Ojectives H P T R 9 Rtios nd similrity To divide quntity in given rtio To determine the rtio in which quntity hs een divided To pply the trnsformtions which re expnsions from the origin To define similrity of two figures To determine when two tringles re similr y using the conditions equl ngles () equl rtios (PPP) corresponding sides hving the sme rtio nd the included ngle equl (PP) To pply similrity to solve prolems To determine nd pply expnsion fctors for res nd volumes 9.1 Rtios This section is revision of work of previous yers. Severl exmples re presented. xmple 1 ivide 300 in the rtio3:. Solution one prt = 300 5 = 60 two prts = 60 = 10 three prts = 60 3 = 180 xmple SMPL ivide 3000 in the rtio 3 : : 1. mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird 7
8 ssentil dvnced Generl Mthemtics Solution one prt = 3000 6 = 500 two prts = 500 = 1000 SMPL three prts = 500 3 = 1500 xmple 3 dy is divided into 10 new-hours, ech new-hour is divided into 100 new-minutes nd ech new-minute is divided into 100 new-seconds. Wht is the rtio of new-second to n ordinry second? Solution There re 10 10 10 new-seconds in dy nd 4 60 60 ordinry seconds in dy the rtio of new-seconds : ordinry seconds = 1 10 : 1 5 8.64 10 4 = 864 : 1000 = 108 : 15 xmple 4 Two positive integers re in the rtio:5.iftheproduct of the integers is 40 find the lrger integer. Solution Let nd denote the integers =... 1 5 nd = 40... From 1 = Sustitute in 5 5 = 40 = 100 =±10 nd s is positive integer, = 10 nd = 4 The lrger integer is 10. xercise 9 xmple 1 1 ivide 9000 in the rtio : 7. xmple ivide 15 000 in the rtio : : 1. mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
hpter 9 Rtios nd similrity 9 3 x :6= 9:15. Find x. 4 The rtio of the numers of ornge flowers to pink flowers in grden is 6 : 11. There re 144 ornge flowers. How mny pink flowers re there? 5 15 : = x :3.Find x. 6 The ngles of tringle re in the rtio 6 : 5 : 7. Find the sizes of the three ngles. 7 Three men X, Y nd Z shre n mount of money in the rtio : 3 : 7. If Y receives $ more thn X,how much does Z otin? 8 n lloy consists of copper, zinc nd tin in the rtios 1 : 3 : 4 (y weight). If there is 10 g of copper in the lloy, find the weights of zinc nd tin. 9 In g the rtio of red eds to white eds to green eds is 7 : : 1. If there re 56 red eds, how mny white eds nd how mny green eds re there? 10 On mp the length of rod is represented y 45 mm. If the scle is 1 : 15 000, find the ctul length of the rod. 11 Five thousnd two hundred dollrs ws divided etween mother nd dughter in the rtio 8 : 5. Find the difference etween the sums they received. 1 Points,, nd re plced in tht order on line so tht = =. xpress s frction of. 13 If the rdius of circle is incresed y two units, find the rtio of the new circumference to the new dimeter. 14 In clss of 30 students the rtio of oys to girls is : 3. If six oys join the clss, find the new rtio of oys to girls in the clss. 15 If : = 3:4nd :( + c) = :5, find the rtio : c. 16 The scle of mp reds 1 : 50 000. Find the distnce, in kilometres, etween two towns which re 3.5 cm prt on the mp. SMPL 9. n introduction to similrity The two tringles nd re similr. Note: O = O, O = O, O = O. Tringle cn e considered s the imge of tringle under mpping of the plne in which the coordintes re multiplied y. This mpping is clled n expnsion from the origin of fctor. This cn e written in trnsformtion nottion: (x, y) (x, y). 1 10 8 6 4 0 (, 6) '(4, 1) '(10, 6) (5, 3) (4, 1) '(8, ) 4 6 8 10 1 mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
