Chapter 4 Topic 1 Module PDF

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Chapter 4: EUCLIDEAN GEOMETRY OF THE POLYGON AND CIRCLE Topic 1: Fundamental Concepts and Theorems _______________________________________________________________________________________________Learning OutcomesAt the end of the lesson, you should be able to:  prove and discuss the fundamental conc...

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Modern Geometry

Chapt er4:EUCLI DEAN GEOMETRYOFTHE POLYGON AND CI RCLE Topi c1:Fundament alConcept sandTheor ems _______ _________ ________ ________ _________ ________ ________ ________ _________ ________ ________ _____ Learni ngOut comes Att heendoft hel esson,youshoul dbeabl et o:  pr oveanddi scusst hef undame nt alconcept sandt heor emsofEucl i deanGeomet r yoft he Pol ygonandCi r cl e. _______ _________ ________ ________ _________ ________ ________ ________ _________ ________ ________ _____ Learni ngCont ent I nChapt er s2and3,muchoft hemat er i alofEucl i deanGeomet rywasst udi edf r om t wo di ffer entpoi nt sofvi ew.Chapt er2emphasi zedt hei deaofEucl i de anGeomet ryasas t udyoft he i nv ar i antpr oper t i esofsetofpoi nt sunderEucl i dean orsi mi l ari t yt r ans f or mat i ons.Chapt er3 concent r at ed on t he concept of conv exi t y , one of t he pr ope r t i es pr eserved by si mi l ar i t y t r ans f or mat i ons ( as we l lasby somewhatmor e gener alt r ansf or mat i ons) .Thepr esentchapt er begi ns wi t h as ur vey of some f undame nt al concept s and t heor ems concer ni ng pol ygons ( par t i cul ar l yt ri angl es) and ci r cl esconce pt s and t heor ems t hat have l ong been a par t of t r adi t i onalEucl i deangeomet ry .Theni tpr ogr essesr api dl yt ot heweal t hofmoder nmat er i alt hat has bee n di scover ed si nce 1800 aboutt hese basi c figur es ofsynt het i c Eucl i dean geomet r y ( geomet ryt hatdoesnotusecoor di nat es) .Thel astsect i oni ncl udest hr eesi gni ficantappl i cat i ons oft heEucl i deangeomet ryoft hepol ygonandci r cl e-t hegol denr at i o,t ess el l at i onsandcar oms. I ti sessent i alt hatt hest udentofmoder n col l egegeome t ryunder st andsever alusef ulkey concept sandt heor emsf r om el ement arygeomet ry .Amongt hesear et heconcurr e nc et he or ems, somet i mesf ound i n hi gh schoolgeome t ryt ext s,t hati dent i f yf oursi gni ficantpoi nt sconnect ed wi t hat ri angl e. THEOREM 4. 1 Theper pendi cul arbi sect oroft hesi desofat r i angl ear econcur r entata poi nt cal l edt heci r cumcent er . Theor em 4. 1i si l l ust r at ed i n Fi gur e4. 1.I ti ssi gni ficantwhen anyt hr eel i neshaveacommon· poi nti n a geomet r y ,because any t hr ee l i nes gener al l yi nt e r sectby pai r st o det er mi ne t hr ee di st i nctpoi nt s ,r at hert hanasi ngl eone.

The pr oofofTheor em 4. 1 depends on showi ng t hatt he poi ntofi nt er sect i on oft wo oft he per pendi c ul arbi sect or s,say B ´' O and C ´' O i n Fi gur e4. 1 al so l i eson t heperpendi cul ar ti sal so bi sect oroft het hi r dsi de.Poi nt0i sequi di st antf r om A andC becausei ti son B ´' O ;i equi di st antf r om A andB becausei ti son C ´' O .Ther ef or e,O i sequi di st antf r om B andC on

