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This ooklet is out ientifing n mnipulting stright sie shpes using their unique properties Mn lever people ontriute to the evelopment of moern geometr inluing: Thles of Miletus (ppro. 6- C) Pthgors (ppro. 6- C) Euli of lenri (ppro. -6 C) (often referre to s the "Fther of moern geometr') rhimees of Srus (ppro -0 C) pollonius of Perg (ppro. 6-0 C) fter n ttk on the it of lenri, mn of the works of these mthemtiins were lost. Look up these people sometime n re out their ontriution to this sujet. New isoveries in geometr re still eing me with the vent of omputers, in prtiulr frtl geometr. The most fmous of these eing enoit Mnelrot Frtl pttern. Q Write own how ou woul esrie this shpe over the phone to frien who h to rw it urtel. Tr it with frien/fmil memer n see if the rw this shpe from our esription. Work through the ook for gret w to o this P Lerning TPIC
ow oes it work? re just n lose shpe with stright lines whih on t ross. Like squre or tringle. ll polgons nee t lest three sies to form lose pth. Polgon? - ll sies re stright - Shpe is lose Polgon? - ll sies re stright - Shpe is NT lose Polgon? - ll sies re NT stright - Shpe is lose Polgon? - Sies ross Prts of polgon: Sie Interior ngle Eterior ngle Digonl (line tht joins two verties n is not sie) Eh orner is lle Verte (verties plurl) There re mn si tpes of polgons. ere re the ones we will e looking t in this ooklet: Conve polgon ll interior ngles re 0 Conve polgon s n interior ngle 0 Equilterl polgon ll sies re the sme length Equingulr polgon ll interior ngles re equl Cli polgon ll verties/orner points lie on the ege (irumferene) of the sme irle. Regulr polgon ll interior ngles re equl ll sies re the sme length The re li polgons ere is nother ifferene etween onve n onve polgons. Conve stright line rwn through the polgon n onl ross mimum of sies Conve stright line rwn through the polgon n ross more thn two sies. TPIC P Lerning
ow oes it work? n polgon n e nme using Greek prefies mthing the numer of stright sies it hs. = Trio = Tetr = Pent = e = ept = t = Non = De = ene = Doe = Trie = Tetre Polgon nming n lssifition hrt Sies Nme Conve Conve Equilterl Equingulr Cli Regulr Tringle (Trigon) N/ Qurilterl (Tetrgon) Pentgon 6 egon eptgon tgon ere re some more polgon nmes. Sies Polgon nme Sies Polgon nme Nongon Enneegon 0 Degon 0 Iosgon enegon 0 Triontgon Doegon 0 Tetrontgon Triegon 0 Pentontgon Tetregon 60 eontgon Pentegon 0 eptontgon 6 eegon 0 tontgon eptegon 0 Enneontgon tegon 00 etogon Nongon sies Ennegon Mn of these polgons hve more thn one nme. Look them up sometime! P Lerning TPIC
ow oes it work? Ientif whih of these shpes re polgons or not. Polgon Not polgon Polgon Not polgon Polgon Not polgon Polgon Not polgon e f g h Polgon Not polgon Polgon Not polgon Polgon Not polgon Polgon Not polgon Tik ll the properties tht eh of these polgons hve n then nme the shpe: Conve Conve Equilterl Equingulr Cli Regulr Conve Conve Equilterl Equingulr Cli Regulr Conve Conve Equilterl Equingulr Cli Regulr Conve Conve Equilterl Equingulr Cli Regulr e Conve Conve Equilterl Equingulr Cli Regulr f Conve Conve Equilterl Equingulr Cli Regulr Drw n lel: regulr tetrgon. onve nongon. TPIC P Lerning
ow oes it work? PLYGNS * PLYGNS * PLYGNS * Drw n lel:.../.../ 0... onve, equilterl hegon. onve, li tetrgon whih is not equilterl. n equingulr, pentgon whih is not equilterl. onve, equilterl heptgon with two refle ngles ( 0 ngle 60). Eplin wh it is not possile to rw li, equilterl, onve otgon. 6 ow woul ou esrie these polgons to someone rwing them in nother room? P Lerning TPIC
