Polygons POLYGONS.

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Polgons PLYGNS www.mthltis.o.uk

ow os it work? Solutions Polgons Pg qustions Polgons Polgon Not polgon Polgon Not polgon Polgon Not polgon Polgon Not polgon f g h Polgon Not polgon Polgon Not polgon Polgon Not polgon Polgon Not polgon Conv Conv Equiltrl Equingulr Cli Rgulr Conv Conv Equiltrl Equingulr Cli Rgulr Conv Conv Equiltrl Equingulr Cli Rgulr Conv pntgon Conv, li trigon (or tringl) Conv, li, quingulr hgon Conv Conv Equiltrl Equingulr Cli Rgulr Conv Conv Equiltrl Equingulr Cli Rgulr f Conv Conv Equiltrl Equingulr Cli Rgulr Conv quriltrl Conv, quiltrl, quingulr, rgulr ttrgon Conv, quiltrl hgon rgulr ttrgon. onv nongon. Polgons Solutions Mthltis Pssport P Lrning TPIC

ow os it work? Solutions Polgons Pg qustions Polgons onv, quiltrl hgon. onv, li ttrgon whih is not quiltrl. n quingulr, pntgon whih is not quiltrl. onv, quiltrl hptgon with two rfl ngls ( ngl 0 ). li quriltrl nnot hv n onv ngls. Rfltion Trnsltion Rottion img ojt Mthltis Pssport P Lrning Polgons Solutions img Rfltion Trnsltion Rottion ojt TPIC 0 img 0 Rfltion Trnsltion Rottion 0 ojt Pg qustions Trnsformtions pnut-shp quiltrl onv otgon with two intrnl rfl ngls. ttrgon pir of with on qulsis nt to h othr (jnt). Th ngl igonll opposit th ngl form th two qul sis is right 0 ngl (or qul to 0).

ow os it work? Solutions Polgons Pg qustions Trnsformtions is of iltion Y ojt img Z W ojt img Rfltion Trnsltion Rottion Rfltion Trnsltion Rottion Y Z W ojt Y Rfltion Trnsltion Rottion Y img ntr of iltion Fiv squrs horizontll to th lft. Four squrs vrtill up. Eight squrs to th right, thn si squrs own. squrs squrs squrs squrs Polgons Solutions Mthltis Pssport P Lrning TPIC

ow os it work? Solutions Polgons Pg qustions Trnsformtions n hlf turn ( rottion). Thr qurtr turn (0 rottion). n qurtr turn (0 rottion). Trnslt tn units to th right first thn rflt own out th givn is of rfltion. Rott out th ntr of rottion, thn trnslt si units up. Rflt out th givn is first, thn trnlst two units to th lft. Thr qurtr turn (0 rottion) first, thn rflt out th givn is of iltion. TPIC Polgons Solutions Mthltis Pssport P Lrning

ow os it work? Solutions Polgons Pg qustions Trnsformtions ENTRY EIT (i) Trnslt shp horizontll to th right. (ii) Rflt shp horizontll to th right using th g BC s th is of rfltion. (iii) Trnslt shp vrtill own. (iv) Trnslt shp horizontll to th right. (v) Rott shp on qurtr turn (0) out th vrt. (vi) Trnslt shp vrtill own. (vii) Rflt shp vrtill own using th g BC s th is of rfltion. (viii) Trnslt shp horizontll to th right. (i) Trnslt shp vrtill up. () Rflt shp horizontll to th right using th g DE s th is of rfltion. (i) Trnslt shp vrtill up. (ii) Rott shp on qurtr turn (0) out th vrt. (iii) Trnslt shp horizontll to th right. Polgons Solutions Mthltis Pssport P Lrning TPIC

