CAMBRIDGE IGCSE MATHEMATICS Additional Practice

Transcription

1 MRIDGE IGSE MTHEMTIS itionl Prtie Shpe, Spe n Mesure 1 Geometri terms 1. Fin the size of the ngle mrke with letter in eh of these tringles. e h e h 2. In the tringle on the right, ll the interior ngles re the sme. Wht is the size of eh ngle? Wht is the nme of speil tringle like this? Wht is speil out the sies of this tringle? 3. In the tringle on the right, two of the ngles re the sme. Work out the size of the lettere ngles. Wht is the nme of speil tringle like this? 40 Wht is speil out the sies n of this tringle? 4. In the tringle on the right, the ngles t n re the sme. Work out the size of the lettere ngles Fin the size of the eterior ngle mrke with letter in eh of these igrms

2 6. Fin the size of the ngle mrke with letter in eh of these qurilterls e f f e g h 72 g h Work out the size of the ngle mrke with letter in eh of the polgons elow e f e f g h 143 h 150 g

3 8. For eh of the trpeziums, lulte the sizes of the lettere ngles. 125 f e 9. For eh of these prllelogrms, lulte the sizes of the lettere ngles. g h i 10. For eh of these kites, lulte the sizes of the lettere ngles e f 11. For eh of these shpes, lulte the sizes of the lettere ngles e g f 3

4 12. Sketh the nets of these shpes. uoi 3 m 1 m 1 m 4 m 4 m 4 m 4 m 4 m E Squre-se prmi 5 m D 5 m 5 m Squre-se prmi, with point E iretl ove point 3 m 6 m 4 m Right-ngle tringulr prism 13. Drw, on squre pper, n urte net for eh of these uois. 2 m 3 m 3 m 4 m 4 m 5 m 5 m 4 m 14. Drw n urte net for eh of these prmis. For eh prmi, the se is retngle n the sloping eges re ll the sme length. 4 m 6 m 5 m 4 m 4 m 3 m 5 m 15. The shpe on the right is tringulr prism. Its ens re isoseles tringles n its other fes re retngles. Drw n urte net for this prism. 4 m 5 m 16. Write own the re of eh tringle. 3 m 26 m 4 m 5 m 10 m 24 m 3 m 13 m 5 m 12 m 4

5 17. Fin the re of the she tringle RST. S 8 m R 11 m Q 10 m T 18. lulte the re of eh of these tringles. 7 m 6 m 3 m 8 m 7 m 4 m e 12 m f 10 m 11 m 15 m 20 m 14 m 19. Fin the re of eh of these shpes. 6 m 5 m 6 m 4 m 4 m 12 m 10 m 13 m 4 m 20. lulte the re of eh prllelogrm elow. 10 m 8 m 12 m 10 m 7 m 5 m 21. lulte the perimeter n the re of eh trpezium. 4 m 6.5 m 12 m 7.6 m 7 m 8 m 5 m 6 m 7 m 8.4 m 6 m 9 m 10 m 22. Whih of the following shpes hs the lrgest re? 12 m 6 m 7 m 4 m 3 m 9 m 10 m 5.5 m 5

6 23. Whih of the following shpes hs the smllest re? 12 m 7 m 8 m 7 m 3 m 11.5 m 2.5 m 2 ngle properties 1. lulte the size of the ngle mrke in eh of these emples e f g h lulte the vlue of in eh of these emples lulte the vlue of in eh of these emples lulte the vlue of first n then lulte the vlue of in eh of these emples

7 5. Eh igrm shows one verte of regulr polgon. For eh polgon, nswer the following. i ii Wht is its eterior ngle? How mn sies oes it hve? iii Wht is the sum of its interior ngles? Stte the sizes of the lettere ngles in eh igrm i g h 70 j k n m 80 l 7. Stte the sizes of the lettere ngles in eh igrm. l j k p s r m 75 n o q 125 7