30 ssentil dvnced Generl Mthemtics There is lso mpping from to which is n expnsion from the origin of fctor 1. ( 1 The rule for this is (x, y) x, 1 ) y. Two figures re similr if one is congruent to n imge of the other under n expnsion from the origin of fctor k. Forexmple, the rectngle of side lengths 1 nd is similr to the rectngle with side lengths 3 nd 6. 9 (3, 9) (6, 9) 8 Note here the expnsion fctor is 3 nd the rule is 7 (x, y) (3x, 3y). 6 5 Note: 4 ny two circles re similr (1, 3) (, 3) 3 (3, 3) (6, 3) ny two squres re similr ny two equilterl tringles re similr 1 (1, 1) (, 1) Fortringle with side lengths,, c nd similr 0 1 3 4 5 6 tringle with corresponding side lengths,, c it cn e seen tht = = c c = k where k is the pproprite expnsion fctor. Similr sttements cn e mde out other pirs of similr polygons. Note lso tht the mesure of n ngle does not chnge under n expnsion: i.e., for two similr figures, corresponding ngles re equl. Similr tringles Two tringles re similr if one of the following conditions holds: tringles hve equl ngles () 45 100 35 corresponding sides re in the sme rtio (PPP) = = = k, where k is the expnsion (enlrgement) fctor two pirs of corresponding sides hve the sme rtio nd the included ngles re equl, (PP) ' SMPL 45 ' 100 ' 35 ' 45 ' 45 ' = mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
hpter 9 Rtios nd similrity 31 two pirs of corresponding sides hve the sme rtio nd two corresponding non-included ngles re equl, provided these ngles re right ngles or otuse. ' 10 ' 10 Tringle is similr to tringle cn e written symoliclly s The tringles re nmed so tht ngles of equl mgnitude hold the sme position i.e., corresponds to, corresponds to, corresponds to. i.e. xmple 5 = or = Give the reson for tringle eing ' similr to tringle. Find the vlue of x. 0 6.5 cm Solution 5 cm 0 3.75 cm 3 cm Tringle is similr to tringle s x cm ' ' 5 3.013 cm 6.5 = 3 3.75 = 0.8 nd the mgnitude of = mgnitude of = 0 PPisthe condition for similrity. x 3.013 = 5 6.5 x = 5 6.5 3.013 =.4104 SMPL xmple 6 Give the reson for tringle eing similr to tringle XY. Find the vlue of x. x cm 6 cm ' X 3 cm.5 cm Y mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
3 ssentil dvnced Generl Mthemtics xmple 5 Solution orresponding ngles re of equl mgnitude (). X = Y x i.e., x + 6 = 3 5.5 5.5x = 3(x + 6).5x = 18 x = 7. xercise 9 1 Give resons why the following pirs of tringles re similr nd find the vlue of x in ech cse. c 4 cm 13 cm 56 8 14 cm 1 cm 5 cm 10 cm x cm x cm ' 6 cm Q x cm SMPL d 9 cm 4 cm 6 cm ' cm 8 cm x cm P ' 10 cm R mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
hpter 9 Rtios nd similrity 33 xmple 6 Give resons why the following pirs of tringles re similr nd find the vlue of x in ech cse. c P x cm 1 cm cm x cm P Q 6 cm 8 cm 16 cm Q 8 cm 3 Given tht = 14, = 1, = 15 nd = 4, find, nd. d x cm cm x cm 1.5 cm 3 cm cm 10 cm 14 1 4 tree csts shdow of 33 m nd t the sme time stick 30 cm long csts shdow 4 cm long. How high is the tree? tree SMPL 5 0mhigh neon sign is supported y 40 m steel cle s shown. n nt crwls long the cle strting t.how high is the nt when it is 15 m from? 0.3 m 6 hill hs grdient of 1 in 0, i.e. for every 0 m horizontlly there is 1mincrese in height. If you go 300 m horizontlly, how high up will you e? 40 m cm 15 4 0 m mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
