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I : Constructions " & Compass stature of has basic - PowerPoint PPT Presentation

Sep 11, 2023 •240 likes •481 views

Applications " Ruler Geometric I : Constructions " & Compass stature of has basic Motivation Geometry Euclidean : , axioms , and prepositions definitions can draw segment Fa and Q , Axioms P given two points :

  1. Applications " Ruler Geometric I : Constructions " & Compass

  2. stature of has basic Motivation Geometry Euclidean : , axioms , and prepositions definitions can draw segment Fa ① and Q , Axioms P given two points : in either dint in extend arbitrarily ① given POT , segment can circle with can draw ③ givin points P and Q a , Oi ( passing though center P why this is congruent geometry generated ④ all angles right are ① parallel ruler & compass postulate by

  3. with tuna ATgEmfm ) tools ? create What we can can crate a point ( Pnp 2) Ginn ED and c segment , so that CT segment a given a given " length point " to transport > CI Thu with ATT AT ED ④ up 3) and we can Givin = CI Ti that find ATB AT E so ou code for : can construct with ruler & compass

  4. angle Chop 9) Bisect given a a perpendicular live l " from P ( Prep to 12 ) " Drop I " . There new 113 lots of were able being Despite to prove " That couldn't solve " natural questions ancients a few

  5. include Then ? a given angle you bisect ① Can a cube of 1 , volume " ⑦ Gain cube " Duplicating : a of volume 2 cube create a 352 given a length of size create ( ie , we can length ) of segment unit a Area A , of circle ⑦ " circle squaring the Given " : a , given leyth r et A ? a sgvun ( ie create area , rift ? ) length create can we

  6. constructible ? ④ regular what are , you n - create equilateral Eg Prep I says you can . triangles - gon ) regular lie , 3 a unresolved for -2000 questions Then were years . The negate ) today ④ - ③ address we'll Lin .

  7. , we'll take questions algebraic view . these more To resolve a numbers ) Petn ( constructible ruler , compass conductible if using a EIR is A number a , , a segment at leythkl create unit length , and . can we constructible ? leyths Question what are : a field , numbs conductible we'll That on see is subset of c. uslmtible Q got Sime 1 is we constructible , the .

  8. field . term censtntible numbers Tim The a , Then ceastuchbte That it xp Pf are we'll show Cd ) Xp constant idle a - p ⑨ art p clap ④ are . T THO with Cal , given , C , D A , B For g. to T eo D •@ D A

  9. AT , extend 2 Axum By Fe = ED so that : find E Prop 2 BI centred @ D , radius curate Axiom 3 : draw a r - - Efi - - . . . poof • I - j - i , % D c , • 1 A at P . length AI has Then

  10. with HIP still given Ares . c. D one T o & D c • A between AIB find E some Prep 3 we can says with FEZ ED Then FB ~ has length • § if Noooo • - P D c • D L E . A

  11. with A. B. c , D him coats one T o % D c • A perpendicular Thgwh A dreg First , length ' unit . " " Mark off . . fi E•¥t E IB Connect " Mark off " B length goop lion thigh F G ¥9 paw ; parallel l to oooo , At Similar Ingles " AI xp says .

  12. (d) Mp another similar Ingle argument . Comdt text . Th DNA fold of constantine about this what say we can numbers ? constricting then Fact : if is > O Important a Ta is so . our Q . degree infinite numbers constructible are Cor : ( Vz ) E ⑥ ( %) E ( THE GQ E PI Consider - - -

  13. coastlines , into deeper take delve we a new To of view . point ( constructible point ) Def 's from Cachao constructible ) . k¥1 A- point lay ) EIR ' it , the Hlaing ;d holds old of keyed if one ⇐ LIB ) n LLC , D ) UAB ) t LHP ) l " ( x , y ) when , " line connecting " A- & B

