ZERO DIMENSION / NO DIMENSION

Transkript

1 ownloaded rom - GOMTRY Note-5 : dksbz Hkh rhu vlajs[k (Non collinear points) fcunq, ges kk,do`ùkh; (oncyclic) gksrk gs ZRO IMNSION / NO IMNSION 1) OINT (fcunq) : fcunq og gs ftlesa u ykkbz u pksm+kbz vksj u eksvkbz gksrk gs Note-1 : fdlh,d fcunq ls vuar js[kk, xqtjrh gs Note-6 :,d o`ùk ges'kk rhu vlajs[k fcunqvksa ls gksdj xqtjrh gs ON IMNSION Note - : nks fofhkuu fcunqvksa ls,d vksj dsoy,d js[kk [khapk tk ldrk gs Note 3 : rhu ;k vf/d fcunq, lajs[k (ollinear points) dgykrh gsa ;fn os,d gh js[kk ij flkr gks rhu ;k vf/d fcunq, tks,d js[kk ij flkr ugha gks vlajs[k fcunq, (Non-collinear points) dgykrh gsa, & lajs[k fcunq gsa 1) LIN (js[kk) :,slh ljy js[kk tks nksuksa rji vuur nwjh rd c<+h gksrh gs js[kk ( ) ) LIN SGMNT (js[kk[k.m) : js[kk[k.m] js[kk dk,d VqdM+k gksrk gs ftldk nks vur fcunq gksrk gs js[kk[k.m ( or ) 3) RY (fdj.k) :,d js[kk dk og Hkkx ftldk,d var fcunq gks,d fdj.k dgykrh gs fdj.k () => nks fcunq, lnk lajs[k gksrh gs, & vlajs[k fcunq gsa Note- 4 : pkj ;k pkj ls vf/d fcunq,,do`ùkh; (oncyclic) dgykrh gsa ;fn buls gksdj,d o`ùk xqtjsa 4) RLLL LINS lekukarj ;k vizfrnsnh js[kk, ) :,d ry dh nks fhkuu js[kk, lekurj dgykrh gs ;fn muesa dksbz Hkh fcunq mhk;fu"k ugha gks pkgs os nksuksa rji fdruh Hkh c<+kbz tk;,, &,do`ùkh; fcunq, gsa MTHMTIS Y- RHUL RJ Note-1: nks lekarj js[kkvksa ds chp ycor~ nwjh ges'kk leku gksrk gs GOMTRY & MNSURTION

2 ) Right ngle (ledks.k) : og dks.k ftldh eki 90 0 gks ledks.k dgykrk gs Note- : ;fn nks js[kk[k.m,d gh lrg ds lkk leku dks.k cuk;s rks nksuksa js[kk, lekarj gksxh 90 0 ledks.k ( = 90 0 ) x 0 x 0 TWO IMNSION 1) NGL (dks.k ) : ;fn nks js[kk[k.mksa ;k fdj.kksa dk,d gh vur fcunq (end point) gks rks blls cuh vkd`fr dks dks.k dgrs gsa vur fcunq dks.k dk 'kh"kz (Vertex) dgykrk gs 3) Obtuse ngle (vf/ddks.k) : og dks.k ftldh eki 90 0 ls vf/d rkk ls de gks vf/d dks.k dgykrk gs vf/d dks.k ( 90 0 < < ) ) Straight ngle (½tqdks.k) : og dks.k ftldh eki gks mls ½tq ;k ljy dks.k dgrs gsa dks.k ( ) dks.k ds izdkj (Types of ngle) (ccording to measurement of angle) 1) cute ngle (U;wudks.k) : og dks.k ftldh eki 90 0 ls de gks U;wudks.k dgykrk gs ½tq dks.k ( = ) 5) Reflex ngle (iqu;zqr dks.k ;k izfroùkhz dks.k): og dks.k ftldh eki nks ledks.k (180 0 ) ls cm+k rkk pkj ledks.k (360 0 ) ls NksVk gks mls iqu;qzr dks.k dgrs gsa U;wudks.k ( < 90 0 ) iqu;qzr dks.k (180 0 < ) MTHMTIS Y- RHUL RJ 3 GOMTRY & MNSURTION

3 6) omplete ngle (liw.kz dks.k) : laiw.kz dks.k dk eku gksrk gs 5) Vertically Opposite ngles ('kh"kkzfhkeq[k dks.k) : nks dks.kksa dks 'kh"kkzfhkeq[k dks.k ;qxe dgrs gs ;fn mudh Hkqtk, foijhr fdj.kksa ds nks ;qxe gksa o TRMS RLT TO NGL (dks.k ls lacaf/r 'kn) 1) omplementary ngles (iwjd dks.k): ;fn nks dks.kksa dh ekiksa dk ;ksx 90 0 gks rks mugsa iwjd dks.k dgrs gsa ) Supplementary ngles (liwjd dks.k) : ;fn nks dks.kksa dh ekiksa dk ;ksx gks] rks mugsa liwjd dks.k dgrs gsa 3) djacent ngles (vkluu dks.k) : nks dks.kksa dks vkluu dks.k dgrs gsa ;fn mudk 'kh"kkz,d gh fcunq gks] mudh,d mhk;fu"k Hkqtk (ommon arm) gks vksj,d dks.k dh nwljh Hkqtk mhk;fu"k Hkqtk ds,d vksj gks vksj nwljs dks.k dh nwljh Hkqtk mhk;fu"k Hkqtk ds nwljh vksj gks O mhk;fu" 'kh"kz mhk;fu" Hkqtk R O vksj OR vkluu dks.k gsa 4) Linear air ngles (jsf[kd ;qxe dks.k) : nks vkluu dks.kksa (djacent ngles) ftudh fhkuu Hkqtk, nks foijhr fdj.ksa gksa] jsf[kd ;qxe dks.k dgykrk gs jsf[kd ;qxe dks.k (nksuksa dks.k dks feykdj ) dk gksrk gs R ( O vksj O) 'kh"kkzfhkeq[k dks.k dk,d ;qxe gs vksj ( O vksj O) 'kh"kkzfhkeq[k dks.k dk nwljk ;qxe gs Note : tc nks js[kk,,d nwljs dks dkvrh gs rks 'kh"kkzfhkeq[k dks.k leku gksrk gs O = O vksj O = O 6) erpendicular (yc) : nks js[kk,,d nwljs ij yc dgykrh gs ;fn muds chp dk dks.k 90 0 gs 90 0 vksj,d nwljs ds ycor~ gsa 7) erpendicular isector (yc lef}hkktd) : ;fn,d js[kk fdlh js[kk[k.m ij yc gks vksj js[kk[k.m dks nks cjkcj Hkkx esa ck Vs rks og js[kk yc lef}hkktd dgykrk gs O 90 0 OR + OR = MTHMTIS Y- RHUL RJ 4, dk yc lef}hkktd gs GOMTRY & MNSURTION

4 Note : yc lef}hkktd ij flkr izr;sd fcunq nksuksa var fcunq ls lenwjlk (quidistant) gksrk gs = = 8) ngle isector (v¼zd) : ;fn,d js[kk fdlh dks.k dks nks cjkcj Hkkxksa esa ck Vs rks og js[kk v¼zd dgykrk gs,d fr;zd js[kk gs ** fr;zd 8 dks.k cukrk gs 1) xterior ngles (ckg~; dks.k) : 1,, 7, & 8 ) Interior ngles (var% dks.k) : 3, 4, 5 & 6 3) our pairs of corresponding angles (laxr dks.k): (, 6), (1,5), (3,7) & (4,8) Note : v¼zd ij flkr izr;sd fcunq nksuksa Hkqtkvksa ls leku nwjh ij flkr gksrk gs 4) Two pairs of lternate Interior ngles (,dkarj var% dks.k) : ( 3, 5) & (4, 6) S 5) Two pairs of lternate xterior ngles (,dkarj ckg~; dks.k) : pair - (, 8) & (1, 7) X Y lekukarj js[kkvksa ds fr;zd R X = X RY = SY TRNSVRSL (fr;zd) Transversal (fr;zd): ;fn,d js[kk nks ;k nks ls vf/d js[kkvksa dh fhkuu&fhkuu fcunqvksa ij dkvs rks mls mu nh gqbz js[kkvksa dh fr;zd js[kk dgrs gsa fr;zd ds }kjk cuk;k x;k dks.k (ngle formed by transversal) ,oa fr;zd gs MTHMTIS Y- RHUL RJ 5 GOMTRY & MNSURTION

5 1) laxr dks.k ds ;qxe cjkcj gksrs gsa 1 = 5, = 6, 3 = 7 & 4 = 8 ),dkarj dks.k ds ;qxe (var%,oa ckg~;) cjkcj gksrs gsa 3 = 5, 4 = 6, = 8 & 1 = 7 3) fr;zd ds,d dh rji ds var% dks.kksa ;k ckg~; dks.kksa dk ;ksxiy gksrk gs = = + 7 = = Note : ;fn nks js[kkvksa dks,d fr;zd dkvrh gs,oa mijksr esa ls dksbz,d flkfr dks lgh ik;k tkrk gs rks nksuksa js[kk, lekarj gksxh 4) var% dks.kksa dk v¼zd,d nwljs dks 90 0 ij dkvrk gs 7) 8) roof : x x + y y x + y x x y y x y ( dks.k dh eki ) MSURMNT O NGL dks.kksa dks ekius dk rhu rjhdk iz;ksx fd;k tkrk gs ) var% dks.kksa dk v¼zd vk;r cukrk gs vk;r gs 1) "kkf"vd i¼fr (Sexagesimal System or nglish System) (egree) ) xfrd i¼fr (entesinal System or rench System) (Grade) 3) o`ùkh; eki (ircular measurement) (radian) 1) "kkf"vd i¼fr (Sexagesimal or nglish System) (egree) : bl i¼fr esa,d ledks.k dks 90 cjkcj Hkkxksa esa fohkkftr fd;k tkrk gs] ftls va'k dgk tkrk gs izr;sd fmxzh dks 60 cjkcj Hkkxksa esa fohkkftr fd;k tkrk gs ftls ^feuv* dgrs gsa vksj blh izdkj izr;sd feuv dks 60 cjkcj Hkkxksa esa ck Vk tkrk gs ftls lsds.m dgrs gsa 6) ;fn fr;zd yacor~ gks rks var% dks.kksa dk v¼zd oxz cukrk gs MTHMTIS Y- RHUL RJ oxz gs 6 60 lsds.m ( 60 ) = 1 feuv (1 ) 60 feuv (60 ) = 1 fmxzh (1 0 ) 90 fmxzh (90 0 ) = 1 ledks.k ) xfrd i¼fr (entesimal system or rench System) (Grade) : bl i¼fr esa,d ledks.k dks 100 cjkcj Hkkxksa esa ck Vk tkrk gs ftls xzsm dgrs gsa izr;sd xzsm dks 100 cjkcj Hkkxksa esa ck Vk tkrk gs ftls ^feuv* dgrs gs vksj izr;sd feuv dks 100 cjkcj Hkkxksa esa ck Vk tkrk gs ftls lsd.m dgrs gsa GOMTRY & MNSURTION