34 ssentil dvnced Generl Mthemtics 7 mn stnds t nd looks t point Y cross the river. He gets friend to plce stone t X so tht, X nd Y re colliner. He then mesures, X nd X to e 15 m, 30 m nd 45 m respectively. Find Y, the distnce cross the river. 15 m 45 m X 30 m Y 8 Find the height, h m, of tree tht csts shdow 3 m long t the sme time tht verticl stright stickmlong csts shdow 6. m long. 9 plnk is plced stright up stirs tht re 0 cm wide nd 1 cm deep. Find x,where x cm is the width of the widest rectngulr ox of height 8 cm tht cn e plced on stir under the plnk. 10 The sloping edge of technicl drwing tle is 1 m from front to ck. lculte the height ove the ground of point,which is 30 cm from the front edge. 11 Two similr rods 1.3 m long hve to e hinged together to support tle 1.5 m wide. The rods hve een fixed to the floor 0.8 m prt. Find the position of the hinge y finding the vlue of x. 80 cm x m plnk 30 cm 8 cm 0 cm 1 m 1.5 m 0.8 m x cm 1 cm (1.3 x) m 1 mn whose eye is 1.7 m from the ground, when stnding 3.5 m in front of wll 3 m high, cn just see the top of tower tht is 100 m wy from the wll. Find the height of the tower. 9 cm SMPL 13 mnis8mup10mldder, the top of which lens ginst verticl wll nd touches it t height of9move the ground. Find the height of the mn ove the ground. 14 spotlight is t height of 0.6 m ove ground level. verticl post 1.1 m high stnds 3mwy nd 5 m further wy there is verticl wll. How high up the wll does the shdow rech? spotlight 0.6 m verticl post 1.1 m 3 m 5 m wll mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
15 Mesurements in the digrm shown re in cm. Prove tht. Find x. c Use Pythgors theorem to find y nd z. d Verify y : z = :. 16 Find. ' 10 ' 7 hpter 9 Rtios nd similrity 35 17 mn who is 1.8 m tll csts shdow of 0.76 m in length. If t the sme time telephone pole csts3mshdow, find the height of the pole. 18 In the digrm shown, RT = 4 cm, ST = 10 cm. Find the length NT. 19 is tringulr frme with = 14 m, = 10 m, = 7m.point P on, 1.5 m from,islinked y rod to point Q on,3mfrom. lculte the length of PQ. 0 Using this digrm, find, x nd y. 4 9.3 res, volumes nd similrity 1 S 5 x z ' 6 y x If two shpes re similr nd the expnsion (enlrgement) fctor is k, i.e., for ny length of one shpe, the length of the corresponding length of the similr shpe hs length k, then the re of similr shpe = k re of the originl shpe Fortwo tringles nd which re similr, i.e., with = k, re of tringle = k re of tringle ' SMPL R N 4 y T c h ' c' ' h' ' ' ' mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
36 ssentil dvnced Generl Mthemtics This cn e shown y oserving tht nd re of tringle = 1 h = 1 k kh, (where = nd = ) ( ) 1 = k h = k re of tringle Some exmples of similr shpes nd the rtio of their res re considered in the following. 3 cm re =.3 3 cm re = 6cm cm Similr circles Scle fctor = 4 3 ( Rtio of res =.4.3 = 4 4 3 = 3 Similr rectngles Scle fctor = Rtio of res = 4 6 = 4 = re =.4 ) 4 cm 6 cm re = 4 cm 10 cm 5 cm 3 cm Similr tringles 4 cm Scle fctor = 8 cm re = 6cm re = 4 cm SMPL xmple 7 Rtio of res = 4 6 = 4 = 4 cm The two rectngles shown elow re similr. The re of rectngle is 0 cm.find the re of rectngle. 6 cm mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
hpter 9 Rtios nd similrity 37 ' ' 3 cm 5 cm Solution The rtio of the length of their ses = = 5 3 The rtio of their res = re of ( ) 5 re of = = 5 3 9 re of = 5 0 cm 9 = 55 5 9 cm Two solids re considered to e similr if they hve the sme shpe nd the rtio of their corresponding liner dimensions re equl. 3 cm F cm G 1 cm H Scle fctor =.5 ' ' 7.5 cm ' ' F'.5 cm SMPL The cuoids FGH nd F G H re similr. For similr solids, if the scle fctor is k then the Forexmple, 5 cm volume of the similr solid = k 3 volume of the originl solid Volume of FGH = (3 1) cm 3 = 6cm 3 Volume of F G H = (5.5 7.5) cm 3 = 93.75 cm 3 The rtio of volumes = 93.75 6 = 15.65 =.5 3 ' ' H' ' G' mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