Modern Geometry

´ .Si nceO i st hepoi nti n at ri angl eequi di s t antf r om t het hr ee t heper pendi cul arbi sec t orof BC vert i ces,i ti s cal l ed t he ci rcumcent er,t he uni que ci r c l e cont ai ni ng al lt hr ee ver t i ces ofa t ri angl e. I nFi gur e4. 2,t r i angl eA’ B’ C’i sf ormedbyj oi ni ngt hemi dpoi nt soft hesi desoft heor i gi nal t ri angl e.Si ncet hesegment sj oi ni ngt hemi dpoi nt soft wosi desofat r i angl ear epar al l elt ot he t hi r dsi de,t heper pendi cul arbi sect or soft hesi desoft ri angl eABC ar eal soper pendi cul art ot he si desoft r i angl eA’ B’ C’ .Thi smeanst hat A ´' O i sper pendi cul art o B '´C ' and s oon;t hus, t heper pendi cul arbi sect oroft hesi desoft r i angl eABC ar et heal t i t udesoft r i angl eA’ B’ C’ .

THEOREM 4. 2Theal t i t ude sofat r i angl ear econcur r entatapoi ntcal l edt heor t hoc ent er . Fi gur e4. 3 shows t ri angl e ABC and i t sor t hocent erH.The f ourpoi nt sA.B.C,and H const i t ut ean or t hocent r i csetoff ourpoi nt s ,s onamed becauseeach oft hef ourpoi nt si st he or t hoc ent eroft het r i angl ef or medbyt heot hert hr ee.

Twoaddi t i onalt heor emsaboutconcur r encyi nvol vet hei nt ernalangl ebi s ect or sand t he medi ansofat r i angl e. THEOREM. 4. 3.The i nt er nalbi sec t or s oft he angl es ofa t r i angl e mee tata poi ntcal l ed t he i nc ent er . Thepr oofofThe or em 4. 3dependsont hef actt hatev erypoi ntont hei nt er nalbi sect orof anangl ei sequi di st antf r om t headj acentsi desoft heangl e. ´ ≅ IX ´ .Si Forexampl e,i n Fi gur e4. 4a,i fLi son t heangl ebi sect orofangl eB,t hen IY nceIi s equi di st antf r om al lt hr eesi desoft het r i angl e,i ti st hecent e roft hei nci r c l e ,aci r c l ei nscr i bedi n t het r i angl e.Thi smeanst hatt het hr eesi desoft het r i angl ear et angentt ot hei nc i r c l e.

Modern Geometry

THEOREM 4. 4.Themedi ansofat ri angl emeetatapoi ntcal l edt hec ent r oi d. Thecent r oi di st hec ent eroft hegr avi t yf orat r i angl e.Recal lt hatmedi ansj oi nt hever t ex and t hemi dpoi ntoft heopposi t esi deofat r i angl e.I n Fi gur e44b,t ri angl esCBG andGC’ B’ar e si mi l arwi t har at i oofsi mi l ar i t yoft wot oone.Thec ompl et i onoft hedet ai l soft hepr oofi sl ef tas anexer ci se . Sof ar ,f ourpoi nt sofc oncur r encyhavebeen i nt r oduced.I ti snat ur alt owonderwhet her t hef ournew poi nt sconst i t ut easi gni ficantsetofpoi nt si nt he i rown r i ght .Themat t erwi l lbe di scuss edi nSect i on4. 3,butyoumaypr ofitf r om conj ec t ur i ngaboutt hel ocat i onoft hesepoi nt s now. Much oft hest udyoft het r i angl ei n Eucl i dean geomet ryi nvol ve swor k wi t h pr opor t i ons, and or di nar i l yt hi sconcepti sr el at ed t osi mi l art r i angl es.A basi cpr opor t i on used asat ooli n col l egegeomet ryi st hepr oper t yt hati sconnect edwi t hi nt er nalangl ebi s ect or soft heangl esofa t ri angl e,st at edasTheor em 4. 5. THEOREM 4. 5.Thei nt er nalbi sect or sofal langl eofat r i angl edi vi det heopposi t esi dei nt ot wo segment spr opor t i onalt ot headj acentsi desoft het r i angl e. ´ st hei nt er nalangl ebi sect orofangl eAoft ri angl eABC I nFi gur e4. 5,assumet hat AD i