ow oes it work? Trnsformtions Trnsformtions re ll out re-positioning shpes without hnging n of their imensions. There re three min tpes: Refletions (Flip) Refleting n ojet out fie line lle the is of refletion. is of refletion (or is of iltion) n st ojet (efore) imge (fter) Keep equl sping from is. orizontl refletion to the right. Vertil refletion up followe horizontl refletion left. Trnsltions (Slie) This trnsformtion involves sliing n ojet either horizontll, vertill or oth. Ever prt of the ojet is move the sme istne. m ojet (efore) imge (fter) m m trnsltion horizontll to the right m Two trnsltions: m horizontll right, n then m vertill up Rottions (Turn) trnsformtion of turning n ojet out fie point ounter-lokwise. ounter-lokwise ojet (efore) Centre of rottion (or entre of iltion) 0 rottion (or turn) 0 rottion (or turn) imge (fter) 0 rottion (or turn) 0 rottion (or turn) 0 rottion (or turn) 6 TPIC P Lerning
Refletion Trnsltion 6 Rottion 0 6K 0 K Q Q Q 0 K 0 Q 0 K 6 6 Refletion Trnsltion Rottion 0 0 Refletion Trnsltion Rottion 0 0 imge ojet X Q K Q K Refletion Trnsltion Rottion Y Y imge ojet is of iltion 6 Y Y X imge ojet Z Z W W X 6 0 Q 6 K Eh of these ojets hs unergone two ifferent trnsformtions. Tik them oth. 0 6 6 X imge 6 6 ojet K 6 entre of iltion Drw the imge on the gris elow when eh of these ojets re reflete out the given is. Drw the imge on the gris elow when eh of these ojets re trnslte the given mounts. Five squres horizontll to the left. Four squres vertill up. P Lerning Eight squres to the right, then si squres own. TPIC 6 Q 0 0 6 Q Refletion 6 Trnsltion Rottion 6 6 imge 6 6K ojet 0 6 6 0 6 6 6 Q 0 0 K 6 6 0 Refletion 0 Trnsltion Q Rottion 0 imge 0 Q Q K6 0 0 0 Q Q 0 Q K 0 6 6 Q K 6 0 0 6 K K Q 6 0 0 6K K Q ojet Q K Q Q 6 eh of6 these pling Ientif whih0tpe of trnsformtion rs hs unergone: K Q 0 K 6 0 Q 0 Q K6 Trnsformtions Q K 6 0 K Q 0 K K K6 K 0 Q 0 K Q Q 0 Q K ow oes it work? K K
ow oes it work? Trnsformtions Drw the imge on the gris elow when eh of these ojets re rotte the given mounts. ne hlf turn (0 rottion). Three qurter turn (0 rottion). ne qurter turn (0 rottion). 6 Drw the imge on the gris elow when eh of these ojets unergo the trnsformtions given. Trnslte ten units to the right first then reflet own out the given is of refletion. Rotte 0 out the entre of rottion, then trnslte si units up. Reflet out the given is first, then trnlste two units to the left. Three qurter turn (0 rottion) first, then reflet out the given is of iltion. TPIC P Lerning
ow oes it work? Trnsformtions Ern ourself n wesome pssport stmp with this one. The ojet (CDE) requires thirteen trnsformtions to move long the white proution line elow. It nees to leve in the position shown t the eit for the net stge of proution. The ojet must not overlp the she prt roun the proution line pth. n of the sies, C, DE n E n e use s n is of refletion. The verte is the onl entre of rottion use t the two irle points long the pth. Desrie the thirteen trnsformtion steps use to nvigte this ojet long the pth, inluing the iretion of trnsformtion n the sies/points use s es of iltion where pproprite. ENTRY EXIT (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (i) () (i) (ii) (iii) P Lerning TPIC
T ow oes it work? Trnsformtions For the igrm shown elow, esrie four ifferent ws the finl imge of the ojet n e hieve using ifferent trnsformtions. I N S * T R M.../.../ 0... R F S N Metho Metho Metho Metho 0 TPIC P Lerning