8 3 onstrutions In this eerise, it is importnt to leve in ll our onstrution lines. 1. Drw line 7 m long. iset it using pir of ompsses n ruler onl. hek our ur mesuring to see if eh hlf is 3.5 m. 2. Drw n tringle whose sies re etween 5 m n 10 m. On eh sie onstrut the perpeniulr isetor s on the igrm. ll our perpeniulr isetors shoul interset t the sme point. Use this point s the entre of irle tht touhes eh verte of the tringle. Drw this irle. This irle is known s the irumsrie irle of the tringle. 3. Drw n ngle of 50. onstrut the ngle isetor. Use protrtor to hek how urte ou hve een. Eh ngle shoul e Drw n tringle whose sies re etween 5 m n 10 m. t eh ngle onstrut the ngle isetor s on the igrm. ll three isetors shoul interset t the sme point. Use this point s the entre of irle tht touhes eh sie of the tringle one. Drw this irle. This irle is known s the insrie irle of the tringle. 5. onstrut these tringles urtel without using protrtor m 7 m 6. Without using protrtor, onstrut squre of sie 6 m. See how urte ou hve een onstruting n ngle isetor on n of the right ngles n seeing whether this lso uts through the opposite right ngle. 7. onstrut n ngle of 90. iset this ngle to onstrut n ngle of onstrut these ngles. i 30 ii 15 iii 22.5 iv 75 hek our ur mesuring with protrtor. (The llowle error is ±1.) 9. onstrut trpezium whose prllel sies re 8 m n 6 m, n hving n ngle of 60 t eh en of the longer sie. 10. onstrut the tringle, where = 7 m, = 60 n = 45. Mesure the lengths of n. 8

9 11. onstrut the tringle PQR, where PQ = 8 m, RPQ = 30 n PQR = 45. Mesure the lengths of PR n RQ. 12. Drw stright line n mrk point ove the line. onstrut the perpeniulr whih psses through tht point to the line. 4 Mps n sle rwings 1. ship sils for 75 km on ering of 078. How fr est hs it trvelle? How fr north hs it trvelle? 2. plne sets off from n irport n flies ue est for 120 km, then turns to fl ue south for 70 km efore lning t Seeth. Wht is the ering of Seeth from the irport? 3. heliopter leves n rm se n flies 60 km on ering of 278. How fr west hs the heliopter flown? How fr north hs the heliopter flown? 4. ship sils from port on ering of 117 for 35 km efore heing ue north for 40 km n oking t ngle. How fr south h the ship sile efore turning? How fr north h the ship sile from the port to ngle? How fr est is ngle from the port? Wht is the ering from the port to ngle? 5. Mountin is ue west of wlker. Mountin is ue north of the wlker. The guieook ss tht mountin is 4.3 km from mountin, on ering of 058. Using sle of 2 m for 1 km, rw igrm to show this informtion. Use our igrm to fin how fr the wlker is from mountin. 9

10 5 ongruene n similrit 1. Stte whether eh pir of tringles in to h is ongruent. If pir is ongruent, give the onition whih shows tht the tringles re ongruent m 5 m 3 m 5 m 4 m m 5 m 7 m 7 m m 4 m 5 m 5 m e f 7 m 65 5 m 7 m m 8 m 35 5 m 7 m 7 m g 4 m 6.5 m 6 m 6.5 m 4 m 6 m h 6.5 m m m 2. Tringle is ongruent to tringle PQR, = 60, = 80 n = 5 m. Fin these. i P ii Q iii R iv PQ 3. D is ongruent to PQRS, = 110, = 55, = 85 n RS = 4 m. Fin these. i P ii Q iii R iv S v D 4. re these pirs of shpes similr? If so, give the sle ftor. If not, give reson. 5 m 20 m 3 m 12 m 5 m 12 m 22 m 15 m 5. Eplin wh these shpes re similr. P Give the rtio of the sies. Whih ngle orrespons to ngle? Whih sie orrespons to sie QP? 5 m 70 6 m m 18 m Q R 6. It is known tht ftor himne is 330 feet high. Ptrik pes out istnes s shown in the igrm, so tht the top of the himne n the top of the flg pole re in line with eh other. How high is the flg pole? 400 pes 200 pes 10

11 7. The length rtio etween two similr solis is 2 : 5. Wht is the re rtio etween the solis? Wht is the volume rtio etween the solis? 8. The length rtio etween two similr solis is 4 : 7. Wht is the re rtio etween the solis? Wht is the volume rtio etween the solis? 9. shpe hs n re of 15 m 2. Wht is the re of similr shpe whose lengths re three times the orresponing lengths of the first shpe? 10. rik hs volume of 300 m 3. Wht woul e the volume of similr rik whose lengths re twie the orresponing lengths of the first rik? three times the orresponing lengths of the first rik? 11. n of pint, 6 m high, hols hlf litre of pint. How muh pint woul go into similr n whih is 12 m high? 12. moel sttue is 10 m high n hs volume of 100 m 3. The rel sttue is 2.4 m high. Wht is the volume of the rel sttue? Give our nswer in m 3. 6 Smmetr 1. op these shpes n rw on the lines of smmetr for eh one. If it will help ou, use tring pper or mirror to hek our results. Isoseles tringle Equilterl tringle Squre Prllelogrm e f g Rhomus Kite Trpezium 2. Fin the numer of lines of smmetr for eh of these regulr polgons. i ii iii Regulr pentgon Regulr hegon Regulr otgon How mn lines of smmetr o ou think regulr egon hs? 11