B ow os it work? Solutions Polgons Pg qustions Trnsformtions B Thr r mn iffrnt orrt ws of oing this, hr r just four possil solutions. Mtho Trnslt shp horizontll. units to th right. Rflt th shp out th is B horizontll to th right. Rott shp of turn (0) out th point B. Trnslt shp vrtill own. units. Mtho Trnslt shp horizontll. units to th right. Rott shp of turn (0) out th point B. Trnslt shp vrtill own. units. Rflt th shp horizontll to th right out vrtil is pssing through B. Mtho Rflt th shp vrtill own out horizontl is pssing through B. Trnslt shp vrtill own. units. Rott shp of turn (0) out th point B. Trnslt shp horizontll. units to th right Mtho Rott shp of turn (0) out th point. Trnslt shp horizontll. units to th right. Rflt th shp horizontll to th right out vrtil is pssing through B. Trnslt shp vrtill. units own. TPIC Polgons Solutions Mthltis Pssport P Lrning

ow os it work? Solutions Polgons Pg qustions Rfltion smmtr Smmtri smmtri Smmtri smmtri Smmtri smmtri Smmtri smmtri f g h Smmtri smmtri Smmtri smmtri Smmtri smmtri Smmtri smmtri i j k l Smmtri Smmtri Smmtri Smmtri smmtri smmtri smmtri smmtri Y Z Y Z YZ = m Z = m Distn from to Y = m Distn from to Y = m Polgons Solutions Mthltis Pssport P Lrning TPIC

ow os it work? Solutions Polgons Pg qustions Rfltion smmtr nswr ths qustions out th smmtri w low: Y ow mn s of smmtr os th w hv? L M G Wht pir of points r quiistnt to LM? Psst: quiistnt mns th sm istn P n P B Brifl plin low how ou i this ws th orrt nswr. Shps with rfltiv smmtr hv points quiistnt on th othr si of th is. Points P n r just lik LM, th strt on n is twn th r n th urv irl, mov prpniulr to th is n n on point of th fifth irl thr s ross. Biltrl smmtr. Two fol smmtr. Thr s of smmtr. Two s of smmtr. Fiv-fol smmtr. (show th othr four s) f Eight-fol smmtr. (show th othr svn s) TPIC Polgons Solutions Mthltis Pssport P Lrning

ow os it work? Solutions Polgons Pg qustions Rottionl n point smmtr Rottionll smmtri Rottionll smmtri Rottionll smmtri Rottionll smmtri Rottionll smmtri Rottionll smmtri f Rottionll smmtri Rottionll smmtri Rottionll smmtri Rottionll smmtri Rottionll smmtri Rottionll smmtri (i) (ii) (iii) s point smmtr No point smmtr s point smmtr No point smmtr s point smmtr No point smmtr (iv) (v) (vi) s point smmtr No point smmtr s point smmtr No point smmtr s point smmtr No point smmtr If thr is n vn numr of ls, th shp hs point smmtr, if thr is n o numr of ls, it osn t. Th numr of ls on th vn l propllrs is qul to th orr of rottionl smmtr. Polgons Solutions Mthltis Pssport P Lrning TPIC

ow os it work? Solutions Polgons Pg qustions Rottionl n point smmtr Rottionl smmtr of orr n lso point smmtr. Rottionl smmtr of orr n lso point smmtr. Rottionl smmtr of orr n no point smmtr. Rottionl smmtr of orr n lso point smmtr. B C W S T R V U P TPIC Polgons Solutions Mthltis Pssport P Lrning

ow os it work? Solutions Polgons Pg qustions Como tim: Rfltion, rottion n point smmtr Cn Mlsi Rfltion smmtr with fols Rfltion smmtr with fols Rottionl smmtr of orr. Rottionl smmtr of orr. Point of smmtr. No smmtr Point of smmtr. No smmtr Ini ustrli Rfltion smmtr with fols Rfltion smmtr with fols Rottionl smmtr of orr. Point of smmtr. No smmtr Rottionl smmtr of orr. Point of smmtr. No smmtr f mi Pkistn Rfltion smmtr with fols Rfltion smmtr with fols Rottionl smmtr of orr. Rottionl smmtr of orr. Point of smmtr. No smmtr Point of smmtr. No smmtr g h South fri Unit Stts of mri Rfltion smmtr with fols Rfltion smmtr with fols Rottionl smmtr of orr. Point of smmtr. No smmtr Rottionl smmtr of orr. Point of smmtr. No smmtr Polgons Solutions Mthltis Pssport P Lrning TPIC