  14. n c(c,D)_ LIA , B ) ( x. y ) C " E circle with center c radios and CDT (3) CCC , D ) CLA , B ) n ( x. g) E constructible iff or 11,0 ) Hy ) :( go ) we say ( x. g) is 1901.4.01 , R , of . . , Pk points or if fun is sequence a constructible Pi - ( x. y ) and is so that each Pk ' subset at { ego ) , 4,03 , Pa . - - - ' Pi . . ) relative to some

  15. la , .ae ) , Ibi , but , Cayce ) , @ , .dz ) kegthem If in are ( a.aybi.bz , c. end . .dz ) ) for ⑥ E F E IR ( say Fh : F ' Sonu AB , c , and D constructible relation to lay ) Then ay extension of quadratic F lives in some . x. yet ( re ) ) that ( ie , exists ZEF so ( sketch ) Ef LCGD ) LL AB ) of and intersection hey ) case is 1 . : LCA , B ) ABEFZ , Then The gratin for Note

  16. Bs , TEF . form =D for rxtsytt takes The some - O ferris , Iet Sean q' fxtsytt has LLC , D ) - Likewise a solved for solution be them the ilntoseltian , and can F . is in we get , then LL A. B) n Ccc , D ) Hy ) If t an equation r , Stef fer rxtsytt - O UAB ) - has ' for qc.pe# - EY ' t ( y ( x - g) equation CCC , D) - p has - and an system t find EEF with y ,xE FIRE ) . Solve this

  17. ⇒ (3) CCA .is/ncC4D ) if Cx , y ) same idea E TBT useful ? Hew is this austmtibk numbers . coustmtibte if Tim x. you IX. g) is " : @ 3=-2 [ ① Ca ) implies constable Cos is a n EIN for some . constructible HI Cao ) constable is . 2 is

  18. . " , lxk-i.gr ) , ko ) ) a sequence { 10,9 , 4,0 ) , Hi , y . ) , hail Hence we cealtrufibk in the so that term sequence is each in the squirm preceding terms to relative some . s . but of to constable relative some Hence ( Xi , y ; ) is ( Xinyi ) is . . , ( Xi . . , you , ) Sinn 199,407 , IX. , Yi ) ; . coordmh canstmtihk in an when point , relink to know - > Xia , Yin ) ④ ( KH . , we . , Xii , Yi - D) c. 2 : ⑥ ( Xi , Yy [ ④ ( x , ,yy , Xi , yi ) . . , Xia , Tia . . .

  19. g th degree along th tower ferula ' Have us - - EQ ( 14,9 , , E ④ ( kyyyxayr ) E . - yxk.ik.in ) . ) ④ ( Xi ⑧ E , y - : Q ) = 21 ( ⑥ ( x , ,y . , . " Yak ) get . ,Xk we . . . we get formula By the degree again Q ] net [ Ak ) - 2 for " some - : . DA

  20. triseotnbk . all angles Cos are Not unit create a can If I we Prep says . ) length guilty Ingle . I Internal angles 60 are is equivalent to couslnetibil.ly angle trisecting this so angle triple However the . ) af cos 120 . fermin says fee µ f¥ cos 1307=4 cos ' la ) On - 3 cos O c. ska ) salish So .

  21. . ] =3 [ Ql cos 200 ) t 2 " : Q get So we . constrict ble . cos 1207 isn 't Hence can 't duplicate cube . Cos a you is equivalent to aeustuetig Vz . duplication of cube PI - 2 - x3 irra ( Vz ) , so has V2 - But " =3 t 2 [ atra ) : - Dappy

  22. circle . Cos you can 't square a constructible : if you have PI ' r Equitant te , , ft is coustutibk ) lie constructible then is rift . so , then [ ④ ( fu ) : Q ) for some " 2 If algebraic , be fries fit to . This n . But algebraic is and is * so it transcendental , so → ← DH .

  23. ioastuetihilitg of regular If keen on are you 137 . you'll find consult pg gens , n - th ( Gauss ) If odd pain , then you is an p . get * i - goniff regular p . constrict p a can - Fermat t > 0 primes for sewn . or I 't 1=5 , 22 't 1=17 , 2%1 ' - 3 2 C. g. , - - th - = -

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