12 3. op these ptterns n rw in ll the lines of smmetr for eh one. e f 4. Write own the numer of lines of smmetr for eh of these flgs. ustri n Ieln Switzerln Greee 5. These ro signs ll hve lines of smmetr. op them n rw on the lines of smmetr for eh one. Drw skethes of other ommon signs tht lso hve lines of smmetr. Stte the numer of lines of smmetr in eh se. 6. op these shpes n write elow eh one the orer of rottionl smmetr. If it will help ou, use tring pper. Squre Retngle Prllelogrm e Equilterl tringle Regulr hegon 7. Fin the orer of rottionl smmetr for eh of these shpes. e 12

13 8. op these shpes on tring pper n fin the orer of rottionl smmetr for eh one. e f 9. Fin the numer of plnes of smmetr in eh of these 3-D shpes. ue Tringulr prism Squre-se prmi 10. This 3-D shpe hs five plnes of smmetr. Drw igrms to show where the re. 11. How mn plnes of smmetr oes eh of the following hve? rik shovel hir spoon e milk ottle f kettle 7 ngle properties of irles 1. Fin the sizes of the lettere ngles in eh of these irles g h 105 p O 136 q 13

14 2. Fin the vlues of n in eh of these irles. Where shown, O mrks the entre of the irle O Fin the vlues of n in this irle O 4. Fin the vlues of n in this irle with entre O In eh igrm, TP n TQ re tngents to irle with entre O. Fin eh vlue of. P T O 52 O 5 O 3 P P T Q T 6. Eh igrm shows tngents to irle with entre O. Fin eh vlue of. 8 m O 10 m O 4 m 10 m 7. In eh of the igrms, TP n TQ re tngents to the irle with entre O. Fin eh vlue of. T T 28 P Q P 71 Q O O 14

15 8 Lous of points 1. is fie point. Sketh the lous of the point P in eh of these situtions. P = 2 m P = 4 m P = 5 m 2. n re two fie points 5 m prt. Sketh the lous of the point P for eh of these situtions. P = P P = 4 m n P = 4 m P is lws within 2 m of the line 3. D is squre of sie 4 m. In eh of the following loi, the point P moves onl insie the squre. Sketh the lous in eh se. P = P P < P P = P P < 4 m e P > 2 m f P > 5 m 4 m D For questions 4 to 7, ou shoul strt skething the piture given in eh question on 6 6 gri, eh squre of whih is 1 m 1 m. The sle for eh question is given. 4. got is tethere rope, 7 m long, in orner of fiel with fene t eh sie. Wht is the lous of the re tht the got n grze? Use sle of 1 m = 2 m. Fene Stke Fene 5. In fiel horse is tethere to stke rope 6 m long. Wht is the lous of the re tht the horse n grze? Use sle of 1 m = 2 m. Stke 6. ow is tethere to ril t the top of fene 6 m long. The rope is 3 m long. Sketh the re tht the ow n grze. Use sle of 1 m = 2 m. Fene 7. horse is tethere to stke ner orner of fene fiel, t point 4 m from eh fene. The rope is 6 m long. Sketh the re tht the horse n grze. Use sle of 1 m = 2 m. Fene Stke Fene 15

16 9 Lengths, res n volumes 1. op n omplete the following tle for eh irle. Give our nswers to 3 signifint figures. Rius Dimeter irumferene re 4 m 2.6 m 12 m 3.2 m 2. The roller use on riket pith hs rius of 70 m. Wht is the irumferene of the roller? riket pith hs length of 20 m. How mn omplete revolutions oes the roller mke when rolling the pith? 3. Wht is the totl perimeter of semiirle of imeter 15 m? 4. lulte the perimeter n the re of this trpezium. 8 m 9.1 m 9 m 9.4 m 12 m 5. Wht perentge of this shpe hs een she? 12 m 6 m 9 m 8 m 15 m 6. The shpe of most of Egpt (see mp) roughl pproimtes to trpezium. The north ost is out 900 km long, the south ounr is out 1100 km long, n the istne from north to south is out 1100 km. iro Wht is the pproimte re of this prt of Egpt? 7. lulte the volume of eh of these prisms. 7 m 4 m 3 m 3 m 4 m 15 m 11 m 4 m 6 m 4 m 6 m 5 m 9 m 6 m 3 m 3 m 12 m 16