38 ssentil dvnced Generl Mthemtics Here is nother exmple. V' 3 cm xmple 8 3 cm V 3 cm Scle fctor = 5 3 The two squre pyrmids re similr. VO = 9 cm. 9 cm V O 4 cm ' 5 cm 5 cm ( Rtio of volumes = 53 5 3 = 3 3 ' ' V' O' ) 3 ' 5 cm ' ' 5 cm Find the rtio of the length of their ses, nd hence the height, V O,ofthe pyrmid V. The volume of V is 48 cm 3.Find the rtio of their volumes, nd hence find the volume of V. Solution The rtio of the length of their ses = V O = 5 4 9 = 45 4 = 5 4 SMPL The length of V O is 11.5 cm. The volume of V is 48 cm 3 The rtio of their volumes = Volume of V ( ) 5 3 Volume of V = = 15 4 64 Volume of V = 15 48 cm3 64 = 93.75 cm 3 ' mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
hpter 9 Rtios nd similrity 39 xmple 7 xercise 9 1 These four rectngles re similr. c Write down the rtio of the lengths of their ses. y counting rectngles, write down the rtio of their res. Is there reltionship etween these two rtios? These four prllelogrms re similr. c Write down the rtio of the lengths of their ses. y counting prllelogrms, write down the rtio of their res. Is there reltionship etween these two rtios? 3 The two rectngles shown re similr. The re of rectngle is 7cm. 3 cm Find the re of rectngle. ' 5 cm SMPL 4 Tringle is similr to tringle XYZ. XY = YZ = ZX =.1 The re of tringle XYZ is 0 cm.find the re of tringle. 5 Tringles nd re equilterl tringles. Find the length of F. Find. c Find the rtio re of tringle re of tringle ' cm F cm cm ' ' ' ' cm F' cm ' mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
40 ssentil dvnced Generl Mthemtics xmple 8 6 The res of two similr tringles re 16 nd 5. Wht is the rtio of pir of corresponding sides? 7 The res of two similr tringles re 144 nd 81. If the se of the lrge tringle is 30, wht is the corresponding se of the smller tringle? 8 These three solids re similr. Write down the rtio of the lengths of the ses. Write down the rtio of the lengths of the heights. c y counting cuoids equl in shpe nd size to the cuoid given in, write down the rtio of the volumes. d Is there reltionship etween the nswers to, nd c? 9 These re two similr rectngulr locks. 8 cm 4 cm 3 cm 1 cm 6 cm 4 1 cm Write down the rtio of their i longest edges ii depths iii heights. y counting cues of side 1 cm, write down the rtio of their volumes. c Is there ny reltionship etween the rtios in nd? 10 These three solids re spheres. Write down the rtio of the rdii of the three spheres. The volume of sphere of rdius r is given y the formul V = 4 3 r 3. xpress the volume of ech sphere s multiple of. Hence write down the rtio of their volumes. c Is there ny reltionship etween the rtios found in nd? SMPL In 11 to 0, ojects referred to in the sme question re mthemticlly similr. 11 The sides of two cues re in the rtio : 1. Wht is the rtio of their volumes? 3cm cm 5cm 1 The rdii of two spheres re in the rtio 3 : 4. Wht is the rtio of their volumes? 13 Tworegulr tetrhedrons hve volumes in the rtio 8 : 7. Wht is the rtio of their sides? mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
hpter 9 Rtios nd similrity 41 14 Two right cones hve volumes in the rtio 64 : 7. Wht is the rtio of their heights their se rdii? 15 Two similr ottles re such tht one is twice s high s the other. Wht is the rtio of their surfce res their cpcities? 16 ch liner dimension of model cr is 1 of the corresponding cr dimension. 10 Find the rtio of the res of their windscreens the cpcities of their oots c the widths of the crs d the numer of wheels they hve. 17 Three similr jugs hve heights 8 cm, 1 cm nd 16 cm. If the smllest jug holds 1 litre, find the cpcities of the other two. 18 Three similr drinking glsses hve heights 7.5 cm, 9 cm nd 10.5 cm. If the tllest glss holds 343 millilitres, find the cpcities of the other two. 19 toy mnufcturer produces model crs which re similr in every wy to the ctul crs. If the rtio of the door re of the model to the door re of the cr is 1 : 500, find the rtio of their lengths the rtio of the cpcities of their petrol tnks c the width of the model, if the ctul cr is 150 cm wide d the re of the rer window of the ctul cr if the re of the rer window of the model is 3 cm. 0 The rtio of the res of two similr lels on two similr jrs of coffee is 144 : 169. Find the rtio of the heights of the two jrs their cpcities. 