´ .Becauseoft ´ i spar al l elt o AD hepar al l el i sm, ∠ ECA ≅∠ CAD ≅∠ andt hat CE ´ ≅ AC ´ .Now t CEA.Thi smeanst hat ∆ ECA i si s osce l esand t hat EA hi nk oft hefigur eas

´ f or medbyt wot r ans ver sal sf r om B i nt er sec t i ngapai rofpar al l ell i nes, EC t hef ol l owi ngi sat r uepr opor t i on: CD DB = EA AB

or

´ ,sot and AD hat

CD DB = CA AB

sot hat ,

CD CA = DB AB aswast obees t abl i shed.Not et hatt hedi r ect edsegment sar enotempl oyedi nt hi sdevel opment .

Modern Geometry

THEOREM 4. 6.Theext er nalbi sect or soft woangl esofat ri angl emeett hei nt er nalbi sec t orof t het hi r dangl eatapoi ntcal l edt heexc ent r e. Ofc our se,when t al ki ngaboutt heext er nalbi sect or sofat r i angl e,onemustt hi nk oft he si desas“ ext ended”orconsi dert hesi desasl i nesr at hert han segment s.I n Fi gur e4. 6,assume t hatE i st heexcent r e.I tbecomesappar enti nt hepr oofofTheor em 4. 6,t hatE i sequi di s t ant f r om al lt hr ees i desoft het r i angl e;henceE i st hecent eroft heci r c l eext ernal l yt angentt oal l t hr e esi desoft het r i angl e.Thi sci r cl ei soneoft hee xci r cl esoft hegi vent ri angl e .

THEOREM 4. 7.I ft hedi st ancesf r om apoi ntPt ot wofixedpoi nt shaveagi venr at i o,t hent heset ofal ll ocat i onsf ort hepoi ntPi saci r c l e,cal l edt heci r c l eofApol l oni us. Theci r c l eofApol l oni usi snamed af t ert heGr eekmat hemat i ci an Apol l oni us,whowr ot ea compr ehensi ve t r eat menton coni c sect i ons pr i ort o 200 B. C.Assume usi ng t he not at i on of Fi gur e4. 7a,t hatPA/PB=c,agi ven c onst ant .I tmustbeshown t hatt hesetofal ll ocat i onsf or ´ poi ntPi saci r cl e.The r ear et wopoi nt son AB whosedi st ancesf r om A andB ar et hecor r ect r at i owi t houtr eg ar dt odi r ect ed di st ances.Thesear ei ndi cat ed bypoi nt sC and D i nt hefigur e. ´ ´ ei nt ernalandext er nalangl ebi sec t or soft heangl eat Then,i nt r i angl eAPB, PC and PD ar P.Thi scanbeshownf orpoi ntC.Forexampl e,si nce

AC PA = CB PB

Modern Geometry

Butt he i nt er naland ext ernalangl e bi sect or s ata ver t ex ofa t r i angl e ar e per pendi cul ar ,s o sari ghtangl e .Tri angl eCPD i si nscr i bedi nasemi ci r c l e,soPl i esonaci r cl ewi t hCD ∠CPD i asdi amet er .

5 . 3

Fi gur e4. 7bshowsaspeci ficexampl eoft heci r c l eofApol l oni us,f orwhi chP’ A’ /P’ B’ =

Eachpoi ntont hec i r cl ei s5/3asf arf r om A'asi ti sf r om B' ,andt hepoi nt sont heci r cl ear et he onl ypoi nt si nt hepl anewi t ht hi spr oper t y .