17 8. len-to is prism. lulte the volume of ir insie the len-to with the imensions shown in the igrm. Give our nswer in litres. 3 m 1.5 m 1.7 m 2 m 9. The top n ottom of the ontiner shown here re the sme size, oth onsisting of retngle, 4 m 9 m, with semiirle t eh en. The epth is 3 m. Fin the volume of the ontiner. 10. Fin i the volume n ii the totl surfe re of eh of these liners. Give our nswers to 3 signifint figures. 3 m 1 m 6 m 8 m 19 m 3.5 m 9 m 15 m 11. The imeter of mrle, linril olumn is 60 m n its height is 4.2 m. The ost of mking this olumn is quote s $67.50 per ui metre. Wht is the estimte totl ost of mking the olumn? 12. linril ontiner is 65 m in imeter. Wter is poure into the ontiner until it is 1 metre eep. How muh wter is in the ontiner? Give our nswer in litres. 13. lulte the volume of eh of these prmis, ll with retngulr ses. 6 m 7 m 12 m 4 m 8 m 13 m 7 m 9 m 20 m e 9 m 6 m 4 m 8 m 7 m 10 m 14. lulte the volume of prmi hving squre se of sie 9 m n vertil height of 10 m. 15. rstl is in the form of two squre-se prmis joine t their ses (see igrm). The rstl hs mss of 31.5 grms. Wht is its ensit? 7 m 17 3 m

18 16. For eh one, lulte i its volume n ii its totl surfe re. Give our nswers to 3 signifint figures m 18 m 34 m 5 m 12 m 13 m 12 m 15 m 20 m 17. The moel shown on the right is me from luminium. Wht is the mss of the moel, given tht the ensit of luminium is 2.7 g/m 3? 4 m 6 m 3 m 18. lulte the volume n the surfe re of sphere with imeter of 50 m. 19. sphere fits etl into n open uil o of sie 25 m. lulte the following. the surfe re of the sphere the volume of the sphere 20. lulte, orret to one eiml ple, the rius of sphere whose surfe re is 150 m 2 whose volume is 150 m The length rtio etween two similr solis is 4 : 7. Wht is the re rtio etween the solis? Wht is the volume rtio etween the solis? 22. shpe hs n re of 15 m 2. Wht is the re of similr shpe whose lengths re three times the orresponing lengths of the first shpe? 23. Thirt ui entimetres of l were use to mke moel sheep. Wht volume of l woul e neee to mke similr moel sheep with these lengths? five times the orresponing lengths of the first moel one hlf of the orresponing lengths of the first moel 24. moel sttue is 10 m high n hs volume of 100 m 3. The rel sttue is 2.4 m high. Wht is the volume of the rel sttue? Give our nswer in m Eh of these epressions represents length, n re or volume. Inite whih it is writing L, or V. Eh letter represents length. 2 z 4 e ( + ) f g

19 10 r, setor n segment 1. For eh of these setors, lulte i the r length ii the setor re m m 12 m m 2. lulte the totl perimeter of eh of these setors. 11 m m 3. lulte the re of eh of these setors m m 4. lulte the segment re in this igrm 50 8 m 0 5. lulte the re of the she shpe giving our nswer in terms of π. 12 m 12 m 6. D is squre of sie length 8 m. P n Q re rs of the irles with entres D n. lulte the re of the she prt. P Q D 19

20 11 Pthgors theorem in two imensions 1. For eh of the following tringles, lulte the length of the hpotenuse,, giving our nswers to one eiml ple. 4.8 m 4 m 9 m 7 m 3 m 8 m 5 m 6 m 2. For eh of the following tringles, lulte the length, giving our nswers to one eiml ple. 6.4 m 17 m 24 m 9 m 25 m 31 m 19 m 8 m 3. For eh of the following tringles, lulte the length, giving our nswers to one eiml ple. 17 m 17 m 19 m 11 m 23 m 9 m 12 m 8.5 m 4. lulte the res of these isoseles tringles. 9 m 9 m 3 m 7 m 10 m 8 m 2 m 5. ler, 12 metres long, lens ginst wll. The ler rehes 10 metres up the wll. How fr w from the foot of the wll is the foot of the ler? 12 m 10 m 6. How long is the igonl of squre with sie of 8 metres? 7. ship going from port to lighthouse stems 15 km est n 12 km north. How fr is the lighthouse from the port? 20