1 In the figure, if M is the midpoint of F nd K is the midpoint of, the re of F is how mny times s gret s the re of KM? If the re of F is 15, find the re of KM. SMPL In the digrm, is equilterl. = F nd is the midpoint of.find the rtio of re of : re of F. M F K F mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
4 ssentil dvnced Generl Mthemtics 3 The res of two similr tringles re 144 cm nd 81 cm.ifthe length of one side of the first tringle is 6 cm, wht is the length of the corresponding side of the second? 9.4 Geometric representtion of rithmetic opertions Simple rithmetic opertions correspond to elementry geometricl constructions. In mny cses the vlidity of these constructions cn e estlished through similr tringles. If two segments re given with lengths nd (s mesured y given unit segment) then +,, r (where r is ny rtionl numer),, nd cn e constructed. onstruction of + rw stright line nd mrk off with compss, s shown in the digrm, the distnce O nd where O = nd =. Then O = +. onstruction of -- rw stright line nd mrk off with compss the distnce O nd where O = nd =, ut this time is constructed in the other direction. Then O =. onstruction of r O O + To construct 3 = + +, three copies of the line segment of length re constructed. For n = + + +,where n is nturl numer, n copies of the line segment of length re constructed. SMPL onstruction of Mrk off line segments O nd O of length units nd units respectively. onstruct O of length 1 unit. Join points nd nd drw line prllel to the line through. The line segment O hs length. Note tht tringle O is similr to tringle O nd O = O. Therefore O = O =. O 1 mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
hpter 9 Rtios nd similrity 43 onstruction of 1 This will e done for = 5. Line segment is of unit length. rw ny line X. hoose line segment nd then replicte this line segment four times to form line segments,, nd. rw line segment nd then prllel line segments Y, Y, Y nd Y to divide line segment into five equl segments. ch of these segments hs length 1 5 of unit. ' ''' '' Y Y' Y" Y"' Note tht tringle Y is similr to tringle. nd 5 =. Hence = 5Y onstruction of One wy of constructing is to mrk off line segments O nd O of length units nd units respectively. onstruct O of length 1 unit. Join points nd nd drw line prllel to the line through. The line segment O hs length O Note tht tringle O is similr to tringle O nd O = O. Therefore O = O nd this implies O =. onstruction of onstruct line segments of length nd 1, nd circle of dimeter + 1. In the digrm O = nd = 1. ngle O is right ngle (right ngle sutended t the circle y dimeter), nd O is right ngle y construction. Therefore tringle O is similr to tringle O nd to tringle. O Hence = = 1 O Therefore = nd hence = SMPL xercise 9 1 X 1 onstruct line segment of length 3 units. onstruct line segment of length 5 units. mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