ow os it work? Solutions Polgons Pg qustions Como tim: Rfltion, rottion n point smmtr k l Vitnm Unit ingom Rfltion smmtr with fols Rfltion smmtr with fols Rottionl smmtr of orr. Point of smmtr. No smmtr Rottionl smmtr of orr. Point of smmtr. No smmtr m n Gorgi Nw Zln Rfltion smmtr with fols Rfltion smmtr with fols Rottionl smmtr of orr. Rottionl smmtr of orr. Point of smmtr. No smmtr Point of smmtr. No smmtr o p Lttr 'D' signl flg Lttr 'L' signl flg Rfltion smmtr with fols Rfltion smmtr with fols Rottionl smmtr of orr. Rottionl smmtr of orr. Point of smmtr. No smmtr Point of smmtr. No smmtr q r Lttr 'Y' signl flg Lttr 'N' signl flg Rfltion smmtr with fols Rfltion smmtr with fols Rottionl smmtr of orr. Rottionl smmtr of orr. Point of smmtr. No smmtr Point of smmtr. No smmtr TPIC Polgons Solutions Mthltis Pssport P Lrning

Whr os it work? Solutions Polgons Pg 0 qustions Spil tringl proprtis ut-ngl sln tringl. Right-ngl isosls tringl. tus-ngl isosls tringl. Equiltrl tringl. Right-ngl sln tringl. Right-ngl isosls tringl Sln tringl Equiltrl tringl Isosls tringl Pg qustions Spil quriltrl proprtis Rtngl Isosls trpzium Rhomus it Squr f it Polgons Solutions Mthltis Pssport P Lrning TPIC

Whr os it work? Solutions Polgons Pg qustions Spil quriltrl proprtis Squrs hv ll sis qul, not just opposit sis. Rtngls hv ll intrnl ngls qul. Rtngls hv rfltiv smmtr. Squrs hv mor fols of rfltiv smmtr. Rhomus hv ll sis qul, not just opposit sis. Digonls of rhomus r prpniulr. Digonls of rhomus ist th ngls. pposit sis prlll in rhomus. ll sis qul in rhomus. Both igonls ist th ngls in rhomus. it hs onl on pir of qul opposit ngls. Rhomus hs ll sis qul, not just opposit sis Digonls of rhomus r prpniulr. Squrs hv fols of smmtr, rhomus onl. f pposit sis prlll in rhomus. Both igonls ist th ngls in rhomus pposit ngls qul in rhomus, jnt ons qul in isosls trpzium. is of smmtr igonl Squr Rhomus it TPIC Polgons Solutions Mthltis Pssport P Lrning

Whr os it work? Solutions Polgons Pg qustions Como tim: Spil quriltrl n tringls Rott on shp hlf-turn n trnslt until th non-prpniulr sis r ommon. r rflt on horizontll thn trnspos. Rott on shp hlf-turn n trnslt until th prpniulr sis r ommon. r rflt on horizontll n thn vrtill, thn trnslt. Rflt on vrtill up/own n thn trnspos until on pir of qul sis r shr. Prlllogrm Rflt on vrtill own n thn trnspos until th shortr sis r shr. Rhomus it Rtngl Prlllogrm Polgons Solutions Mthltis Pssport P Lrning TPIC

Wht ls n ou o? Solutions Polgons Pg qustions Trnsformtion on th Crtsin numr pln ojt - - - - 0 - img - - - ojt img - - - - 0 - - - - Rflt Vrtill orizontll Digonll Rflt Vrtill orizontll Digonll img ojt - - - - 0 - - - - ojt - - - - 0 - img - - - Rflt Vrtill orizontll Digonll Rflt Vrtill orizontll Digonll img - - - - 0 - ojt - - - f ojt - - - - 0 - - img - - Rflt Vrtill orizontll Digonll Rflt Vrtill orizontll Digonll TPIC Polgons Solutions Mthltis Pssport P Lrning