21 8. mst on silot is strengthene wire (lle st), s shown on the igrm. The mst is 10 m tll n the st is 11 m long. How fr from the se of the mst oes the st reh? 10 m 11 m 9. How long is the line tht joins the two oorintes (13, 6) n (1, 1)? Is the tringle with sies 7 m, 24 m n 25 m, right-ngle tringle? 24 m 7 m 25 m 12 Sine, osine n tngent in right-ngle tringles 1. Fin these vlues, rouning off our nswers to three signifint figures. sin 43 sin 56 sin 67.2 sin 90 e sin 45 f sin 20 g sin 22 h sin 0 2. Fin these vlues, rouning off our nswers to three signifint figures. os 43 os 56 os 67.2 os 90 e os 45 f os 20 g os 22 h os 0 3. Use our lultor to work out the vlues of tn 43 tn 56 tn 67.2 tn 90 e tn 45 f tn 20 g tn 22 h tn 0 4. Fin the ngle mrke in eh of these tringles. 8 m 4 m 1 m 5 m 100 m 160 m 21

22 5. Fin the sie mrke in eh of these tringles. 9 m 42 m m 6. Fin the vlue of in eh of these tringles. 6.5 m m 16 m 13 m 17 m m 7. Fin the ngle mrke in eh of these tringles. 6 m 20 m 9 m 15 m 35 m 45 m 8. Fin the sie mrke in eh of these tringles m m 300 m Fin the vlue in eh of these tringles m 8 m 9 m 9 m 4 m 3.5 m 10. Fin the length mrke in eh of these tringles e f

23 11. Fin the ngle mrke in eh of these tringles e f Fin the ngle or length mrke in eh of these tringles e f g h i j In these questions, give nswers involving ngles to the nerest egree. 13. ler, 6 m long, rests ginst wll. The foot of the ler is 2.5 m from the se of the wll. Wht ngle oes the ler mke with the groun? 14. lulte the ngle tht the igonl mkes with the long sie of retngle whih mesures 10 m 6 m. 23

24 15. hme sees n irrft in the sk. The irrft is t horizontl istne of 25 km from hme. The ngle of elevtion is 22. How high is the irrft? 16. pssenger in n irrft hers the pilot s tht the re fling t n ltitue of 4000 m n re 10 km from the irport. If the pssenger n see the irport, wht is the ngle of epression? 17. mn stning 200 m from the se of television trnsmitter looks t the top of it n noties tht the ngle of elevtion of the top is 65. How high is the tower? 18. From the top of vertil liff, 200 m high, ot hs n ngle of epression of 52. How fr from the se of the liff is the ot? 19. ship sils for 75 km on ering of 078. How fr est hs it trvelle? How fr north hs it trvelle? 20. Lophm is 17 km from Wth on ering of 210. How fr south of Wth is Lophm? How fr est of Lophm is Wth? 21. plne sets off from n irport n flies ue est for 120 km, then turns to fl ue south for 70 km efore lning t Seeth. Wht is the ering of Seeth from the irport? 22. heliopter leves n rm se n flies 60 km on ering of 278. How fr west hs the heliopter flown? How fr north hs the heliopter flown? 13 Sine n osine rules 1. Fin the length in eh of these tringles m 12 m 32 m Fin the ngle in eh of these tringles m 7 m 14 m 58 6 m 13 m 15 m In tringle, the ngle t is 38, the sie is 10 m n the sie is 8 m. Fin the two possile vlues of the ngle t. 24

25 4. To fin the height of tower stning on smll hill, Mr me some mesurements (see igrm). From point, the ngle of elevtion of is 20, the ngle of elevtion of is 50, n the istne is 25 m. h lulte these ngles. i 25 m ii Using the sine rule n tringle, lulte the height h of the tower n ol uiling is unsfe, so it is protete fene. To work out the height of the uiling, nnie me the mesurements shown on the igrm. Use the sine rule to work out the istne. h lulte the height of the uiling, D D 20 m 6. Fin the length in eh of these tringles. 15 m 6 m 65 8 m 22 m m m 7. Fin the ngle in eh of these tringles. 12 m 10 m 7 m 15 m 24 m 9 m 50 m 30 m 40 m Eplin the signifine of the nswer to prt. 8. tringle hs two sies of length 40 m n n ngle of 110. Work out the length of the thir sie of the tringle. 9. The igrm shows trpezium D. = 6.7 m, D = 7.2 m, = 9.3 m n ngle D = 100. lulte these. D 7.2 m m length D ngle D 9.3 m ngle D length D e re of the trpezium 25