44 ssentil dvnced Generl Mthemtics 3 rw line segment of length 10 cm nd use construction descried ove to divide it into three equl intervls. 4 rw line segment of length 0 cm nd use construction descried ove to divide it into nine equl intervls. 5 rw two line segments O nd O of lengths 4 cm nd 14 cm respectively. Use construction descried ove to construct line segment of length 7 units. 6 rw two line segments O nd O of lengths 9 cm nd 13 cm respectively. Use construction descried ove to construct line segment of length 9 13 units. 7 escrie the method for constructing line of length 10 3 units. 8 Illustrte the construction of line segment of length 3 4 units, given line segments of length 3 units, 4 units nd 1 unit. 9.5 Golden rtio If = c then is sid to e the geometric men of c nd (or sometimes the men proportionl of nd c). Let e line segment length units nd point on such tht =. Let = x. Therefore = x x x The reltion x = x holds. is the geometric men of nd. x If x = x then ( x) = x x Which implies tht x + x = 0 Therefore using the generl qudrtic formul x = ± 4 1 1 = ± 5 1 x = 1 ± 5 SMPL Only one of these is possile s is length. Thus = 1 + 5 (which is positive) Therefore = ( 1 + 5) = 1 + 5 = 1 + 5 1 + 5 1 + 5 = 1 + 5 mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
hpter 9 Rtios nd similrity 45 Hence the rtio is independent of the length of nd is lwys the sme numer. This numer is known s the golden rtio or section nd is denoted y, i.e., = 1 + 5 is the only numer which when diminished y one ecomes its own reciprocl, i.e., 1 = 1 This is shown s 1 = 1 + 5 1 = 1 + 5 = 1 + 5 1 + 5 1 + 5 = 1 + 5 construction of the golden rtio is s follows. Let e segment of unit length. rw of length, perpendiculr to. rw line segment. With centre drw n rc of rdius cutting t. rw n rc of rdius with centre cutting t. = 1 x = The golden rectngle The rectngle HF shown is known s the golden rectngle. The rtio of the side lengths :F= 1 + : ( 1 + nd = 1 + 1 + ) ( 5 1 + ) 5 = 3 + 5 1 + 5 φ F x x φ 1 x SMPL = 3 + 5 5 + 1 = + 5 4 = 1 + 5 = L G 1 φ 1 K 1 H Tht is, the rtio of the side lengths is. This rectngle hs some very plesnt properties, s oserved in the following explortions. mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
46 ssentil dvnced Generl Mthemtics Forming sequence of similr golden rectngles In the golden rectngle HF, construct squre GF with side length. The remining rectngle HG hs side lengths 1 nd. onstruct the squre LKHG with side length 1. The sides of the remining rectngle KL re 1 nd 1. It ws estlished erlier in the section tht 1 = 1. Thus the rectngles HF, HG, KL re ll similr s they ll hve sides in the rtio :1.This pttern continues. onsider the golden rectngle KL. Y Now rectngle YKX hs sides 1 nd 1 1. 1 The rtio :1 1 = 1: 1 nd s shown previously 1 = 1 + 5 1 = 1 + 5 = 1 Therefore 1 : 1 = :1 It cn e shown tht ll the rectngles formed in this wy re similr to ech other. φ F φ L G 1 L X K Forming sequence of squres nd rectngles, the res of which re in geometric sequence with common rtio 1 The rtio of the res of the squres nd rectngles is lso worth considering. The res in sequence re re rectngle HF = ( + 1) = 3 re of squre GF = re of rectngle HG = L X re of squre LKHG = 1 re of rectngle KL = 1 φ 1 K 1 H Y SMPL re of squre YXL = 1 re of rectngle YKX = 1 3 F G K H mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
hpter 9 Rtios nd similrity 47 xercise 9 1 For the golden rtio show tht 1 = 1 c = ( 1) = 1 3 = + 1 is right-ngled tringle with the right ngle t. X is the ltitude of the tringle from. Prove tht X X = X ; i.e., the length X is the X geometric men of lengths X nd X. Find X if i X = nd X = 8 ii X = 1 nd X = 10. 3 squre is inscried in semicircle s shown. Prove tht = = 1. 4 regulr decgon is inscried in circle with unit rdius s shown. c d Find the mgnitude of ngle i O ii O The line X isects ngle O. Prove tht i tringle X is isosceles ii tringle XO is isosceles iii tringle O is similr to tringle X Find the length of,totwo deciml plces. escrie construction for i regulr decgon ii regulr pentgon. 5 lculte 0, 1,, 3, 4 nd 1,, 3, 4. Show tht ech power of is equl to the sum of the two powers efore it, i.e., n+1 = n + n 1 SMPL 6 The Fioncci sequence is defined y t 1 = t = 1 nd t n+1 = t n 1 + t n. The sequence is 1, 1,, 3, 5,...onsider the sequence t, t 3, t 4, t 5...nd show tht s n gets very lrge t 1 t t 3 t 4 (n pproches infinity), t n+1 pproches. t n O X X mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