Wht ls n ou o? Solutions Polgons Pg qustions Trnsformtion on th Crtsin numr pln ojt - - - - 0 - img- - - - - - - 0 - img ojt - - - 0 0 rottion 0 0 rottion img img ojt - - - - 0 - - - - 0 - - - - - - ojt - - 0 0 rottion 0 0 rottion - - - - 0 - ojt img - - - f img - - - - 0 - - ojt- - 0 0 rottion 0 0 rottion g img - - - - 0 - ojt - - - h ojt - - - - 0 - img - - - 0 0 rottion 0 0 rottion Polgons Solutions Mthltis Pssport P Lrning TPIC

Wht ls n ou o? Solutions Polgons Pg qustions Trnsformtions on th Crtsin numr pln Rflt ojt out th lin =. Trnslt th ojt four units in th positiv irtion. = img - - - img ojt - 0 - - - - - 0 - - - - - ojt - Nw oorints for ot = ( -, ) - Nw oorints for ot = ( -, ) Rott th ojt out th ^00, h. Trnslt th ojt four units in th ngtiv irtion. - - img - - 0 - ojt - ojt img - - - 0 - - - - - - Nw oorints for ot = ( -, ) - Nw oorints for ot = (., 0 ) Rflt ojt out th -is. f rflt ojt out th givn is lin, =. img ojt - - - - 0 - - - - 0 - - img - - ojt - - - (, ) (., -.) Nw oorints for ot = Nw oorints for ot = - = TPIC Polgons Solutions Mthltis Pssport P Lrning

Wht ls n ou o? Solutions Polgons Pg qustions Trnsformtions on th Crtsin numr pln Trnslt ojt units in th positiv -irtion n thn rflt out th lin =. Rott th ojt on qurtr turn out th point (-, ) thn trnslt. units in th ngtiv -irtion. ojt img = - - - - 0 - - - - Nw oorints for ot = - - Nw oorints for ot = Rott ojt 0 out th point (-, ) n thn rflt out th -is. ojt - - - - 0 - img - - (, ) ( -, 0. ) - Rflt th ojt out th -is, n thn rflt out th lin =. ojt - - - - 0 - - img - - Nw oorints for ot = (, - ) ojt - - - - 0 - img - - - = Nw oorints for ot = (., - ) Rflt ojt out th -is thn rott out th origin ^00, h. f Trnslt th ojt. units in th ngtiv -irtion n thn rflt out th lin = -. ojt - - - - 0 - - - img - Nw oorints for ot = (., -. ) ojt - - - - 0 - - - - img = - Nw oorints for ot = (,. ) Polgons Solutions Mthltis Pssport P Lrning TPIC

D Wht ls n ou o? Solutions Polgons Pg 0 qustions Trnsformtions on th Crtsin numr pln B C D Strt hr - - - - - - 0 - B C - Finish hr - - Rflt horizontll out th si B. Coorints of : ^, h Trnslt horizontll in th ngtiv -irtion unit. Coorints of : ^, h Trnslt vrtill in th ngtiv -irtion. units. Coorints of : ^,. h Trnslt horizontll in th ngtiv -irtion units. Coorints of : ^,. h Rflt horizontll out th si CD. Coorints of : ^-,. h Trnslt vrtill in th positiv -irtion. units. Coorints of : ^-, h Trnslt horizontll in th ngtiv -irtion units. Coorints of : ^-, h Rott out th ntr of th plr squr. Coorints of : ^-, h Trnslt vrtill in th ngtiv -irtion units. Coorints of : ^-, h Rflt vrtill out th si BC. Coorints of : ^-, -h Trnslt horizontll in th positiv -irtion unit. Coorints of : ^-, -h Trnslt vrtill in th ngtiv -irtion unit. Coorints of : ^-, -h Trnslt horizontll in th positiv -irtion units. Coorints of : ^, -h Trnslt vrtill in th positiv -irtion unit. Coorints of : ^, -h 0 TPIC Polgons Solutions Mthltis Pssport P Lrning

* CMB TIME:REFLECTIN,RTTINNDPINTSYMMETRY I N S * T T M.../.../ 0... R R F S N SPECIL TRINGLE PRPERTIES.../.../ 0... * TRNSFRMTIN N TE CRTESIN NUMBER PLNE.../.../ 0... PLYGNS * PLYGNS * PLYGNS *.../.../ 0....../.../0....../.../0...

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