26 10. qurilterl D hs D = 6 m, D = 9 m, = 10 m n = 12 m. ngle D = 120. lulte ngle. 11. Fin the re of eh of the following tringles. Tringle where = 7 m, = 8 m n ngle = 59 Tringle where ngle = 86, = 6.7 m n = 8 m Tringle PQR where QR = 27 m, PR = 19 m n ngle QRP = 109 Tringle XYZ where XY = 231 m, XZ = 191 m n ngle YXZ = 73 e Tringle LMN where LN = 63 m, LM = 39 m n ngle NLM = The re of tringle is 27 m 2. If = 14 m n ngle = 115, fin. 13. In qurilterl D, D = 4 m, D = 11 m, ngle D = 32, ngle D = 48 n ngle D = 61. lulte the re of the qurilterl. 14. or is in the shpe of tringle with sies 60 m, 70 m n 80 m. Fin the re of the or. 14 Three-imensionl trigonometr 1. o mesures 8 m 12 m 5 m. H G lulte the lengths of the following. i ii G iii E lulte the igonl istne H. E D 8 m F 5 m 12 m 2. grge is 5 m long, 3 m wie n 3 m high. n 7 m long pole e store in it? 3. Spike, spier, is t the orner S of the wege shown in the igrm. Fre, fl, is t the orner F of the sme wege. lulte the two istnes Spike woul hve to trvel to get to Fre if she use the eges of the wege. S R 12 m E P F 4 m Q 8 m lulte the istne Spike woul hve to trvel ross the fe of the wege to get iretl to Fre. F 4. Fre is now t the top of ke-ens n n Spike is iretl elow him on the se of the n. To th Fre surprise, Spike tkes igonl route roun the n. How fr oes Spike trvel? S 10 m 5. The igrm shows squre-se prmi with se length 8 m n sloping eges 9 m. M is the mi-point of the sie, X is the mi-point of the se, n E is iretl ove X. lulte the length of the igonl. lulte EX, the height of the prmi. Using tringle E, lulte the length EM. 6 m D X E 9 m 8 m M 26

27 6. The igrm shows uoi with sies of 40 m, 30 m, n 22.5 m. M is the mi-point of the sie FG. lulte (or write own) these lengths, giving our nswers to three signifint figures if neessr. E H F M G 40 m H G M HM D 7. The igrm shows prmi. The se is horizontl retngle D, 20 m 15 m. The length of eh sloping ege is 24 m. The pe, V, is over the entre of the retngulr se. lulte these. the size of the ngle V the height of the prmi the volume of the prmi 24 m D 20 m V 22.5 m 15 m 30 m the size of the ngle etween the fe VD n the se D 8. The igrm shows the roof of uiling. The se D is horizontl retngle 7 m 4 m. The tringulr ens re equilterl tringles. Eh sie of the roof is n isoseles trpezium. The length of the top of the roof, EF, is 5 m. lulte these. the length EM, where M is the mipoint of the size of ngle E the size of the ngle etween the fe E n the se D E M 4 m 5 m 4 m D F 7 m the surfe re of the roof (eluing the se) 9. D is vertil retngulr plne. ED is horizontl tringulr plne. ngle DE = 90, = 10 m, = 4 m n ED = 9 m. lulte these. ngle ED ngle DE D 9 m 10 m 4 m E ngle E 15 Vetors n vetor properties 1. On this gri, O is n O is. E E H J K Nme three other vetors equivlent to. Nme three other vetors equivlent to. D G I Nme three vetors equivlent to. Nme three vetors equivlent to. O F 2. Using the sme gri s in question 1, give the following vetors in terms of n. O OE OD OG e OJ f OH g G h K i K j DI k GJ l DK 27