ow oes it work? Refletion smmetr There re mn tpes of smmetr n in this ooklet we will just e fousing on three of them. If the is of refletion splits shpe into two ientil piees, then tht shpe hs refletion smmetr. The is of refletion is then lle the is of smmetr. is of refletion = is of smmetr Smmetri Shpe hs refletion smmetr smmetri Shpe oes not hve refletion smmetr The istnes from the ege of the shpe to the is of smmetr re the sme on oth sies of the line. X Y C Z = C n XY = YZ This shpe hs onl one is of smmetr. When this hppens, we s the shpe hs ilterl smmetr. Mn nimls/plnts or ojets in nture hve nerl perfet ilterl smmetr. ther shpes n hve more thn one is of smmetr (es of smmetr for plurl). 6 Regulr egon There re 6 ifferent ws this shpe n e fole in hlf with oth sies of the fol fitting over eh other etl. So we n s it hs si-fol smmetr. P Lerning TPIC
ow oes it work? Refletion smmetr (i) (ii) Ientif whih of these shpes hve refletion smmetr tiking smmetri or smmetri. Drw ll the es of smmetr for those tht o. Smmetri smmetri Smmetri smmetri Smmetri smmetri Smmetri smmetri e f g h Smmetri smmetri Smmetri smmetri Smmetri smmetri Smmetri smmetri i j k l Smmetri smmetri Smmetri smmetri Smmetri smmetri Smmetri smmetri ow mn es of refletion smmetr woul these nture items hve if perfetl smmetril? These shpes ll hve refletion smmetr. Clulte the istne etween X n Y. Z Y X Z Y X YZ = m XZ = m Distne from X to Y = Distne from X to Y = TPIC P Lerning
ow oes it work? Refletion smmetr nswer these questions out the smmetri we elow:.../.../ 0... REFLECTIN SYMMETRY REFLECTIN SYMMETRY X Y ow mn es of smmetr oes the we hve? L M G Wht pir of points re equiistnt to LM? Psst: equiistnt mens the sme istne K P Q riefl eplin elow how ou eie this ws the orret nswer. Complete these igrms to proue n imge with s mn es of refletive smmetr s inite. ilterl smmetr. Two fol smmetr. Three es of smmetr. Two es of smmetr. e Five-fol smmetr. (show the other four es) f Eight-fol smmetr. (show the other seven es) P Lerning TPIC
ow oes it work? Rottionl smmetr When n ojet is rotte 60 ( full irle), it looks the sme s it ws efore rotting. If the ojet looks the sme gin efore ompleting full irle, it hs rottionl smmetr. The numer of times the ojet repets efore ompleting the full irle tells us the orer of rottionl smmetr. 0 (three qurter turn) 0 (hlf turn) 0 (qurter turn) 0 (hlf turn) Rottionl Smmetr of orer i.e. it looks the sme times in one full rottion. Rottionl Smmetr of orer i.e. it looks the sme times in one full rottion. Point smmetr This is when n ojet hs prts the sme istne w from the entre of smmetr in the opposite iretion. stright line through the entre of smmetr will ross t lest two points on the ojet. Eh pir of points rosse on opposite sies of the entre of smmetr re n equl istne w from it. Point smmetr for one ojet Point smmetr for piture with two ojets X X Y Y For oth igrms: = n X = Y These oth hve point smmetr euse for ever point on them, there is nother point opposite the entre of smmetr () the sme istne w. jets n pitures n often hve oth rottionl n point smmetr. TPIC P Lerning
ow oes it work? Rottionl n point smmetr Ientif whih of these ojets re rottionll smmetri or smmetri. Rottionll smmetri Rottionll smmetri Rottionll smmetri Rottionll smmetri Rottionll smmetri Rottionll smmetri e f Rottionll smmetri Rottionll smmetri Rottionll smmetri Rottionll smmetri Rottionll smmetri Rottionll smmetri Write the orer of rottionl smmetr eh of these mthemtil smols hve: ll these propellers hve rottionl smmetr. Ientif whih ones lso hve point smmetr. (i) (ii) (iii) s point smmetr No point smmetr s point smmetr No point smmetr s point smmetr No point smmetr (iv) (v) (vi) s point smmetr No point smmetr s point smmetr No point smmetr s point smmetr No point smmetr Desrie the reltionship etween the numer of les n the point smmetr of these propellers. Desrie the reltionship etween the numer of les n the orer of point smmetr for the smmetri les. P Lerning TPIC