I n t he pr oof of Theor em 4. 7 and e l sewher e, not et hat even t hough par al l el i sm ( per pe ndi cul ari t y)i sa r el at i onshi p usual l ydefined f orl i nes,useoft heconcepti sext ended t o segment sandr ayswi t houtdi fficul t y .Forexampl e,al t houghi ti st echni cal l yi mpr eci set osayt hat ´ ´ ,t ´ of i spar al l e l( per pendi c ul ar )t o CD hatexpr essi oni sunder st ood t omean t hat AB AB

´ AB

´ ofwhi ´ i sasubseti spar al l el( per pendi cul ar )t o CD ch CD i sas ubset . Twofinalt he or emsaboutt hesegment sr el at edt oaci r cl ear esomewhatconnect edt ot he pr evi oust heor e m andal soar eusef uli npr ovi ngmor eadvancedt he or ems.

whi ch

THEOREM 4. 8. I f a quadr i l at er al i si nscr i bed i n a ci r cl e, t he n t he oppos i t e angl es ar e suppl ement ary . Thet er m cy cl i cquadri l at er ali sal soused f ora quadr i l at er ali nsc r i bed i n aci r c l e.St udy Fi gur e4. 8.You shoul dr ecal lt hatt hemeas ur eofani nscr i bedangl ei naci r cl ei shal ft hatofi t s i nt er cept edar c.Forexampl e,t hemeasur eofangl eB i shal ft hatofar cADC.Si nceangl esB and D t oget her i nt er ce pt ar cs wi t h a t ot al measur ement of 2 π degr ees, t he angl es ar e suppl ement arysi ncet hesum oft hei rmeasur esi s π .

Opposi t esi desofani nscr i bedquadri l at er alar esomet i mescal l e d ant i par al l elwi t hr espect t ot her emai ni ngpai rofsi des.Thepr efix" ant i "suggest sac r ossf r om oropposi t e .I nFi gur e4. 8,i f ´ ´ woul angl esD andA wer esuppl ement ar y .t hen DC and AB dbepar al l el .Buti nst e ad,i ti s t he angl e opposi t e or ac r oss f r om A t hati ss uppl ement ar yt o D,hence t he segment s ar e ant i par al l eli nst e adofpar al l e l . THEOREM 4. 9.Thepr oductoft hel engt hsoft hesegment sf r om anext er i orpoi ntt ot hepoi nt s ofi nt e r sect i on ofasecantwi t h aci r cl ei sequalt ot hesquar eoft hel engt h oft het angentf r om t hepoi ntt ot hec i r cl e. lagai n Usi ngt henot at i on ofFi gur e4. 9,t het heor em sayst hatAC ∙ AD= ( AB)2 .Recal

´ .Tri t hatt henot at i on AC i ndi cat e sa number ,t hemeasur eof AC angl esABC and ADB cue si mi l arbecausecor r espondi ngangl e sar econgr uent ;t her ef or e,AC/AB =AB/AD f r om whi cht he t heor em f ol l ows.

Modern Geometry

LEARNI NG ACTI VI TI ES 1.Uset henot at i onofFi gur e4. 3t onameal loft het r i angl eswhosever t i cesar et hr eeort he gi venpoi nt sandwhoseor t hocent eri st hef ourt hpoi nt . 2.Wher ei st heor t hocent erofari ghtt r i angl e ? 3.How manyexcent er sdoesat r i angl ehave ? 4.Coul dant i par al l else gment sal sobepar al l el ? 5.Pr ov et hatt hesegmentj oi ni ngt hemi dpoi nt soft wosi desofat ri angl ei spar al l e lt ot he t hi r dsi de. 6.Pr ov et hatt hei nt e rnalandext er nalangl ebi sec t or sataver t exofat r i angl ear e per pe ndi cul ar .

ASSESSMENT TASK Cr eat eavi deoofy our se l fr eci t i ngt het he or emsonEucl i deangeomet ryoft hepol ygonandci r cl e . ( Theor em 4. 14. 9)

REFERENCES JAMESR.SMART Cal i f or ni aSt at eUni ver si t y ,SanJose

Pr epar edby:PAULJOHN M.AGCAOI LI ,LPT I nst ruc t or...