28 3. On op of this gri, mrk on the points to P to show the following. O = OD = 2 + OE = + 2 OF = 3 e OG = 4 f OH = g OI = O h OJ = + i OK = j OM = k ON = l OP = On this gri, O is n O is. D E Give the following vetors in terms of n. OH OK OJ OI e O f O g K h DI H I J i JE j k K l DK O F K G 5. On op of this gri, mrk on the points to P to show the following. O = 2 OD = 2 + O OE = 2 OF = 2 e OG = f OH = 2 g OI = 2 2 h OJ = + i OK = j OM = 3 2 k ON = l OP = The points P, Q n R lie on stright line. The vetor PQ is 2 +, where n re vetors. Whih of the following vetors oul e the vetor PR n whih oul not e the vetor PR (two of eh) The points P, Q n R lie on stright line. The vetor PQ is 3, where n re vetors. Write own n other vetor tht oul represent PR. How n ou tell from the vetor PS tht S lies on the sme stright line s P, Q n R? 28

29 8. The igrm shows the vetors O = n O =. M is the mipoint of. i Work out the vetor. ii Work out the vetor M. iii Eplin wh OM = O + M. M iv Using our nswers to prts ii n iii, work out OM in terms of n. i Work out the vetor. ii Work out the vetor M. iii Eplin wh OM = O + M. iv Using our nswers to prts ii n iii, work out OM in terms of n. op the igrm n show on it the vetor O whih is equl to +. O Desrie in geometril terms the position of M in reltion to O,, n. 9. The igrm shows the vetors O = n O =. N is the mipoint of. i Work out the vetor. O ii Work out the vetor N. iii Eplin wh ON = O + N. iv Using our nswers to prts ii n iii, work out ON in terms of n. i Work out the vetor. ii Work out the vetor N. iii Eplin wh ON = O + N. iv Using our nswers to prts ii n iii, work out ON in terms of n. op the igrm ove n show on it the vetor OD whih is equl to. N Desrie in geometril terms the position of N in reltion to O,, n D. 10. The igrm shows the vetors O = n O =. The point ivies the line in the rtio 1:2 (i.e. is 1 3 the istne from to ). i Work out the vetor. ii Work out the vetor. iii Work out the vetor O in terms of n. O If now ivies the line in the rtio 1:3 (i.e. is 1 4 the istne from to ), write own the vetor tht represents O. 29

30 11.DEF is regulr hegon. is represente the vetor, n the vetor. mens of igrm, or otherwise, eplin wh D =. Epress these vetors in terms of n. i DE ii EF iii F F Work out the nswer to + + D + DE + EF + F E D Eplin our nswer. Epress these vetors in terms of n. i D ii E iii F iv E v DF 12. DEFGH is regulr otgon. is represente the vetor, n the vetor. mens of igrm, or otherwise, eplin wh D = 2. H mens of igrm, or otherwise, eplin wh DE = 2. Epress the following vetors in terms of n. i EF ii FG iii GH iv H v H vi D vii E viii F G F E D 16 Trnsformtions 1. op the igrm elow n rw the refletion of the given tringle in the following lines. = 2 = 1 = 3 = 2 e = 1 f -is

31 2. Drw eh of these tringles on squre pper, leving plent of spe on the opposite sie of the given mirror line. Then rw the refletion of eh tringle. 3. Drw pir of es n the lines = n =, s shown. Drw the tringle with oorintes (2, 1), (5, 1), (5, 3). e Drw the refletion of tringle in the -is n lel the imge P. Drw the refletion of tringle P in the line = n lel the imge Q. Drw the refletion of tringle Q in the -is n lel the imge R = 4 5 = f Drw the refletion of tringle R in the line = n lel the imge S. g Drw the refletion of tringle S in the -is n lel the imge T. h Drw the refletion of tringle T in the line = n lel the imge U. i Drw the refletion of tringle U in the -is n lel the imge W. j Wht single refletion will move tringle W to tringle? 4. op the igrm n reflet the tringle in these lines. = = 1 = = On squre pper, rw eh of these shpes n its entre of rottion, leving plent of spe ll roun the shpe. Rotte eh shpe out its entre of rottion i first 90 lokwise (ll the imge ) ii then 90 ntilokwise (ll the imge ). Desrie, in eh se, the rottion tht woul tke i k to its originl position ii to. 31