ow oes it work? Rottionl n point smmetr Complete eh of the hlf rwn shpes elow to mth the given smmetries. Rottionl smmetr of orer n lso point smmetr. Rottionl smmetr of orer n lso point smmetr. Rottionl smmetr of orer n no point smmetr. Rottionl smmetr of orer n lso point smmetr. ll the verties shown elow represent hlf of ll the verties of shpes whih hve point smmetr out the entre of rottion (). (i) Mrk in the other verties. (ii) Drw the ounr of the whole shpe. K C W S T Q R P V U 6 TPIC P Lerning
.../.../0... ow oes it work? Como time: Refletion, rottion n point smmetr 6 Ientif if these flgs of the worl hve smmetr n wht tpe. Inlue the numer of fols or orer of rottions for those flgs with the relevnt smmetr. Cn Mlsi Refletion smmetr with fols Refletion smmetr with fols Rottionl smmetr of orer. Rottionl smmetr of orer. Point of smmetr. No smmetr Point of smmetr. No smmetr Ini ustrli Refletion smmetr with fols Refletion smmetr with fols Rottionl smmetr of orer. Rottionl smmetr of orer. e Point of smmetr. No smmetr mi f Point of smmetr. No smmetr Pkistn.../.../0... * CM TIME:REFLECTIN,RTTINNDPINTSYMMETRY Refletion smmetr with fols Refletion smmetr with fols Rottionl smmetr of orer. Rottionl smmetr of orer. Point of smmetr. No smmetr Point of smmetr. No smmetr g h South fri Unite Sttes of meri Refletion smmetr with fols Refletion smmetr with fols Rottionl smmetr of orer. Point of smmetr. No smmetr Rottionl smmetr of orer. Point of smmetr. No smmetr P Lerning TPIC
ow oes it work? Como time: Refletion, rottion n point smmetr 6 Ientif if these flgs of the worl hve smmetr n wht tpe. Inlue the numer of fols or orer of rottions for those flgs with the relevnt smmetr. k l Vietnm Unite Kingom Refletion smmetr with fols Refletion smmetr with fols Rottionl smmetr of orer. Rottionl smmetr of orer. Point of smmetr. No smmetr Point of smmetr. No smmetr m n Georgi New Zeln Refletion smmetr with fols Refletion smmetr with fols Rottionl smmetr of orer. Rottionl smmetr of orer. Point of smmetr. No smmetr Point of smmetr. No smmetr o p Letter 'D' signl flg Letter 'L' signl flg Refletion smmetr with fols Refletion smmetr with fols Rottionl smmetr of orer. Rottionl smmetr of orer. Point of smmetr. No smmetr Point of smmetr. No smmetr q r Letter 'Y' signl flg Letter 'N' signl flg Refletion smmetr with fols Refletion smmetr with fols Rottionl smmetr of orer. Point of smmetr. No smmetr Rottionl smmetr of orer. Point of smmetr. No smmetr TPIC P Lerning
Where oes it work? Speil tringle properties Tringles ome in numer of ifferent tpes, eh with their own speil fetures (properties) n nmes. ere the re summrise in this tle: SPE TRINGLES Slene Isoseles PRPERTIES Three stright sies n internl ngles. ll three sies hve ifferent length. ll three internl ngles re ifferent size. Two of the intenl ngles hve the sme size. The two sies opposite the equl ngles hve equl lengths. -fol refletive smmetr. No rottionl smmetr. Equilterl ll of the internl ngles hve the sme size of 60. ll sies hve the sme length. -fol refletive smmetr. s rottionl smmetr of orer. ute ngle tringle 0 ll of the interl ngles re smller thn 0. Right ngle tringle tuse ngle tringle =0 : ne of the internl ngles is equl to 0 (i.e. one pir of sies re perpeniulr to eh other). 0 0 ne of the internl ngles is etween 0 n 0. Determine wht tpe of tringle is esrie from the informtion given. (i) ll internl ngles re less thn 0, n it hs one is of refletion smmetr. Isoseles tringles hve one is of refletion smmetr. ` It is n ute ngle isoseles tringle. (ii) ll internl ngels re equl n it hs point smmetr. ` It is n equilterl tringle. Ientifing properties n nming shpes tht mth is lle lssifing. P Lerning TPIC