48 ssentil dvnced Generl Mthemtics Review hpter summry Two figures re similr to ech other if one is congruent to the other under n expnsion from the origin of fctor k.nexpnsion of fctor k from the origin hs rule (x, y) (kx, ky) Similr tringles Two tringles re similr if one of the following conditions holds. Tringles hve equl ngles () orresponding sides re in the sme rtio (PPP) Two pirs of corresponding sides hve the sme rtio nd the included ngles re equl (PP) 45 = '' '' ' If tringle is similr to tringle XYZ, this cn e written symoliclly s XYZ. The tringles re nmed so tht ngles of equl mgnitude hold the sme position, i.e., corresponds to X, corresponds to Y, corresponds to Z. If two shpes re similr nd the scle fctor is k, i.e. for ny length of one shpe, the corresponding length of the similr shpe hs length k, then the re of the similr shpe = k re of the originl shpe. For similr solids, if the scle fctor is k, then the volume of the similr solid is k 3 volume of the originl solid. Multiple-choice questions 1 If 5:3= 7:x then x is equl to 1 35 5 1 5 3 5 1 rss is composed of mixture of copper nd zinc. If the rtio copper : zinc is 85 : 15, then the mount of copper in 400 kg of rss is 60 kg 340 kg 360 kg 380 kg 150 kg 3 If the totl cost of P rticles is Q dollrs, then the cost of R rticles of the sme type is P PQR PQ QR R QR R P PQ 4 cr is 3. m long. The length in cm of model of the cr if the scle is 1 : 100 is 0.03 0.3 3. 30 3 5 n thlete runs 75 m in 9 seconds. If she were to mintin the sme verge speed for 100 m her time for 100 m in seconds would e 11.6 1.0 11.8 1. 1.4 45 ' SMPL ' mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
hpter 9 Rtios nd similrity 49 6 If 50 is divided into three prts in the rtio 1 : 3 : 6 then the lrgest prt is 5 15 50 30 3 3 7 Two similr cylinders re shown. The rtio of the volume of the smller cylinder to the lrger cylinder is 15 cm 45 cm 1:3 1:9 1:7 1:5 :9 10 cm 8 The rdius of sphere is 4 times the rdius of 5 sphere. Hence, the rtio of the volume of sphere to the volume of sphere is 16 : 5 4:5 5:4 5 : 16 64 : 15 9 Tringles nd XYZ re similr isosceles tringles. The length of XY is 4cm 5cm 4. cm.5 cm 3.6 cm 10 cm 10 YZ is prllel to Y Z nd Y Y = 1 YX. The re of tringle 3 XYZ is 60 cm. The re of tringle XY Z is 0 cm 30 cm 0 9 cm 0 3 cm 80 3 cm Short-nswer questions (technology-free) 3 cm 10 cm Y Y' X Z 30 cm 1 cm 1 cm 1 In tringle XYZ, P is point on XY nd Q is point on XZ such tht PQ is prllel to YZ. Show tht the two tringles XYZ nd XPQ re similr. If XY = 36 cm, XZ = 30 cm nd XP = 4 cm, find i XQ ii QZ c Write down the vlues of XP : PY nd PQ : YZ. Tringles nd F re similr. If the re of tringle is 1.5 cm, the re of tringle F is 4.5 cm nd = 5 cm, find the length of the vlue of : F c the vlue of F :. 3 If 1mstke csts shdow.3 m long, find the height of tree (in metres) which csts shdow 1 m long. SMPL X Z' Z Y Review mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