32 6. op the igrm n rotte the given tringle the following. 90 lokwise out (0, 0) 180 out (3, 3) 90 ntilokwise out (0, 2) 180 out ( 1, 0) e 90 lokwise out ( 1, 1) Wht other rottions re equivlent to these rottions? 270 lokwise 90 lokwise 60 ntilokwise 100 ntilokwise 8. Show tht refletion in the -is followe refletion in the -is is equivlent to rottion of 180 out the origin. 9. Show tht refletion in the line = followe refletion in the line = is equivlent to rottion of 180 out the origin. 10. Drw tringle on squre pper. Mrk four ifferent entres of enlrgement on our igrm s follows. one ove our tringle one to the left of our tringle one elow our tringle one to the right of our tringle Enlrge the tringle sle ftor of 2 from eh entre. Wht o ou notie out eh enlrge shpe? 11. Enlrge eh of these shpes sle ftor of 1 2 out the given entre of enlrgement. 12. op this igrm onto squre pper. Enlrge the retngle sle ftor 1 3 out the origin. Lel the imge. Write own the rtio of the lengths of the sies of retngle to the lengths of the sies of retngle. Work out the rtio of the perimeter of retngle to the perimeter of retngle. Work out the rtio of the re of retngle to the re of retngle

33 13. op the igrm onto squre pper. Enlrge sle ftor of 3 out entre (4, 5). Enlrge sle ftor 1 2 out entre ( 1, 3). Enlrge sle ftor 1 2 out entre ( 3, 1). e f Wht is the entre of enlrgement n sle ftor whih mps onto? Wht is the entre of enlrgement n sle ftor whih mps onto? Wht is the entre of enlrgement n sle ftor whih mps the nswer to prt to the nswer to prt? g h Wht is the entre of enlrgement n sle ftor whih mps the nswer to prt to the nswer to prt? Wht is the onnetion etween the sle ftors n the entres of enlrgement in prts n e, n in prts f n g? 14. Use vetors to esrie the following trnsltions. i to ii to iii to D iv to E v to F vi to G i to ii to iii to D iv to E v to F vi to G i to ii to iii to D iv to E v to F vi to G i D to E ii E to iii F to iv G to D v F to G vi G to E 8 7 F G E 1 0 D

34 15. Drw the tringle with oorintes (1,1), (2,1) n (1,3). 2 Drw the imge of fter trnsltion with vetor (). Lel this tringle P. 3 1 Drw the imge of fter trnsltion with vetor ( ). Lel this tringle Q. 2 3 Drw the imge of fter trnsltion with vetor ( ). Lel this tringle R. 2 2 e Drw the imge of fter trnsltion with vetor ( ). Lel this tringle S The tringle is mppe onto the tringle. The line is invrint. Fin the sher ftor for the trnsformtion. Wht n ou s out the re of tringle n tringle? 17. Fin the imge of the retngle D uner sher ftor 2 with the is invrint. D D 18. Desrie full the trnsformtion whih mps D onto D. D 34

35 19. The shpe D is strethe with streth ftor 2 n the -is invrint. op the igrm n show the imge D. D 20. Desrie full the trnsformtions tht will mp the she tringle onto eh of the tringles F E F 3 2 D Desrie full the trnsformtions tht will result in the following movements. T 1 to T 2 T 1 to T 6 T 2 to T 3 T 6 to T 2 e T 6 to T 5 f T 5 to T T 1 2 T T T 5 T 3 T The point P(3, 4) is reflete in the -is, then rotte 90 lokwise out the origin. Wht re the oorintes of the imge of P? 23. point Q(5, 2) is rotte 180, then reflete in the -is. Wht re the oorintes of the imge point of Q? Wht single trnsformtion woul hve tken point Q iretl to the imge point? 24. Fin the oorintes of the imge of the point (3, 5) fter lokwise rottion of 90 out the point (1, 3). 35

36 17 Mtries Fin: If = ( ) n = ( ) 7 8 e f g 1 h 1 2. If = ( ) fin Fin the eterminnt n inverse of the following mtries: 3 0 ( 2 1 ) 2 2 ( ) ( ) If = (2 4) = ( ) = ( ) D = ( ) Whih of the following mtri lultions is not possile? i iid iii ivd lulte D. Fin the inverse of If M = ( ) fin i 2M ii M If M = ( ) Fin the 2 2 mtri, suh tht M + = ( 0 0). 1 0 Fin the 2 2 mtri, suh tht M = ( 0 1). 7. The point P with oorintes ( 3, 4) is reflete in the line =. Use mtries to fin the oorintes of P, the imge of P uner this trnsformtion. 8. The tringle with oorintes (4,2), (3,5) n ( 1,4) is given trnsformtion the mtri n ientif the trnsformtion. Fin the imges of the points, n uner this trnsformtion ( ) 36