Where oes it work? Speil tringle properties Clssif wht tpe of tringle is esrie from the informtion given in eh of these: ll internl ngles re less thn 0n it hs no es of refletion. ne internl ngle is equl to 0n two sies re equl in length. ne internl ngle is otuse n there is one is of refletion. s rottionl smmetr n ll internl ngles equl to 60. e No internl ngles re the sme size n one sie is perpeniulr to nother. Clssif wht tpe of tringle hs een rwn elow with onl some properties shown. SPECIL TRINGLE PRPERTIES.../.../ 0... 0 TPIC P Lerning
Where oes it work? Speil qurilterl properties Qurilterls eist in mn ifferent forms, eh with their own speil properties n nmes. ere the re summrise in this tle: SPE QUDRILTERL Four stright sies n internl ngles. PRPERTIES Slene onve or onve qurilterl ll four sies hve ifferent length. ll four internl ngles re ifferent size. No smmetr. Trpezium onve qurilterl nl one pir of prllel sies. No smmetr. Isoseles Trpezium Non-prllel sies re the sme length. Digonls ut eh other into equl rtios. Two pirs of equl internl ngles with ommon rms. is of refletive smmetr. pposite sies re prllel. pposite sies re equl in length. Digonll opposite internl ngles re equl. Prllelogrm onve Qurilterl Digonls iset eh other (ut eh other etl in hlf). No is of refletive smmetr. Rottionl smmetr of orer n point smmetr t the intersetion of the igonls. pposite sies re prllel. pposite sies re equl in length. ll internl ngles = 0. Retngle onve, equingulr qurilterl Digonls re equl in length. Digonls iset eh other (ut eh other etl in hlf). -fol refletive smmetr. Rottionl smmetr of orer n point smmetr t the intersetion of the igonls. P Lerning TPIC
Where oes it work? Speil qurilterl properties SPE Squre regulr qurilterl PRPERTIES pposite sies re prllel. pposite sies re the sme length. ll internl ngles = 0. Digonls iset eh other. Digonls iset eh internl ngle. Digonls ross t right ngles to eh other (perpeniulr). -fol refletive smmetr. Rottionl smmetr of orer n point smmetr t the intersetion of the igonls. Rhomus onve qurilterl pposite sies re prllel. ll sies re the sme length. Digonll opposite internl ngles re the sme. Digonls iset eh other. Digonls iset eh internl ngle. Digonls ross t right ngles to eh other (perpeniulr). -fol refletive smmetr. Rottionl smmetr of orer n point smmetr t the intersetion of the igonls. Two pirs of jent, equl sies. Internl ngles forme unequl sies re equl. Kite onve qurilterl Shorter igonl is isete the longer one. Longer igonl isets the ngles it psses through. Digonls re perpeniulr to eh other. -fol refletive smmetr. No Rottionl smmetr. This igrm shows how eh qurilterl reltes to the previous one whih shres one similr propert. Isoseles Trpezium Trpezium Retngle Qurilterl Prllelogrm Squre Kite Rhomus TPIC P Lerning
Where oes it work? Speil qurilterl properties Clssif wht speil qurilterl is eing esrie from the informtion given in eh of these: Two pirs of equl sies, ll internl ngles re right-ngles n hs -fol refletive smmetr. ne pir of prllel sies n one pir of opposite equl sies. Two pirs of equl internl ngles with the igonls the onl es of refletive smmetr. ne pir of prllel sies n one pir of opposite equl sies. e Digonls iset eh other n split ll the internl ngles into pirs of. f Perpeniulr igonls n no rottionl smmetr. Write own two ifferenes etween eh of these speil qurilterls: squre n retngle. retngle n prllelogrm. SPECIL QUDRILTERL PRPERTIES * prllelogrm n rhomus. rhomus n kite..../.../ 0... e rhomus n squre. f kite n n isoseles trpezium. qurilterl hs een prtill rwn elow. Drw n nme the three possile qurilterls this igrm oul hve een the strt of oring to the given informtion. is of smmetr igonl P Lerning TPIC