50 ssentil dvnced Generl Mthemtics Review 4 is right-ngled tringle with = 4 nd = 3. If the tringle is folded long the line XY, vertex coincides with vertex. Find the length of XY. 5 Points, nd lie on stright line. The squres re djcent nd hve side lengths 4, 7 nd x. Find the vlue of x. 6 Find the vlue of y in the digrm on the right. 6.6 Y X 4 7 x 7 n lloy is produced y mixing metl X with metl Y in the rtio of 5 : 3 y volume. The mss of 1 cm 3 of metl X is 8 5 g nd of 1 cm3 of metl Y is 4 g. lculte 3 the mss of solid cue of lloy of edge 4 cm the rtio, in the form n :1,ymss, of metl X to metl Y in the lloy c the volume, to the nerest cm 3,ofcuic lock of lloy whose mss is 1.5 kg d the length, in mm, of the edge of this cuic lock. 8 is rectngle in which = 40 cm nd = 60 cm. M is the midpoint of, nd P is perpendiculr to M. M Prove tht the tringles M nd P re similr. P lculte the rtio of the res of the tringles M 40 cm nd P. c lculte the length of P. 60 cm 9 sculptor is commissioned to crete ronze sttue m high. He egins y mking cly model 30 cm high. xpress, in simplest form, the rtio of the height of the completed ronze sttue to the height of the cly model. If the totl surfce re of the model is 360 cm, find the totl surfce re of the sttue. c If the totl volume of the model is 1000 cm 3, find the volume of the sttue. 10 The rdius of sphericl sop ule increses y 1%. Find, correct to the nerest whole numer, the percentge increse in its surfce re its volume. SMPL. 7. y 6.4 mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
hpter 9 Rtios nd similrity 51 11 is the digonl of rhomus. The line XYZ is prllel to, X = 3cmnd = 9 cm. Find Y X XY Y Y c d YZ re tringle XY re tringle YZ e f re tringle re tringle 1 nd re prllel sides of trpezium nd = 3. The digonls nd intersect t O. Prove tht O = 1 4. 13 Tringles nd PQR re similr. The medins X nd PY re drwn. (X is the midpoint of nd Y is the midpoint of QR.) Prove tht tringles X nd PQY re similr X PY = QR xtended-response questions 1 In this digrm which other tringle is similr to? xplin why h p = y x + y. p c Use nother pir of similr tringles to write h down n expression for h in terms of x nd y. q ( x y 1 d xplin why h p + 1 ) = 1. q e lculte h when p = 4 nd q = 5. is regulr pentgon whose sides re ech 1 unit long. ch digonl is of length d units. In regulr pentgon, ech digonl is prllel to 1 1 one of the sides of the pentgon. d Wht kind of shpe is F nd wht is the length of F? xplin why the length of F is d 1. 1 c Which tringle is similr to F? d Use the pir of similr tringles to write n eqution 1 for d nd show tht the eqution cn e rewritten s d d 1 = 0. e Find d. 3 Plce conditions upon x such tht is prllel to given tht = x 3, = 3x 19, = 4 nd x 3 4 = x 4. SMPL 3x 19 F 1 x 4 F q Z Review mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
5 ssentil dvnced Generl Mthemtics Review 4 If R, S nd T re perpendiculr to, nme the pirs of similr tringles. Which is correct: z y = p q or z y = p p + q? c Which is correct: z x = q p or z x = q p + q? d Show tht 1 x + 1 y = 1 z 5 In the digrm, PQ is prllel to nd PR is prllel to. 3 cm Q = cm, Q = 6cm, P = 3cmndPQ = 4 cm. cm P Q lculte 4 cm i P ii R 6 cm re PQ re PR iii iv re re If the re of tringle PQ is cm,express in terms of : R i re ii re PQ 6 onstruct tringle such tht = 10 cm, = 9cmnd = 6 cm. Find point on nd point on, such tht is prllel to nd the re of is one-ninth the re of. 7 tringulr lot hs oundries of lengths = 130 m, = 40 10 m nd = 150 m. The length of is 10 m. fence is to e erected which runs t right ngles from. If the lot is to e divided into two equl res, find x. 8 The Greek historin Herodotus wrote tht the proportions of the gret pyrmid t Giz in gypt were chosen so tht the re of squre, for which the side lengths re equl to the height of the gret pyrmid, is equl to the re of one of the tringulr fces. Let h methe height of the pyrmid, k m the ltitude of one of the fce tringles, nd methe length of side of the squre se. Show tht Herodotus definition gives k : =. SMPL R x p z x m S fence m V h m q T y k m mridge University Press Uncorrected Smple Pges 978-0-51-615-4 008 vns, Lipson, Jones, very, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird