 1 : n rchitect mkes smll scle model of house out of ls wood with the dimensions shown in the digrm. If the ctul length of the uilding is 6. m, wht is the scle of the model? cm Wht is the rtio of the volume of the uilding to the volume of the model? 6 cm 11 Two similr cylinders hve volumes of 400 cm nd 0 cm respectively. Wht is the liner scle fctor? If the length of the lrger cylinder is 8 cm, wht is the length of the smller one? 1 If cue hs volume of cm nd is then enlrged y liner scle fctor of., wht will the new volume e? 8 cm D cm E Topic 8 Similrity 1

19 1 lculte the liner scle fctor etween two similr drink ottles if one hs volume of 600 ml nd the other hs volume of 1. L. Mster 14 If n re of 71 m is represented on scle drwing y n re of 44. cm, wht is the ctul length tht distnce of. cm on the drwing represents? 1 model cr is n ect replic of the rel thing reduced y fctor of 1. If the ctul surfce re of the cr tht is sprypinted is 4. m, wht is the equivlent pinted re on the model cr? If the ctul storge cpcity of the cr is 1.78 m, wht is the equivlent volume for the model cr? 16 compny sells cnned fruit in two sizes of similr cylindricl cns. For ech size, the dimeter is fourfifths of the height. Write n epression for clculting the volume of cn of fish in terms of its dimeter. If the dimensions of the lrger cns re 1. times those of the smller cns, write n epression for clculting the volume of the lrger cns of fish in terms of the dimeter of the smller cns. 16 Mths Quest 11 GENERL MTHEMTIS VE Units 1 nd

20 ONLINE ONLY 8. Review the Mths Quest review is ville in customisle formt for you to demonstrte your knowledge of this topic. the review contins: Multiple-choice questions providing you with the opportunity to prctise nswering questions using S technology short-nswer questions providing you with the opportunity to demonstrte the skills you hve developed to efficiently nswer questions using the most pproprite methods ONLINE ONLY ctivities to ccess eookplus ctivities, log on to Interctivities comprehensive set of relevnt interctivities to ring difficult mthemticl concepts to life cn e found in the Resources section of your eookplus. Etended-response questions providing you with the opportunity to prctise em-style questions. summry of the key points covered in this topic is lso ville s digitl document. REVIEW QUESTIONS Downlod the Review questions document from the links found in the Resources section of your eookplus. studyon is n interctive nd highly visul online tool tht helps you to clerly identify strengths nd weknesses prior to your ems. You cn then confidently trget res of gretest need, enling you to chieve your est results. Units 1 & Similrity Sit topic test topic 8 similrity 17

21 8 nswers Eercise = 11. = 7. = 1.87, nd ll ngles re equl = = 9. = 1.68 =.6, nd ll ngles.8 re equl = 4.6 =.8, nd ll ngles re equl c d 6 c d = 78. =.7, SS = = 14.8 =., SSS = = 1.8, = 1.7; not similr =.96 =.8 nd ll ngles re equl; similr = 1., = 1.4; not similr =.6, =.6; similr =.1, =.1; similr = 1.09, = 1.08; not similr = 1.4, = 1.4, = 1.4 nd ll ngles re 16 9 equl; similr = 1.9, = 1.9, = 1.9 nd ll ngles re equl; similr = 1.6, = 1.6, = 1.6 nd ll ngles 8 4 re equl; similr = 1., = 1., = 1. nd ll ngles re equl; similr 7 nd nd c nd d nd 8 ll ngles re equl nd side lengths re in proportion. ll mesurements (rdius nd circumference) re in proportion. c ll ngles re equl nd side lengths re in proportion : 1.6 : 1 c.6 : :. : = 6., ll sides re in proportion nd ll 8. ngles equl = 1.18, ll sides re in proportion nd ll 1.6 ngles re equl =.6, y = Eercise c 1. d c.8 d = 1, y = c.8 d =, y = 1.6, z =.76 4 =., y = y = c 0.7 d Mths Quest 11 GENERL MTHEMTIS VE Units 1 nd

22 6 c d e = 7 1 = 6 1. = = = 81 9 = 9. = 1. = = = m 1.67 m m 1 4 m m m 0.67 m from the ottom of the left side Eercise = =.0176 c 49 4 = =.6 49 = c nd 19 squre units 6 4 c d m 8 cm cm 10 1 : : cm cm m 1 00 cm 100 cm 16 V = πd 7πD1 V = 40 Topic 8 Similrity 19