Where oes it work? Como time! Speil qurilterl n tringles These two ientil trpeziums n e trnsforme n omine to mke two speil qurilterls. Eplin the trnsformtion use, n then nme n rw the new qurilterl forme. These two equl isoseles tringles n e trnsforme n omine to mke two speil qurilterls. Eplin the trnsformtion use, then nme n rw the two speil qurilterls forme. Drw ll the ifferent qurilterls tht n e forme using these two ientil right-ngle slene tringles. TPIC P Lerning
D Wht else n ou o? Trnsformtions on the Crtesin numer plne ust s gris were use erlier to help trnsform shpes, the numer plne n lso e use. The oorintes of verties help us lote n move ojets urtel. Positive iretion Trnslte units in the positive iretion Negtive iretion ojet imge ^-, h ^-, h imge - - - - 0 Rotte one qurter turn 0 out the point ^, -h Positive iretion - ^-, -h - ^, -h ^, -h ojet imge - ojet Reflete out the -is - Negtive iretion Sme methos ppl s efore, this time inluing the new oorintes of importnt points. Determine the new oorintes for the points fter these trnsltions (i) The oorintes of fter CD is reflete out the line =. = New oorintes for re (-., ) - - 0 D C - C - 0 D C (ii) The oorintes of E fter the shpe CDEF is rotte 0out the origin (0,0). E F D E D C D E C C - - 0 - - 0 - F - F New oorintes for E re (-, ) P Lerning TPIC
Wht else n ou o? Trnsformtions on the Crtesin numer plne ll these imges re refletions of the ojet. Choose whether the refletion ws vertil (up/own), horizontl (right/left) or oth (igonll). ojet - - - - 0 - imge - - - ojet imge - - - - 0 - - - - Reflete Vertill orizontll Digonll Reflete Vertill orizontll Digonll imge ojet - - - - 0 - - - - ojet - - - - 0 - imge - - - Reflete Vertill orizontll Digonll Reflete Vertill orizontll Digonll e imge - - - - 0 - ojet - - - f ojet - - - - 0 - - imge - - Reflete Vertill orizontll Digonll Reflete Vertill orizontll Digonll 6 TPIC P Lerning
Wht else n ou o? Trnsformtions on the Crtesin numer plne ll these imges re rottions of the ojet. Choose whether the rottion is 0, 0or 0 out the given point of rottion lelle. ojet - - - - 0 - imge- - - - - - - 0 - imge ojet - - - 0 0 0 rottion 0 0 0 rottion imge imge ojet - - - - 0 - - - - 0 - - - - - - ojet - - 0 0 0 rottion 0 0 0 rottion e - - - - 0 - ojet imge - - - f imge - - - - 0 - - ojet- - 0 0 0 rottion 0 0 0 rottion g imge - - - - 0 - ojet - - - h ojet - - - - 0 - imge - - - 0 0 0 rottion 0 0 0 rottion P Lerning TPIC
Wht else n ou o? Trnsformtions on the Crtesin numer plne (i) Drw the imge for the requeste trnsformtions on the numer plnes elow. (ii) Write own the new oorintes for the ot mrke on eh ojet. Reflet ojet out the line =. - Rotte the ojet 0 out the ^00, h. - - - - - - 0 - - - - - 0 - - - - = ojet New oorintes for ot = ojet New oorintes for ot = (, ) (, ) Trnslte the ojet four units in the positive iretion. - Trnslte the ojet four units in the negtive iretion. - - - - - - 0 - - - - - 0 - - - - ojet New oorintes for ot = ojet New oorintes for ot = (, ) (, ) e Reflet ojet out the -is. f reflet ojet out the given is line, =. - - - - 0 - - - - 0 - - TPIC - ojet - - P Lerning ojet (, ) (, ) New oorintes for ot = New oorintes for ot = - - - =
Wht else n ou o? Trnsformtions on the Crtesin numer plne (i) Drw the imge for the requeste oule trnsformtions on the numer plnes elow. (ii) Write own the new oorintes for the ot mrke on eh imge. Trnslte ojet units in the positive -iretion n then reflet out the line =. Rotte the ojet one qurter turn out the point (-, ) then trnslte. units in the negtive -iretion. ojet ojet = - - - - 0 - - - - 0 - - - - - - - New oorintes for ot = New oorintes for ot = Rotte ojet 0 out the point (-, ) n then reflet out the -is. (, ) (, ) - Reflet the ojet out the -is, n then reflet out the line =. ojet - - - - 0 - - - - New oorintes for ot = (, ) ojet - - - - 0 - - - - = New oorintes for ot = (, ) e Reflet ojet out the -is then rotte 0 out the origin ^00, h. f Trnslte the ojet. units in the negtive -iretion n then reflet out the line = -. ojet - - - - 0 - - - - New oorintes for ot = (, ) ojet - - - - 0 - - - - = - New oorintes for ot = (, ) P Lerning TPIC
D Wht else n ou o? Trnsformtions on the Crtesin numer plne pler in snow sports gme n onl use trnsformtions to perform triks n hnge iretion to get through the ourse mrke trees. Points re eute if trees re hit. Points re wre when the orner ot mrke psses iretl over oorintes mrke with flgs on the ourse. The imensions of the pler re squre with sies two units long. Write own the steps (inluing the oorintes of point fter eh trnsformtion) pler n tke to get mimum points from strt to finish. C 6 Strt here * TRNSFRMTIN N TE CRTESIN NUMER PLNE D.../.../ 0... -6 - - - - - 0 6 - C - - - Finish here 0 TPIC P Lerning
Chet Sheet ere is wht ou nee to rememer from this topi on polgons re just n lose shpe with stright lines whih on t ross. Like squre or tringle. ll polgons nee t lest three sies to form lose pth. Interior ngle Eterior ngle Polgon? Polgon? Polgon? Polgon? Tpes of polgons: Shpes whih re/re not polgons Sie Digonl Prts of polgon Verte Conve ll interior ngles re 0 Conve s n interior ngle 0 Equilterl ll sies re the sme length Equingulr ll interior ngles re equl Regulr ll interior ngles re equl ll sies re the sme length The re li polgons Cli ll verties/orner points lie on the ege (irumferene) of the sme irle. Sies Polgon nme Trigon (tringle) 6 egon Nongon Doegon Sies Polgon nme Tetrgon eptgon 0 Degon Pentegon Sies Polgon nme Pentgon tgon enegon 0 Iosgon Trnsformtions Refletions (Flip) Trnsltions (Slie) Rottions (Turn) 0 rottion (or turn) ounter-lokwise ojet 0 rottion (or turn) ojet imge ojet imge imge 0 rottion (or turn) Refletion Smmetr Where n is of refletion splits n ojet into two ientil piees. is of refletion = is of smmetr Smmetri: Shpe hs refletion smmetr smmetri: Shpe oes not hve refletion smmetr The istnes from the ege of the shpe to the is of smmetr re the sme on oth sies of the line. X C Y Z = C n XY = YZ P Lerning TPIC
Chet Sheet Rottionl Smmetr If n ojet looks the sme uring rottion efore ompleting full irle, it hs rottionl smmetr. The numer of times the ojet repets efore ompleting the full irle tells us the orer of rottionl smmetr. 0 (qurter turn) 0 (hlf turn) 0 (three qurter turn) Rottionl Smmetr of orer s it looks the sme four times within one full rottion. Point Smmetr These ojets hve point smmetr euse for ever point on them, there is nother point opposite the entre of smmetr () the sme istne w. Point smmetr for one ojet X Y Point smmetr for two ojet X Y For oth igrms: = n X = Y Speil Tringles n Qurilterls (summr of ke sies n ngle ifferenes onl) Tringles Slene Isoseles Equilterl Right ngle tringle No equl sies or ngles pir of equl sies & ngles ll sies n ngles equl internl ngle = 0 ute ll internl ngles 0 tuse ne internl ngle etween 0n 0 Qurilterls Slene No equl sies or ngles. t lest pir of prllel sies. t lest pir of prllel sies. Non-prllel sies equl in length. Prllelogrm Retngle Squre pposite sies equl in length n prllel to eh other. Trpezium pposite sies equl in length n prllel to eh other. ll internl ngles = 0. Isoseles Trpezium ll sies equl in length n opposite sies prllel to eh other. ll internl ngles = 0. Rhomus ll sies equl in length n opposite sies prllel to eh other. Digonll opposite internl ngles equl. Kite Two pirs of jent equl sies. ngles opposite short igonl equl. For more etile summr, see pges, n of the ooklet. TPIC P Lerning
* CM TIME:REFLECTIN,RTTINNDPINTSYMMETRY I N S * T T M.../.../ 0... R R F S N SPECIL TRINGLE PRPERTIES.../.../ 0... * TRNSFRMTIN N TE CRTESIN NUMER PLNE.../.../ 0... PLYGNS * PLYGNS * PLYGNS *.../.../ 0....../.../0....../.../0...