6 100 lsds.m (100 ) = 1 feuv (1 ) 100 feuv (100 ) = 1 xzsm (1 g ) 100 xzsm (100 9 ) = 1 ledks.k 3) o`ùkh; eki ( ircular measurement or Radian measure) : fdlh o`ùk ds pki (arc) ds }kjk dsunz ij cuk;k x;k dks.k dk jsfm;u pki ds cjkcj gksrk gs f=kt;k Ok`Ùk dh f=kt;k ds cjkcj pkj o`ùk ds dsunz ij tks dks.k varfjr djrk gs] mls,d jsfm;k (1 c ) dgk tkrk gs q = l => r => π = = π 180 x = x π x π = x 180 pki dh yca kbz f=kt;k ' '' xyz x y π π = π z ?kM+h dh lwb;ksa ds }kjk cuk;k x;k dks.k (ngle made by Needles of a lock)?kavk dh lwbz (Hour Needle) 1 Mk;y = ?kaVs = ?kaVk = 30 0 feuv dh lwbz (Minute Needle) 1 Mk;y = x 9 : 3 60 feuv = feuvk = dks.k = [ 3 6] = [ ] [ 19] = = 94 0 OLYGON (cgqhkqt) cgqhkqt (olygon) :,d T;kferh; vkd`fr tks de ls de rhu js[kk[k.mksa ls f?kjk gqvk gks] cgqhkqt dgykrk gs cgqhkqt ds uke (Name of olygons) Name No. of Sides f=khkqt (Triangle) 3 prqhkqzt (uadrilateral) 4 iaphkqt (entagon) 5 "kv~hkqt (Hexagon) 6 lirhkqt (Heptagon) 7 v"v~hkqt (Octagon) 8 uohkqt (Nonagon) 9 nlhkqt (ecagon) 10 cgqhkqt ds izdkj (Types of olygons) i) onvex olygon (mùky cgqhkqt) : ;fn fdlh cgqhkqt ds izr;sd dks.k ls NksVk gks rks mls mùky cgqhkqt dgrs gsa 60 feuv = feuv = 1 MTHMTIS Y- RHUL RJ 7 GOMTRY & MNSURTION

7 ii) iii) oncave olygon (vory cgqhkqt): ;fn fdlh cgqhkqt dk de ls de,d dks.k ls T;knk gks mls vory cgqhkqt dgrs gsa Regular olygon (le cgqhkqt) : ;fn fdlh cgqhkqt dh lhkh Hkqtk,oa lhkh dks.k cjkcj gks rks og cgqhkqt le cgqhkqt dgykrk gs cgqhkqt ls lacaf/r lw=k ORMUL related to olygon 1) lhkh var% dks.kksa dk ;ksxiy = (n ) x ) lhkh ckg~; dks.kksa dk ;ksxiy = ) fod.kksz dk la[;k = ( ) n n n 3 = n 4) rkjkd`fr cgqhkqt ds 'kh"kzdks.kksa dk ;ksxiy (sum of vertices angles of a star-shaped polygon) = (n 4) x iv) leckgq f=khkqt oxz le iaphkqt Non-Regular olygon (fo"ke cgqhkqt) : ;fn fdlh cgqhkqt dh dksbz Hkh Hkqtk cjkcj ugh gks rks og cgqhkqt fo"ke cgqhkqt dgykrk gs = ( 6 4 ) x = fo"keckgq f=khkqt vk;r iaphkqt cgqhkqt ls lacaf/r 'kn (Terms related to olygon) iagonal (fod.kz) : nks vøekxr (Non consecutive) 'kh"kksz (Vertices) dks tksm+us okyh js[kk dks fod.kz dgrs gsa = ( 5 4 ) x = ) var% dks.k + ckg~; dks.k = ) ckg~; dks.k = var% dks.k ** n Hkqtkvksa okys lecgqhkqt ds fy, (or a regular polygon of n sided ) vksj fod.kz gsa 1) izr;sd var% dks.k (ach interior angle) n 180 = ( ) n MTHMTIS Y- RHUL RJ 8 GOMTRY & MNSURTION

8 ) izr;sd ckg~; dks.k (ach exterior angle) = 360 n 3) Hkqtkvksa dh la[;k (Number of Sides) = 4) {ks=kiy (rea) = 360 izr;sd ckg~; dks.k tgk a = Hkqtk dh yackbz n 180 a cot 4 n 5) leckgq f=khkqt dk {ks=kiy = 6) oxz dk {ks=kiy = a 7) "kv~hkqt dk {ks=kiy = 3 3 a 3 a 4 cgqhkqt ls lacaf/r xq.k roperties related to olygon fdlh cgqhkqt (f=khkqt,oa prqhkqzt dks NksM+dj) var% dks.kksa dk ;ksxiy] ckg~; dks.kksa ds ;ksxiy ls cm+k gksrk gs f=khkqt,dek=k,slk cgqhkqt gs ftlesa var% dks.kksa dk ;ksxiy ckg~; dks.kksa ds ;ksxiy dk vk/k gksrk gs prqhkqzt,dek=k,slk cgqhkqt gs ftlesa ckg~; dks.kksa dk ;ksxiy,oa vr% dks.kksa dk ;ksxiy leku gksrk gs f=khkqt (TRINGL) f=khkqt (Triangle) - rhu js[kk[k.mksa ls cuh,d can T;kfefr; vkd`fr dks f=khkqt dgrs gsa,d f=khkqt esa rhu Hkqtk, (sides), rhu dks.k (angles),oa rhu 'kh"kz (Vertex) gksrs gsa f=khkqt ds izdkj (Types of triangle) Hkqtkvksa ds vk/kj ij (ccording to side) 1) quilateral Triangle (leckgq f=khkqt) : f=khkqt] ftlds lhkh rhu Hkqtk, cjkcj gksrs gsa] leckgq f=khkqt dgykrk gs Note (1) : leckgq f=khkqt esa izr;sd dks.k leku gksrk gs Note () : leckgq f=khkqt esa izr;sd dks.k dk eki 60 0 gksrk gs ) Isosceles Triangle (lef}ckgq f=khkqt) : f=khkqt] ftlds dksbz nks Hkqtk cjkcj gks] lef}ckgq f=khkqt dgykrk gs Note (3) : lef}ckgq f=khkqt esa nks dks.k cjkcj gksrs gsa Note (4) : ;fn fdlh f=khkqt ds nks Hkqtk, cjkcj gks rks muds foijhr dks.k Hkh cjkcj gksrs gs Note (5) : ;fn fdlh f=khkqt ds nks dks.k cjkcj gksa rks muds foijhr Hkqtk Hkh cjkcj gksrs gsa 3) Scalene Triangle (fo"keckgq f=khkqt) : f=khkqt ftlds lhkh rhu Hkqtk vleku gks] fo"keckgq f=khkqt dgykrk gs Note (6) : fo"keckgq f=khkqt ds lhkh rhu dks.k vleku gksrs gsa Note (7) : ;fn fdlh f=khkqt ds nks Hkqtk vleku gks rks cm+h Hkqtk dk leq[k dks.k cm+k,oa NksVh Hkqtk dk leq[k dks.k NksVk gksrk gs Note (8) : ;fn fdlh f=khkqt ds nks dks.k vleku gks rks cm+s dks.k dh leq[k Hkqtk cm+k,oa NksVs dks.k dh leq[k Hkqtk NksVk gksrk gs f=khkqt ds izdkj (dks.k ds vuqlkj) Types of Triangle (ccording to angle) 1) cute-angled Triangle (U;wudks.k f=khkqt) : ;fn fdlh f=khkqt ds lhkh dks.k U;wudks.k gks rks f=khkqt U;wudks.k f=khkqt dgykrk gs MTHMTIS Y- RHUL RJ 9 GOMTRY & MNSURTION

9 Note (9) : U;wudks.k f=khkqt esa fdlh Hkh nks dks.kksa ds ;ksxiy 90 0 ls cm+k gksrk gs Note (10) : U;wudks.k f=khkqt esa :- c < a + b (tgk a, b vksj c Hkqtkvksa dh yckbz gs,oa c lcls cm+h Hkqtk gs) ) Right-angled Triangle (ledks.k f=khkqt) : ;fn fdlh f=khkqt dk dksbz,d dks.k ledks.k gks rks og ledks.k f=khkqt dgykrk gs Note (11) : ledks.k f=khkqt ds vu; nks dks.kksa dk ;ksxiy 90 0 ds cjkcj gksrk gs Note (1) : ;fn fdlh f=khkqt ds nks dks.kksa dk ;ksxiy rhljs dks.k ds cjkcj gks rks f=khkqt ledks.k f=khkqt gksxk Note (13) : ledks.k f=khkqt esa :- c = a + b, tgk a, b vksj c Hkqtkvksa dh yckbz gs,oa c lcls cm+h Hkqtk dh yckbz gs 3) Obtuse-angled Triangle (vf/d dks.k f=khkqt): ;fn fdlh f=khkqt dk dksbz,d dks.k vf/d dks.k gks rks og vf/d dks.k f=khkqt dgykrk gs Note (14) : ;fn fdlh f=khkqt esa dksbz nks dks.kksa dk ;ksxiy 90 0 ls de gks rks og vf/d dks.k f=khkqt gksxk Note (15) : vf/d dks.k f=khkqt esa :- c > a + b, tgk a, b vksj c rhuksa Hkqtkvksa dh yckbz gs vksj c lcls cm+h Hkqtk gs (f=khkqt ls tqm+s in ) Terms related to Triangle 1) Median (ekfè;dk) : f=khkqt ds 'kh"kz,oa leq[k Hkqtk ds eè; fcunq dks feykus okyh js[kk dks ekfè;dk dgrs gsa,d f=khkqt esa rhu ekfè;dk gksrk gs Note (17) : lef}ckgq f=khkqt esa nks leku dks.kksa ds 'kh"kksz ls [khaph xbz ekfè;dk dh yckbz leku gksrk gs vr% lef}ckgq f=khkqt dh nks ekfè;dk leku yckbz dh gksrh gs Note (18) : lef}ckgq f=khkqt esa vleku dks.k ds 'kh"kz ls foijhr Hkqtk ij [khaph xbz ekfè;dk foijhr Hkqtk ij yc Hkh gksrk gs Note (19) : lef}ckgq f=khkqt esa vleku dks.k ds 'kh"kz ls foijhr Hkqtk ij [khaph xbz ekfè;dk 'kh"kzdks.k dks lef}hkkftr djrk gs = Note (0) : fo"keckgq f=khkqt dh lhkh rhu ekfè;dk dh yackbz vleku gksrk gs Note (1) : fdlh Hkh f=khkqt esa ekfè;dk ges'kk f=khkqt ds vunj gksrk gs Note () : f=khkqt ds lhkh rhu ekfè;dk,d fcunqxkeh (concurrent) gksrk gs bldk eryc ;g gs fd lhkh rhu ekfè;dk,d nwljs dks fdlh,d gh fcunq ij dkvrh gs Note (3) : ledks.k f=khkqt esa ledks.k okys 'kh"kz ls d.kz (Hypotenuse) ij [khaph xbz ekfè;dk d.kz ds vk/k ds cjkcj gksrk gs ;fn = rks ekfè;dk gs Note (16) : leckgq f=khkqt dh lhkh rhu ekfè;dkvksa dh yckbz leku gksrh gs MTHMTIS Y- RHUL RJ 10 GOMTRY & MNSURTION

10 vkok, = 1 ;fn fdlh f=khkqt esa ekfè;dk vius laxr Hkqtk dh vk/h gks rks f=khkqt ledks.k f=khkqt gksxk gksj ekfè;dk d.kz ij [khaph xbz gksxh ) entroid (dsuæd ;k xq:ro dsuæ) : f=khkqt dh rhuksa ekfè;dk ges'kk,d nwljs dks,d gh fcunq ij izfrnsn djrh gs bls izfrnsn fcunq dks dsunzd dgrs gsa dsunzd yc gs,oa ekfè;dk Hkh gs Note (7) : lef}ckgq f=khkqt esa leku dks.kksa okys 'kh"kksza ls leku Hkqtkvksaa ij [khapk x;k] nksuksa 'kh"kz yac dh yckbz leku gksrk gs,oa vleku dks.k ls vleku Hkqtk ij [khapk x;k 'kh"kz yac ekfè;dk Hkh gksrk gs,oa dks.k dk v¼zd Hkh gksrk gs Note (4) : dsunzd ekfè;dk dks : 1 ds vuqikr esa fohkkftr djrk gs G G : G = : 1 dsunzd 3) ltitude / erpendicular / Height (m yc) : 'kh"kz ls leq[k Hkqtk dks tksm+us okyh ljy js[kk tks Hkqtk ds lkk 90 0 dk dks.k cuk;s yc dgykrk gs ;fn = than = dks.k dk v¼zd,oa ij ekfè;dk gs Note (8) : fo"keckgq f=khkqt esa rhuksa yc vleku yckbz gksrk gs Note (9) : U;wudks.k f=khkqt esa lhkh rhu yc f=khkqt ds vunj gksrk gs Note (30) : ledks.k f=khkqt esa ledks.k cukus okyh nks Hkqtk gh yc gksrk gs vksj ledks.k cukus okyh 'kh"kz ls d.kz ij Mkyk x;k yc f=khkqt ds vunj gksrk gs Hkqtk dk yc gs Note (5) : leckgq f=khkqt dk lhkh rhu ycksa dh yckbz leku gksrk gs Note (6) : leckgq f=khkqt esa fdlh Hkqtk ij 'kh"kz ls Mkyk x;k yc ml Hkqtk dk ekfè;dk Hkh gksrk gs, vksj yc gsa MTHMTIS Y- RHUL RJ 11 GOMTRY & MNSURTION

11 Note (31) : vf/d dks.k f=khkqt esa nksuksa U;wudks.k okys 'kh"kz ls [khapk x;k yc f=khkqt ds ckgj gksrk gs tcfd vf/d dks.k okys 'kh"kz ls [khapk x;k yc f=khkqt ds vunj gksrk gs Note (36) : leckgq f=khkqt esa dsunzd rkk yacdsunz,d gh fcunq gksrk gs Note (37) : lef}ckgq f=khkqt esa dsunzd rkk yc dsunz nks fofhkuu fcunq gksrk gs tks fd vleku dks.k ds 'kh"kz ls vleku Hkqtk ij Mkys x;s yac ;k ekfè;dk ij flkr gksrk gs ycdsunz dsunzd yc, vksj rhuksa Hkqtkvksa Øe'k%, vksj ij [khapk x;k gs Note (3) : lcls cm+h Hkqtk ij lcls NksVk 'kh"kzyc,oa lcls NksVh Hkqtk ij lcls cm+k 'kh"kzyac gksrk gs Note (33) : 'kh"kz ls leq[k Hkqtk dks tksm+us okyh lhkh js[kkvksa esa yco~r js[kk lcls NksVk gksrk gs Note (34) : rhuksa 'kh"kz yac,d fcunqxkeh (concurrent) gksrk gs 4) Orthocentre (ycdsuæ) : rhuksa 'kh"kzyac,d&nwljs dks,d gh fcunq ij dkvrs gsa vksj ;g izfrnsn fcunq ^yc dsunz* dgykrk gs O ycdsunz Note (35) : fdlh Hkqtk ds }kjk ycdsunz ij cuk;k x;k dks.k foijhr dks.k ds liwjd (supplementary) gksrk gs Note (38) : lef}ckgq f=khkqt esa 'kh"kz] dsunzd rkk ycdsunz rhu lajs[k fcunq gksrk gs Note (39) : ycdsunz,oa ekfè;dk dks tksm+usokyh js[kk ;fn fdlh Hkqtk ds lkk 90 0 dk dks.k cuk;s ;k Hkqtk dks lef} Hkkftr djs rks f=khkqt] lef}ckgq gksxk Note (40) : fo"keckgq f=khkqt esa 'kh"kz] dsunz rkk yc dsunz rhu vlajs[k fcunq gksrk gs Note (41) : U;wudks.k f=khkqt esa yacdsunz f=khkqt ds vunj gksrk gs Note (4) : ledks.k f=khkqt esa ycdsunz ledks.k cukus okyk 'kh"kz gksrk gs ycdsunz Note (43) : vf/ddks.k f=khkqt esa ycdsunz f=khkqt ds ckgj gksrk gs x O y ycdsunz x + y = O ycdsunz MTHMTIS Y- RHUL RJ 1 GOMTRY & MNSURTION

12 5) ngle isector (dks.k v¼zd) : 'kh"kz ls leq[k Hkqtk dks tksm+us okyh js[kk tks 'kh"kzdks.k dks nks cjkcj Hkkxkssa esa ck V ns dks.k v¼zd dgykrk gs Note (50) : fdlh Hkqtk ds }kjk var% dks.k ij cuk;k x;k dks.k 90 + foijhr dks.k dk vk/k gksrk gs I var% dsunz Note (44) : leckgq f=khkqt esa lhkh rhu v¼zd leku yackbz ds gksrs gsa Note (45) : leckgq f=khkqt esa] dks.k v¼zd] yc,oa ekfè;dk,d gh js[kk gksrk gs Note (46) : lef}ckgq f=khkqt esa nks leku dks.kksa dk dks.k v¼zd yckbz esa leku gksrk gs vksj vleku dks.k dk v¼zd foijhr Hkqtk ij yc,oa ekfè;dk gksrk gs f=khkqt esa, =,oa, vksj v¼zd gs rks = Note (47) : fo"keckgq f=khkqt esa lhkh rhu v¼zd vleku yckbz ds gksrs gsa Note (48) : v¼zd ges'kk f=khkqt ds vunj flkr gksrk gs Note (49) : lhkh rhu v¼zd,dfcunqxkeh (concurrent) gksrk gs 6) Incentre (var% dsuæ) : lhkh rhu v¼zdksa dk izfrnsn fcunq var% dsunz dgykrk gs 1 I = 90 + Note (51) : var% dsunz rhuksa Hkqtkvksa ls leku nwjh ij flkr gksrk gs I I = I = I var% dsunz Note (5) : leckgq f=khkqt esa dsunzd] yc dsunz rkk var% dsunz,d gh fcunq gksrk gs Note (53) : lef}ckgq f=khkqt esa dsunzd] ycdsunz rkk var% dsunz rhu fofhkuu fcunq gksrk gs tks fd vleku dks.k okys 'kh"kz ls foijhr Hkqtk ij [khaps x;s yc ;k ekfè;dk ;k v¼zd ij flkr gksrk gs Note (54) : lef}ckgq f=khkqt esa dsunzd] ycdsunz rkk var% dsunz rhu fofhkuu lajs[k (collinear) fcunq gksrk gs Note (55) : fo"kekckgq f=khkqt esa dsunzd] ycdsunz rkk var% dsunz rhu fofhkuu vlajs[k (Non-collinear) fcunq gksrk gs Note (56) : fdlh Hkh f=khkqt esa var% dsunz f=khkqt ds vanj gksrk gksrk gs 7) Incircle (var%o`ùk ) : var% o`ùk,d,slk o`ùk gs tks f=khkqt ds vunj bl izdkj flkr gksrk gs rkfd og rhuksa Hkqtkvksa dks Li'kZ dj lds,oa bl o`ùk dk dsunz f=khkqt dk var% dsunz gksrk gs var%dsunz var%dsunz MTHMTIS Y- RHUL RJ 13 GOMTRY & MNSURTION

13 Note (57) : var%f=kt;k = f=khktq dk {k=s kiy f=khktq dk v¼iz fjeki ifjdsunz 8) erpendicular isector (yc lef}hkktd) : ;fn dksbz ljy js[kk f=khkqt ds Hkqtk ds eè; fcunq gksdj xqtjs,oa Hkqtk ds lkk 90 0 dk dks.k cuk;s rks ml ljy js[kk dks yc lef} Hkktd dgrs gsa Note (58) : leckgq f=khkqt esa lhkh rhu yc lef}hkktdksa dh yckbz leku gksrh gs Note (59) : leckgq f=khkqt esa yc lef}hkktd] ekfè;dk] yc,oa v¼zd,d gh js[kk esa gksrk gs Note (60) : lef}ckgq f=khkqt esa leku Hkqtkvksa ij [khapk x;k yc lef}hkktd yckbz esa leku gksrk gs = G Note (61) : lef}ckgq f=khkqt esa leku Hkqtkvksa dk yc lef} Hkktd 'kh"kz gksdj ugha xqtjrk gs Note (6) : lef}ckgq f=khkqt esa vleku Hkqtk ij [khapk x;k yc lef}hkktd ml Hkqtk dk yc] dks.k v¼zd,oa ekfè;dk Hkh gksrk gs Note (63) : fo"keckgq f=khkqt esa lhkh rhu yc lef}hkktd yckbz esa vleku gksrk gs Note (64) : fo"keckgq f=khkqt esa lhkh rhu yc lef}hkktd 'kh"kz gksdj ugha xqtjrk gs Note (65) : lhkh rhu yc lef}hkktd,d fcunqxkeh (concurrent) gksrk gs 9) ircumcentre ( ifjdsuæ ) : f=khkqt ds lhkh rhu yc lef} Hkktd,d nwljs dks,d fuf'pr fcunq ij izfrnsn djrs gsa bl izfrnsn fcunq dks f=khkqt dk ifjdsunz dgrs gsa G Note (66) : ifjdsunz rhuksa 'kh"kksza ls leku nwjh ij flkr gksrk gs O ifjdsunz O = O = O Note (67) : fdlh Hkqtk }kjk ifjdsunz ij cuk;k x;k dks.k leq[k dks.k ds nqxquk gksrk gs O x x ifjdsunz Note (68) : leckgq f=khkqt esa dsunzd] ycdsunz] var%dsunz rkk ifjdsunz,d gh fcunq gksrk gs Note (69) : lef}ckgq f=khkqt esa dsunzzd] ycdsunz] var% dsunz rkk ifjdsunz pkj vyx&vyx fcunq gksrk gs tks,d gh js[kk[k.m ij flkr gksrk gs vksj og js[kk[k.m vleku Hkqtk dk yc ;k ekfè;dk gksrk gs Note (70) : lef}ckgq f=khkqt esa dsunzd] ycdsunz] var%dsunz rkk ifjdsunz pkj lajs[k fcunq gksrk gs Note (71) : fo"keckgq f=khkqt esa dsunzd] ycdsunz] var% dsunz rkk ifjdsunz pkj valjsa[k fcunq gksrk gs Note (7) : U;wudks.k f=khkqt esa ifjdsunz f=khkqt ds vunj gksrk gs Note(73): ledks.k f=khkqt esa ifjdsunz d.kz dk eè; fcunq gksrk gs Note (74) : vf/d dks.k f=khkqt esa ifjdsunz f=khkqt ds ckgj gksrk gs MTHMTIS Y- RHUL RJ 14 GOMTRY & MNSURTION

14 Note(75): ifjdsunz vksj fdlh Hkqtk ds eè; fcunq dks feykus okyh js[kk Hkqtk ij yc gksrk gs] vkok ifjdsunz ls fdlh Hkqtk ij Mkyk x;k yc Hkqtk dks lef}hkkftr djrk gs 10) ircumcircle (ifjo`ùk) :,d o`ùk tks fdlh f=khkqt ds rhuksa 'kh"kksza ls gksdj xqtjrh gs,oa mldk dsunz f=khkqt dk ifjdsunz gksrk gs og o`ùk ml f=khkqt dk ifjo`ùk dgykrk gs (f=khkqtksa dh lokzaxlerk ) ongruence of Triangle nks f=khkqt lokzaxle dgykrk gs] ;fn os vkdkj (shape) vksj eki (size) esa leku gksa ;k] nks f=khkqt lokzaxle gksaxs ;fn vksj dsoy ;fn,d f=khkqt dks nwljs f=khkqt ij bl izdkj vè;kjksfir fd;k tk lds rkfd ;g f=khkqt nwljs f=khkqt dks iw.kz #i ls < d ysa Note (76) : ifjo`ùk dh f=kt;k = rhukas Hkqtkvksa dk x.q kuiy 4 f=khktq dk {ks=kiy Note (77) : ledks.k f=khkqt esa ifjf=kt;k d.kz dk vk/k gksrk gs Note (78) : ledks.k f=khkqt esa d.kz (Hypotenuse) ml f=khkqt ds ifjo`ùk dk O;kl gksrk gs Note (79) : leckgq f=khkqt esa& (i) var%f=kt;k % ifjf=kt;k = 1 : (ii) var% o`ùk dk {ks=kiy % ifjo`ùk dk {ks=kiy = 1 : 4 (iii) var% f=kt;k = (iv) ifjf=kt;k = a a 3 3 (v) Nk;kafdr Hkkx dk {ks=kiy % vuknkfnr Hkkx dk {ks=kiy = 1 : 3 ;fn dks ij bl izdkj j[kk x;k fd ds 'kh"kz ds 'kh"kz ij bl Øe esa im+s&,, (i) (ii) (iii) (iv) Rkks fuufyf[kr Ng lekurk izkir gksxh & =, =, = (laxr Hkqtk leku gksrk gs ) =, =, = (laxr dks.k leku gksrk gs) (laxr Hkqtk) orresponding sides = nks f=khkqtksa esa leku dks.kksa dh leq[k Hkqtkvksa dks laxr Hkqtk dgrs gsa (laxr dks.k) orresponding angles)= nks f=khkqtksa esa leku Hkqtkvksa ds leq[k dks.kksa dks laxr dks.k dgrs gsa ;fn gs rks =, =, = vksj =, =, = ;fn vksj R rks R (nks f=khkqtksa esa lokzaxlerk ds fy, i;kzir izfrcu/ dh dlksfv;k ) (Sufficient onditions (riteria) for ongruence of Triangles) 1) Hkqtk&Hkqtk&Hkqtk (S-S-S) lokzaxlerk izes;: ;fn,d f=khkqt dh rhu Hkqtk, nwljs f=khkqt dh Øe'k% rhu Hkqtkvksa ds cjkcj gksa rks os nksuksa f=khkqt lokzaxle gksrs gsa (i) MTHMTIS Y- RHUL RJ 15 GOMTRY & MNSURTION

15 ;fn =, = & = Rkks x ;fn =, = & =, rks. 7 x 0 (ii) (a) ( lgh ) (b) ( lgh ) (c) ( xyr ) (d) ( xyr ) 6 5 (iii) (a) ( lgh ) (b) ( xyr ) (c) ( xyr ) (d) ( lgh ) (e) ( lgh ) (iii) = LM, R = MN & R = LN (a) R LMN ( lgh ) (b) R LNM ( lgh ) (c) R MNL ( lgh ) (d) R LMN ( xyr ) ) Hkqtk&dks.k&Hkqtk (S--S) lokzaxlerk izes; : nks f=khkqt lokzaxle gksrs gsa ;fn vksj dsoy ;fn,d f=khkqt dh nks Hkqtk, rkk muds vurxzr dks.k] nwljs f=khkqt ds rnuq:ih nksuksa Hkqtkvksa rkk muds vurxzr dks.k ds cjkcj gksa 5 x 0 7 mijksr nksuksa f=khkqt lokzaxle ugha gs ;kasfd varxzr dks.k cjkcj gksuk pkfg, (iv) x 0 = ST, R = TM & = T Rkks R STM 3) dks.k&hkqtk&dks.k (-S-) lokzaxlerk izes; : ;fn,d f=khkqt ds nks dks.k vksj mudh varfjr Hkqtk Øe'k% nwljs f=khkqt ds nks laxr dks.kksa vksj mudh varfjr Hkqtk ds cjkcj gksa rks os f=khkqt lokzaxle gksrs gsa 5 7 (i) ;fn =, = vksj = Rkks] MTHMTIS Y- RHUL RJ 16 GOMTRY & MNSURTION

16 4) dks.k&dks.k&hkqtk (--S) lokzaxlerk izes; : ;fn,d f=khkqt ds nks dks.k vksj,d Hkqtk (tks dks.kksa ds varxzr u gksa) Øe'k% nwljs f=khkqt ds laxr dks.kksa vksj Hkqtk ds cjkcj gksa] rks os f=khkqt lokzaxle gksrs gsa ;fn =, = vksj = Rkks] 5) ledks.k&d.kz&hkqtk (R-H-S) lokzaxlerk izes; : ;fn,d ledks.k f=khkqt dk d.kz vksj,d Hkqtk nwljs ledks.k f=khkqt ds Øe'k% d.kz vksj laxr Hkqtk ds cjkcj gks] rks os ledks.k f=khkqt lokzaxle gksrs gsa f=khkqtksa dh le:irk (Similarity Of Triangles) nks f=khkqt le:i dgykrs gsa ;fn os vkdkj (shape) esa leku gksa ysfdu dksbz vko';d ugha gs fd muds eki (size) Hkh leku gksa ;k nks f=khkqt le:i dgykrs gsa ;fn muds laxr dks.k cjkcj gksa,oa laxr Hkqtk lekuqikrh gks ;fn =, =, = vksj = = rks f=khkqt dh le:irk ds fy, dlksfv;k (riteria Of Similarity ) 1) dks.k&dks.k ( ) /dks.k&dks.k&dks.k ( ) : ;fn fdlh f=khkqt ds nks dks.k] nwljs f=khkqt ds nks dks.kksa ds cjkcj gksa rks nksuksa f=khkqt le:i gksrs gsa =, = vksj = = 90 0 Rkks] lokzaxlerk ls lacaf/r izes; (roperties Related To Triangles) Note-1 : ;fn nks f=khkqt lokzaxle gksaxs rks muds laxr Hkqtk leku gksxk Note- : ;fn nks f=khkqt lokzaxle gksaxs rks muds laxr dks.k leku gksxk Note-3 : ;fn nks f=khkqt lokzaxle gksxk rks os leku dks.k okyk gksxk ysfdu ;fn nks f=khkqt leku dksk okyk gksxk rks dkbz t:jh ugha gs fd os lokzaxle gksxk Note-4 : ;fn nks f=khkqt lokzaxle gksxk rks mudk {ks=kiy vksj ifjeki leku gksxk Note-5 : ;fn nks f=khkqt lokzaxle gksxk rks mlds lhkh laxr Hkkx cjkcj gksxk MTHMTIS Y- RHUL RJ 17 ;fn =, = & = rks. ) Hkqtk&Hkqtk&Hkqtk (S S S) : ;fn nks f=khkqtksa dh laxr Hkqtk, lekuqikrh gks rks nksuksa f=khkqt le:i gksxk (i) ;fn = =, rks GOMTRY & MNSURTION

17 (ii) f=khkqt ds le:irk ij vk/kfjr xq.k ( roperties related to Similarity ) (iii) = =, rks Note-1 : ;fn nks f=khkqt le:i gksaxsa rks mudh laxr Hkqtk lekuqikrh gksxk ;fn rks = = Note- : ;fn nks f=khkqt le:i gksaxs rks muds lhkh laxr Hkkx (dks.k dks NksM+dj) lekuqikrh gksaxs bldk eryc muds laxr Hkqtk dk vuqikr = laxr ekfè;dk dk vuqikr = laxr m pkbz dk vuqikr = laxr dks.k v¼zd dk vuqikr = laxr yc lef}hkktd dk vuqikr Note- 3 : ;fn nks f=khkqt le:i gksaxs rks os dksf.kd gksaxs vksj ;fn nks f=khkqt leku&dksf.kd gksaxs rks os le:i gksaxs ;fn rks =, = & = = = = =, rks 3) Hkqtk&dks.k&Hkqtk (S S) : ;fn,d f=khkqt dk,d dks.k nwljs f=khkqt ds,d dks.k ds cjkcj gks rkk bu dks.kksa dks varxzr djus okyh Hkqtk, lekuqikrh gksa] rks nksuksa f=khkqt le:i gksrs gsa (i) Note- 4 : ;fn nks f=khkqt le:i gksaxs rks muds ifjeki dks vuqikr muds laxr Hkqtk ds vuqikr ds leku gksxk Note-5 : ;fn nks f=khkqt le:i gksaxs rks muds {ks=kiy dk vuqikr muds laxr Hkqtk ds vuqikr ds oxz ds cjkcj gksxk Note-6 : fdlh f=khkqt ds nks Hkqtkvksa dks tksm+us okyh js[kk tks rhljs Hkqtk ds lekarj gks og f=khkqt dks nks Hkkx esa ck Vrh gs vksj,d u;k f=khkqt cukrh gks rks fd oklrfod f=khkqt ds le:i gksrh gs ;fn rks nks f=khkqt vksj esa (ii) = vksj = rks = vksj = Rkks = = = Rkks = = MTHMTIS Y- RHUL RJ 18 GOMTRY & MNSURTION

18 Note-7 : (ewyhkwr lekuqikfrdrk izes; dk ksyl izes;) (asic proportionality theorem or Thales theorem)- ;fn fdlh f=khkqt dh,d Hkqtk ds lekukurj vu; nks Hkqtkvksa dks fhkuu&fhkuu fcunqvksa ij izfrnsn djus ds fy, js[kk [khaph tk, rks vu; nks Hkqtk,,d gh vuqikr esa fohkkftr gks tkrh gsa Note 8 : ;fn rks = Note-10 : fdlh f=khkqt ds,d Hkqtk ds eè; fcunq ls xqtjus okyh js[kk tks nwljs Hkqtk ds lekukurj gks] og rhljs Hkqtk dks lef} Hkkftr djrh gs ;fn vksj Hkqtk dk eè; fcunq gks rks = Note-11 : f=khkqt ds fdlh nks Hkqtkvksa ds eè; fcunqvksa dks tksm+us okyh js[kk rhljh Hkqtk dh vk/h,oaa lekukurj gksrh gs ;fn rks, (i) (ii) = (iii) (iv) = (iv) = = ;fn vksj,oa dk eè; fcunq gks rks (i) (ii) = 1 (iii) dk {ks=kiy : dk {ks=kiy = 1 : 4 (iv) dk {ks=kiy : dk {ks=k0 = 1 : 3 (vi) = (vii) = Note-9 : (ksyl izes; dk foykse& onverse of the basic proportionally theorem) ;fn,d js[kk fdlh f=khkqt dh nks Hkqtkvksa dks,d gh vuqikr esa fohkkftr djs] rks og rhljh Hkqtk ds lekarj gksrh gs ;fn = rks (v) dk {ks=kiy : dk {ks=k0 = 4 : 3 Note-1 : ;fn ledks.k f=khkqt esa ledks.k okys 'kh"kz ls d.kz (Hypotenuse) ij yc Mkyk tkrk gs rks f=khkqt nks Hkkxksa esa c V tkrk gs vksj nks u;k le:i f=khkqt izkir gksrk gs tks fd ewy f=khkqt ds Hkh le:i gksrk gs x 0 y x y 0 MTHMTIS Y- RHUL RJ 19 GOMTRY & MNSURTION

19 (a) (i) = x (b) (c) (ii) (iii) (i) (ii) (iii) (i) (ii) = x = x = = = = + = Note-13 : x,oa y ehvj ds nks iksy,d nwljs ls p ehvj dh nwjh ij flkr gs (x > y),d iksy ds 'kh"kz ls nwljs Ikksy ds ikn,oa nwljs iksy ds 'kh"kz ls igys Ikksy ds ikn dks tksm+us okyh js[kk ds izfrnsn fcunq dh m pkbz z ehvj gks rks x, y,oa z ds chp laca/ gksxk & (i) = + (ii) z = z x y xy x+ y Note-15 : 'kh"kz,oa,d ekfè;dk ds eè; fcunq dks tksm+us okyh js[kk rhljs Hkqtk dks 1 : ds vuqikr ck Vrh gs ;fn ekfè;dk gks,oa, dk eè; fcunq gks rks (i) : = 1 : (ii) = 1 3 Note-16 : fdlh f=khkqt ds,d dks.k dk vkarfjd f}hkktd foijhr Hkqtk dks vkarfjd :i ls mlh vuqikr esa fohkkftr djrk gs tks ml dks.k dks cukus okyh Hkqtk dk vuqikr gksrk gs = x z y Note-17 : ;fn fdlh f=khkqt ds,d dks.k dks ck Vus okyh js[kk foijhr Hkqtk dks mlh vuqikr esa ck Vs tks ml dks.k dks cukus okyh js[kk ds chp vuqikr gks rks og js[kk ml dks.k dk f}hkktd gksrk gs Note-14 : leyc prqhkqzt ds fod.kz prqhkzt dks pkj f=khkqt esa ck Vrs gsa lekurj Hkqtkvksa ls tqm+s nks f=khkqt le:i gksrs gsa tcfd vlekarj Hkqtkvksa ls tqm+s nks f=khkqt {ks=kiy esa leku gksrs gsa Note-18 : fdlh dks.k dk ckg~; f}hkktd foijhr Hkqtk dks ckg~; :i ls mlh vuqikr esa fohkkftr djrh gs tks ml dks.k dks cukus okyh Hkqtkvksa ds chp vuqikr gksrk gs o ;fn rks (i) O O & (ii) O dk {ks=kiy = O dk {ks=kiy MTHMTIS Y- RHUL RJ 0 = Note -19 : ;fn rhu ;k rhu ls vf/d lekarj js[kk, nks fr;zd ds }kjk izfrnsn gksrk gks rks muds }kjk cuk;k x;k var%[k.m lekuqikrh gksrk gs GOMTRY & MNSURTION

20 6) f=khkqt ds fdlh nks dks.kksa ds ckg~; f}hkktd ds }kjk cuk;k x;k dks.k 90 0 rhljs dks.k ds vk/k ds cjkcj gksrk gs R G = R G Note -0 : f=khkqt ds rhu Hkqtkvksa ds eè; fcunqvksa dks feykus ij pkj lokzaxle f=khkqt curk gs ftuesa ls izr;sd f=khkqt ewy f=khkqt ds le:i gksrk gs O 1 O = 90 7) fdlh f=khkqt ds,d dks.k ds vkarfjr f}hkktd,oa nwljs dks.k ds ckg~; f}hkktd ds }kjk cuk;k x;k dks.k rhljs dks.k dk vk/k gksrk gs O f=khkqt ds xq.k (roperties Related To Triangle) 1) fdlh f=khkqt ds rhuksa dks.kksa dk ;ksxiy gksrk gs ) fdlh f=khkqt ds nks Hkqtkvksa dk ;ksxiy rhljs ls cm+k gksrk gs 3) fdlh f=khkqt ds nks Hkqtkvksa dk vurj rhljs ls NksVk gksrk gs 4) ;fn fdlh f=khkqt ds,d Hkqtk dks c<+k;k tk; rks mlls cuk ckg~; dks.k mlds foijhr nks var%dks.kksa ds ;ksxiy ds cjkcj gksrk gs ) f=khkqt ds fdlh nks dks.kksa ds vkarfjr f}hkktd ds }kjk cuk;k x;k dks.k rhljs dks.k ds vk/k ds cjkcj gksrk gs 1 O = 8) fdlh f=khkqt ds,d dks.k ds vkarfjd f}hkktd,oa mlh dks.k ls foijhr Hkqtk ij Mkys x;s yc ds }kjk 'kh"kz ij cuk;k x;k dks.k vu; nks dks.kksa ds vurj ds vk/k gksrk gs ;fn vksj dks.k dk f}hkktd gks rks 1 = ( ) 9) fdlh f=khkqt dk,d var% dks.k,oa mlh Hkqtk ij cuk ckg~; dks.k dk ;ksxiy mlh Hkqtk ij foijhr dks.k ds v¼zd ds } kjk mlh vksj cuk;sa x;s dks.k dk nqxquk gksrk gs O 1 O = 90 + x z y x + y = z MTHMTIS Y- RHUL RJ 1 GOMTRY & MNSURTION

21 10) f=khkqt ds fdlh nks Hkqtkvksa dk ;ksxiy rhljs ij [khaps x;s ekfè;dk ds nqxqus ls cm+k gksrk gs 16) y + > 11) fdlh f=khkqt dk ifjeki mlds rhuksa ekfè;dkvksa ds ;ksxiy ls cm+k gksrk gs 1) f=khkqt ds rhuksa m pkbz;ksa dk ;skxiy mlds rhuksa Hkqtkvksa ds ;ksxiy ls NksVk gksrk gs 13) f=khkqt ds rhuksa ekfè;dk f=khkqt dks leku {ks=kiy okys N% NksVs f=khkqt esa ck Vrh gs x z x+y+z 17) fdlh leckgq f=khkqt ds vunj flkr fdlh fcunq ls rhuksa Hkqtkvksa dh ycor~ nwjh dk ;ksxiy ml f=khkqt ds m pkbz ds cjkcj gksrk gs O R leckgq f=khkqt dh m pkbz = O + O + OR 14) dsunzd,oa fdlh nks Hkqtkvksa ds eè; fcunqvksa ds }kjk cuk;k x;k f=khkqt dk {ks=kiy ewy f=khkqt dk 1 xq.kk gksrk gs 1 o dsunzd O dk {ks=kiy = 1 x dk {ks=kiy 1 15) 'kh"kz ls foijhr Hkqtk dks tksm+us okyh lhkh js[kk, vu; nks Hkqtkvksa ds eè; fcunqvksa dks tksm+us okyh js[kk ds }kjk lef}hkkftr gksrh gs 18) leckgq f=khkqt esa & (i) Hkqtk % m pkbz = : 3 (ii) ( Hkqtk ) : ( m pkbz ) = 4 : 3 (iii) 3 x ( Hkqtk ) = 4 x ( m pkbz ) 19) f=khkqt ds,d 'kh"kz ls foijhr Hkqtk dks tksm+us okyh js[kk f=khkqt dks nks Hkkxksa esa ck Vrh gs vksj bu nksuksa f=khkqt ds {ks=kiy dk vuqikr muds vk/kj ds vuqikr ds cjkcj gksrk gs dk {k=s kiy dk {k=s kiy = 0) ekfè;dk f=khkqt dks nks leku {ks=kiy okys f=khkqt esa ck Vrh gs 1) leku vk/kj,oa leku lekarj js[kkvksa ds chp flkr nks f=khkqt {ks=kiy esa leku gksrs gsa MTHMTIS Y- RHUL RJ GOMTRY & MNSURTION

22 R S (ii) vf/d dks.k (Obtuse ngle) f=khkqt esa ;fn RS rks dk {ks=k0 = dk {ks=k0 ),d gh vk/kj,oa leku lekarj js[kkvksa ds chp flkr f=khkqt dk {ks=kiy lekarj prqhkqzt ds {ks=kiy dk vk/k gksrk gs = + + x (iii) U;wudks.k (cute ngle) f=khkqt esa dk {ks=k0 = 1 x lekarj dk {ks=k0 3) leku ifjeki okys nks f=khkqt esa lekckgq f=khkqt dk {ks=kiy vf/d gksrk gs (iv) = + x dk {ks=k0 > dk {ks=k0 4),d gh o`ùk ds vurxzr nks f=khkqt esa leckgq f=khkqt dk {ks=kiy vf/d gksrk gs 5) ikbkkxksjl izes; (ythagoras Theorem ) (i) ledks.k f=khkqt (Right ngle Triangle) esa & (v) ;fn ekfè;dk gks rks + = ( + ),,oa ekfè;dk gsa rks 3( + + ) = 4 ( + + ) = + MTHMTIS Y- RHUL RJ 3 (vi) ledks.k f=khkqt esa U;wudks.k okys nksuksa 'kh"kksza ls [khaps x;s ekfè;dkvksa ds oxksza ds ;ksxiy dk pkj xq.kk] d.kz ds oxz ds ik p xquk ds cjkcj gksrk gs GOMTRY & MNSURTION

23 f=khkqt ls lacaf/r lw=k (ormula Related To Triangle) 1) f=khkqt dk {ks=kiy = 1 x vk/kj x m pkbz 4 ( + ) = 5 (vii) f=khkqt,d ledks.k f=khkqt gs ftlesa dks.k ledks.k gs vksj, rkk ij flkr nks fcunq gsa rks + = + ) f=khkqt dk {ks=kiy = s(s a)(s b)(s c tgk, s = a + b + c vksj a, b,oa c Hkqtk dh yackbz gs 3) f=khkqt dk {ks=kiy = 4 s(s a)(s b)(s c 3 tgk, s = a + b + c vksj a, b,oa c ekfè;dk dh yackbz gs 4) leckgq f=khkqt dh m pkbz = 3 Hkqtk 5) leckgq f=khkqt dh Hkqtk = 3 m pkbz (viii) asic ythagorean Triplets (3, 4, 5), (5, 1, 13), (7, 4, 5), (8, 15, 17), (9, 40, 41), (11, 60, 61) n n + n + 1 If n = 1 => 1 1 (4) + = + = (3) (3, 4, 5) (5) 6) leckgq f=khkqt dk {ks=kiy = 3 4 ( m p k b Z 7) leckgq f=khkqt dk {ks=kiy = ) 3 Hktq k 8) lef}ckgq f=khkqt dh m pkbz = 1 4b a tgk b leku Hkqtk dh yackbz gs 9) lef}ckgq f=khkqt dk {ks=kiy = a 4 4b a If n = => (1) + = + = (5) (5, 1, 13) + 1 (13) MTHMTIS Y- RHUL RJ 4 GOMTRY & MNSURTION

24 gsa prqhkqzt (uadrilateral ) pkj Hkqtkvksa ls f?kjs T;kferh; vkd`fr dks prqhkqzt dgrs prqhkqzt ls lacaf/r xq.k (roperties Related To uadrilateral ) 7) fdlh prqhkqzt ds eè; fcunqvksa dks tksm+usokyh js[kk ls cuk prqhkqzt dk {ks=kiy ewy prqhkqzt dk vk/k gksrk gs prqhkqzt ds izdkj (Types Of uadrilateral) 1) prqhkqzt ds lhkh var% dks.kksa dk ;ksxiy gksrk gs ) prqhkqzt ds lhkh ckg~; dks.kksa dk ;ksxiy gksrk gs 3) fdlh nks yxkrkj dks.kksa (consecutive angles) ds f}hkktdksa }kjk cuk;k x;k dks.k vu; nks dks.kksa ds ;ksxiy dk vk/k gksrk gs o 1 O = + ( ) 4) prqhkqzt ds fdlh nks foijhr ckg~; dks.kksa dk ;ksxiy vu; nks var% dks.kksa ds ;ksxiy ds cjkcj gksrk gs p r s r + s = p + q 5) prqhkqzt ds fdlh nks lfuudv Hkqtkvksa ds eè;fcunqvksa dks tksm+usokyh js[kk laxr fod.kz ds lekarj vksj mldk vk/k gksrk gs q & = 1 6) fdlh prqhkqzt ds eè;fcunqvksa dks tksm+usokyh js[kk ls cuk prqhkqzt lekarj prqhkqzt gksrk gs 1) arallelogram ( lekarj prqhkqzt ) ) Rectangle ( vk;r ) 3) Square ( oxz ) 4) Rhombus ( leprqhkqzt ) 5) Trapezium ( leyc prqhkzt ) lekarj ;k lekukarj prqhkqzt (arallelogram ),slk prqhkqzt ftlds foijhr Hkqtk dk nksuksa ;qxe lekukurj gks] lekukurj prqhkqzt dgykrk gs vksj dksbz prqhkqzt lekukurj prqhkqzt gksxk ;fn fuu esa ls dksbz,d /kj.k djrk gks& 1) foijhr Hkqtk dk izr;sd ;qxe vkil esa lekukurj gks ;k ) foijhr Hkqtk dk izr;sd ;qxe vkil esa leku gks ;k 3) foijhr dks.k dk izr;sd ;qxe vkil esa cjkcj gks ;k 4) foijhr Hkqtk dk,d ;qxe vkil esa cjkcj rkk lekukurj gks ;k 5) fod.kz,d nwljs dks lef}hkkftr djs MTHMTIS Y- RHUL RJ 5 GOMTRY & MNSURTION

25 lekarj prqhkqzt ds xq.k (roperties related to parallelogram) 1) lekarj prqhkqzt ds nksuksa fod.kz,d nwljs dks lef}hkkftr djrs gsa rkk izr;sd fod.kz lekarj prqhkqzt dks nks lokzaxle f=khkqt esa lef}hkkftr djrk gs 6) leku vk/kj,oa leku lekarj js[kkvksa ds chp flkr lekarj prqhkqzt,oa vk;r dk {ks=kiy leku gksrk gs lekarj dk {ks=k0 = vk;r dk {ks=k0 ) lekarj prqhkqzt ds dks.kksa ds v¼zdksa }kjk vk;r curk gs 7) leku vk/kj,oa leku lekarj js[kkvksa ds chp flkr f=khkqt dk {ks=kiy lekarj prqhkqzt ds {ks=kiy ds vk/k gksrk gs 3) fdlh Hkh nks yxkrkj dks.kksa dk ;ksxiy gksrk gs dk {ks=k0 = 1 lekarj dk {ks=k0 + = + = + = + = ) fdlh Hkh nks yxkrkj dks.kksa ds v¼zd,d nwljs dks 90 0 ij dkvrk gs o 8),d o`ùk ds varxzr (Inscribed in-circle) lekarj prqhkqzt ;k rks vk;r ;k oxz gksrk gs 9),d o`ùk ds cfgxzr (circumscribed in-circle) lekarj prqhkqzt ;k rks leprqhkqzt ;k oxz gksrk gs 10) lekarj prqhkqzt ds Hkqtkvksa ds oxz dk ;ksxiy muds fod.kksaz ds oxz ds ;ksxiy ds cjkcj gksrk gs O = ) leku vk/kj,oa leku lekarj js[kkvksa ds chp flkr nks lekarj prqhkqzt dk {ks=kiy leku gksrk gs lekarj dk {ks=k0 = lekarj dk {ks=k0 (i) + = (ii) + = ( + ) 11) lekarj prqhkqzt dk {ks=k0 = vk/kj x m pkbz 1) ;fn fdlh lekarj prqhkqzzt dk fod.kz cjkcj gks rks mldk izr;sd dks.k 90 0 dk gksrk gs bldk eryc ;g vk;r ;k oxz gksxk MTHMTIS Y- RHUL RJ 6 GOMTRY & MNSURTION

26 vk;r (Rectangle ) vk;r,d,slk lekarj prqhkqzt gs ftldk izr;sd dks.k 90 0 gksrk gs vk;r ds xq.k (roperties of Rectangle) 1) foijhr Hkqtk ds nksuksa ;qxe vkil esa cjkcj gksrs gsa = vksj = ) izr;sd dks.k 90 0 ds cjkcj gksrk gs 3) fod.kz cjkcj gksrs gsa 4) fod.kz,d nwljs dks lef}hkkftr djrs gsa ysfdu ycor~ ugha gksrs gsa 5) fod.kz dks.k dk v¼zd ugha gksrk gs 6) vk;r ds eè; fcunqvksa dks tksm+us okyh js[kk leprqhkqzt gksrk gs 7) vk;r dk {ks=kiy = yckbz x pksm+kbz 8) vk;r dk ifjeki = ( yckbz + pksm+kbz ) 9) vk;r dk fod.kz = l + b oxz (Square ) oxz,d,slk lekarj prqhkqzt gksrk gs ftldk izr;sd Hkqtk cjkcj,oa izr;sd dks.k ledks.k gksrk gs = = = oxz ds xq.k (roperties of Square ) 1) izr;sd Hkqtk cjkcj gksrk gs ) izr;sd dks.k 90 0 ds cjkcj gksrk gs 3) fodz.k cjkcj gksrk gs 4) fod.kz,d nwljs dks ycor~ lef}hkkftr djrk gs 5) fod.kz dks.kksa dk v¼zd gksrk gs 6) {ks=kiy = ( Hkqtk ) 7) ifjeki = 4 x Hkqtk 8) fod.kz = Hkqtk 9) oxz ds eè; fcunqvksa dks tksm+us okyh js[kk oxz gksrk gs leprqhkqzt (Rhombus) leprqhkqzt,d,slk lekarj prqhkqzt gksrk ftldk izr;sd Hkqtk cjkcj gksrk gs = = = leprqhkqzt ds xq.k (roperties of Rhombus) 1) izr;sd Hkqtk cjkcj gksrk gs ) foijhr dks.kksa dk ;qxe vkil esa cjkcj gksrk gs 3) fod.kz cjkcj ugha gksrk gs 4) fod.kz,d nwljs dks ycor~ lef}hkkftr djrk gs 5) fod.kz dks.kksa dk v¼zd gksrk gs 6) {ks=kiy = 1 fod.kksza dk xq.kuiy 7) ifjeki = 4 x Hkqtk 8) leprqhkqzt ds Hkqtkvksa ds eè;fcunqvksa dks tksm+usokyh js[kk vk;r gksrk gs MTHMTIS Y- RHUL RJ 7 GOMTRY & MNSURTION

27 lekarj prqhkqzt] vk;r] oxz,oa leprqhkqzt ds chp laca/ & lekarj prqhkqzt vk;r oxz leprqhkqzt & = lhkh lekarj prqhkqzt ds fod.kz] Hkqtk] dks.k ds xq.k fod.kz Hkqtk dks.k ds xq.k 1) fod.kz,d nwljs dks lef}hkkftr djrk gs ) fod.kz dh yackbz cjkcj gksrk gs 3) fod.kz dks.k v¼zd gksrk gs 4) fod.kz ycor~ gksrk gs 5) fod.kz pkj lokzaxle f=khkqt cukrk gs 6) lhkh Hkqtk cjkcj gksrk gs 7) lhkh dks.k ledks.k gksrk gs lekarj prqhkqzt vk;r le prqhkqzt oxz Trapezium (leyc prqhkqzt ),slk prqhkqzt ftlesa foijhr Hkqtk dk,d ;qxe lekarj gks] leyc prqhkqzt dgykrk gs leyc prqhkqzt ds xq.k ( roperties related to trapezium) 1) nksuksa lekarj Hkqtkvksa ls tqm+s yxkrkj dks.kksa ds ;qxe (consecutive angles along both parallel sides) liwjd gksrk gs ;fn rks + = + = ) leyac prqhkqzt ds fod.kz,d nwljs dks lekuqikfrd [k.mksa esa fohkr djrs gsa o O O = O O 3) ;fn fdlh prqhkqzt ds fod.kz,d nwljs dks lekuqikfrd [k.mksa esa fohkr djsa rks ;g prqhkqzt leyc prqhkqzt gksxk 4) leyac prqhkqzt ds lekarj Hkqtkvksa ds lekarj js[kk vlekarj Hkqtkvksa dks leku vuqikr esa ck Vrh gsa ;fn vlekarj Hkqtk dh yckbz leku gks rks ;g leyac lef}ckgq ( Isosceles trapezium ) prqhkqzt dgykrk gs rks MTHMTIS Y- RHUL RJ 8 GOMTRY & MNSURTION

28 = 5) leyc lef}ckgq prqhkqzt (Isosceles trapezium) ds fod.kz cjkcj gksrs gsa IRL (o`ùk ) o`r,d,slk cun oø gs ftlij flkr izr;sd fcunq,d flkj fcunq (ixed oint) ls leku nwjh ij flkr gksrk gs ;g flkj fcunq (ixed point) o`ùk dk dsunz (entre of circle) dgykrk gs centre (dsuæ) ;fn & = rks = 6) leyc lef}ckgq prqhkqzt ds izr;sd lekarj Hkqtkvksa ls tqm+s yxkrkj dks.k (consecutive angles along each parallel sides) cjkcj gksrs gsa circumference (ifjf/) o`ùk ls lcfu/r 'kn (Terms related to ircle) 1) f=kt;k (Radius) : o`ùk ds dsunz,oa ifjf/ ij ds fcunq dks tksm+us okyh js[kk f=kt;k dgykrh gs = & = 7) leyc lef}ckgq prqhkqzt ds izr;sd foijhr dks.k ds ;qxe liwjd gksrs gsa Radius (f=kt;k) ) thok (hord) : o`ùk ds ifjf/ ij flkr fdlh nks fcunq dks tksm+us okyh js[kk thok dgykrk gs hord (thok) 3) O;kl (iameter) o`ùk ds dsunz xqtjus okyh thok O;kl dgykrk gs O;kl lcls cm+h thok gksrh gs + = + = ) leyc lef}ckgq prqhkqzt ds 'kh"kz,do`ùkh; (oncyclic) gksrs gsa 9) leyc prqhkqzt dk {ks=kiy = 1 (lekarj Hkqtkvksa dk ;ksx) x m pkbz iameter (O;kl) 4) pki (rc of a circle) : o`ùk dk,d [kam pki dgykrk gs MTHMTIS Y- RHUL RJ 9 GOMTRY & MNSURTION

29 pki dks?km+h ds foijhr fn'kk esa (counter clock wise) fu:fir fd;k tkrk gs y?kq pki (Minor arc) : - nh?kz pki (Major arc) : - 5) ldsfuæ; o`ùk (oncentric ircles): ;fn nks ;k nks ls vf/d o`ùkksa ds leku dsunz gks rks ;s o`ùk ldsfunz; o`ùk dgykrs gsa 10) o`ùk[k.m (Segment of a circle) :,d thok,oa,d pki ds }kjk f?kjs Hkkx dks o`ùk[k.m dgrs gsa nh?kz o`ùk[k.m y?kq o`ùk[k.m 11) dsuæh; dks.k (entral ngle) : fdlh pki ;k thok ds } kjk dsunz ij cuk;k x;k dks.k dsunzh; dks.k dgykrk gs 6) izfrnsnh js[kk (Secant of a circle) : izfrnsnh js[kk,d,slh ljy js[kk tks o`ùk dks fdugha nks fcunqvksa ij izfrnsn djsa izfrnsnh js[kk dgykrh gs 1) ifjf/ dks.k (Inscribed ngle) : fdlh pki ;k thok ds }kjk ifjf/ ij cuk;k x;k dks.k ifjf/ dks.k dgykrk gs 7) LiZ'kjs[kk (Tangent of a circle):,d,slh js[kk tks o`ùk dks dsoy,d gh fcunq ij Li'kZ djsa Li'kZ js[kk dgykrh gs o`ùk ls lacaf/r izes; (roperties related to ircle) 1. ;fn fdlh o`ùk ds nks pki lokzaxle gks rks mlds laxr thok leku gksaxsa 8) v¼zo`ùk (Semicircle) : fdlh o`ùk dk O;kl ifjf/ dks nks leku pki esa fohkkftr djrk gs vksj izr;sd pki v¼zo`r dgykrk gs 9) f=kt;[k.m (Sector of a circle) : nks f=kt;k,oa,d pki ds }kjk f?kjs Hkkx dks f=kt;[k.m dgrs gsa nh?kz f=kt;[k.m If = then =. o`ùk ds dsunz ls thok ij Mkyk x;k yc thok dks lef}hkkftr djrk gs O y?kq f=kt;[k.m MTHMTIS Y- RHUL RJ 30 L ;fn OL rks L = L GOMTRY & MNSURTION

30 3. o`ùk ds dsunz vksj thok ds eè; fcunq dks feykus okyh js[kk thok ij yc gksrh gs 4. fdlh thok dk yc lef}hkktd dsunz gksdj xqtjrh gs 5. nks ;k nks ls vf/d thokvksa dk yc lef}hkktd,d nwljs dks dsunz ij dkvrk gs 6. ;fn nks o`ùk,d&nwljs dks nks fcunqvksa ij izfrnsn djsa rks muds dsunzksa ls gksdj tkus okyh js[kk mhk;fu" thok dk yc lef} Hkktd gksrk gs O O' = and O 11. ;fn nks o`ùk,d nwljs dks izfrnsn djsa rks muds dsunzksa dks tksm+us okyh js[kk izfrnsn fcunq ij leku dks.k cukrh gs = 1. leku yackbz dh nks thok, dsunz ls leku nwjh ij flkr gksrh gs 13. ;fn nks thok, dsunz ls leku nwjh ij flkr gks rks nksuksa thok, dh yckbz leku gksxh 14.,d o`ùk dh leku thok, dsunz ij leku dks.k cukrh gs 7. ;fn nks o`ùk,d nwljs dks izfrnsn djsa,oa,d&nwljs ds dsunz gksdj xqtjs rks nksuksa o`ùk lokzaxle gksaxsa vkkzr~ mudh f=kt;k, leku gksxh O O O' If = then O = O 8. ;fn nks o`ùk,d nwljs dks izfrnsn djsa,oa,d nwljs ds dsunz gksdj xqtjs rks muds mhk;fu" thok dh yckbz 3r gksxh 15. ;fn,d o`ùk dh nks thok, dsunz ij leku dks.k cukrh gs] rks thok, leku (cjkcj) gksxh 16. ;fn nks thok, yckbz vleku gksxh rks cm+h thok dsunz ds T;knk utnhd gksxh 17. ;fn nks thok,,d nwljh dks izfrnsn djsa,oa os muds izfrnsn fcunq ls gksdj tkus okyh O;kl ds lkk leku dks.k cuk;s rks nksuksa thok, dh yckbz leku gksxh,oa muds VqdM+ksa dh Hkh yckbz leku gksxh O = 3 r 9. nks lekarj thokvksa dk lef}hkktd dsunz gksdj xqtjrh gs 10. ;fn fdlh o`ùk dk O;kl nks thokvksa dks lef}hkkftr djs rks nksuksa thok, lekarj gksxh = = = = MTHMTIS Y- RHUL RJ 31 GOMTRY & MNSURTION

31 3. ;fn nks fcunqvksa dks feykusokyk js[kk[k.m nks vu; fcunqvksa ij tks bl js[kk[k.m ds,d gh vksj flkr gks] leku dks.k varfjr djrk gks rks ;s pkj fcunq,d o`ùkh; gksrs gsa x 0 x 0 = = = 18. ;fn nks thok,,d nwljs dks lef}hkkftr djs rks nksuksa thok, O;kl gksxh 19.,d pki }kjk dsunz ij varfjr dks.k o`ùk ds 'ks"k Hkkx ds fdlh fcunq ij varfjr dks.k dk nqxquk gksrk gs,,,,do`ùkh; (oncyclic) gs 4. ;fn nks thok,d nwljs dks o`ùk ds vunj ;k ckgj izfrnsn djs rks muds [k.mksa (segment) dk xq.kuiy leku gksrk gs x x x = x 0.,d gh o`ùk[kam ds dks.k cjkcj gksrs gsa x 0 x 0 1. v¼zo`ùk dk dks.k ledks.k gksrk gs 90 0 x = x 5. ;fn nks o`ùk,d nwljs dks Li'kZ djs rks Li'kZ fcunq ls gksdj xqtjus okyh js[kk tks nksuksa o`ùkksa dks Li'kZ djsa] ml js[kk dks Li'kZ fcunq muds f=kt;kvksa ds vuqikr esa fohkkftr djrh gs. fdlh thok ds }kjk NksVs o`ùk[k.m esa cuk;k x;k dks.k vf/ dks.k,oa cm+s o`ùk[kam esa cuk;k x;k dks.k U;wudks.k gksrk gs cute ngle r1 r r1 = r Obtuse ngle 6. nks leku thok, rkk dks tc c<+k;k tkrk gs rks og,d nwljs dks fcunq ij dkvrh gs rks = vksj = MTHMTIS Y- RHUL RJ 3 GOMTRY & MNSURTION

32 ;fn =, rks = vksj = 7. ;fn nks thok,d nwljs dks izfrnsn djs rks muds }kjk izfrnsn fcunq ij cuk;k x;k dks.k & 1 x = X (pki ds }kjk dsunz ij cuk;k x;k dks.k pki ds }kjk dsunz ij cuk;k x;k dks.k ) x 0 1 x = ( pki ds }kjk dsunz ij cuk;k x;k dks.k + pki ds }kjk dsunz ij cuk;k x;k dks.k ) pøh; prqhkqzt ( yclic uadrilateral) yclic quadrilateral ( pfø; prqhkqzt ) :,slk prqhkqzt ftlds lhkh pkj 'kh"kz,d o`ùk ij flkr gks pfø; prqhkqzt dgykrk gs 1. izr;sd foijhr dks.k ds ;qxe dk ;ksxiy gksrk gs x 0. ;fn fdlh prqhkqzt ds izr;sd foijhr dks.k ds ;qxe dk ;ksxiy gks rks og prqhkqzt pfø; prqhkqzt gksxk 3. ;fn fdlh pfø; prqhkq Zt ds,d Hkqtk dks c<+k;k tk; rks mlls cuk cfg"dks.k mlds foijhr var%dks.k ds cjkcj gksrk gs = 4. fdlh pøh; prqhkqzt ds dks.kksa ds v¼zdksa ds }kjk cuk prqhkqzt Hkh pøh; gksrk gs R S RS,d pøh; prqhkqzt gs 5. ;fn fdlh pøh; prqhkqzt ds nks Hkqtk lekarj gks rks 'ks"k nksuksa Hkqtk dh yackbz leku gksxh,oa fod.kz dh Hkh yackbz leku gksxh 6. ;fn fdlh pfø; prqhkqzt ds nks foijhr Hkqtkvksa dh yackbz leku gksxh rks 'ks"k nksuksa Hkqtk lekarj gksxh 7. pøh; prqhkqzt RS esa nks foijhr dks.k] vksj R ds v¼zd o`ùk dks nks fcunqvksa rkk ij dkvs rks js[kk[k.m o`ùk dk O;kl gksxk 8. pøh; prqhkqzt ds lhkh pkj o`ùk[k.mksa ds dks.kksa dk ;ksxiy 6 ledks.k ds cjkcj gksrk gs S S R + = vksj + = MTHMTIS Y- RHUL RJ 33 R = 90 0 x 6 = GOMTRY & MNSURTION

33 9.,d pfø; prqhkqzt gs ;fn Hkqtk vksj dks c<+k;k tk; rkfd os,d nwljs dks ij dkvs rks ~. 10. o`ùk dk O;kl gs rkk thok dh yackbz f=kt;k ds cjkcj gs vksj dks c<+kus ij os,d nwljs dks ij dkvrs gsa rks = ) o`ùk ds ckgj flkr fdlh,d fcunq ls nks Li'kZ js[kk, [khaph tk ldrh gs vksj bu nksuksa Li'kZ js[kkvksa dh yckbz leku gksrh gs 7) = O Li'kZ js[kk,oa mlds xq.k (Tangent nd Its roperties) 1) o`ùk ds fdlh fcunq ij dh Li'kZ js[kk] Li'kZ fcunq Ij [khaph xbz f=kt;k ij yc gksrh gs O O ) og js[kk tks f=kt;k ds Nksj fcunq (end point) ls gksdj tkrh gs vksj bl ij (f=kt;k) yc gs] o`ùk dh Li'kZ js[kk gksrh gs 3) o`ùk ds fdlh,d fcunq ij,d,oa dsoy,d Li'kZ js[kk [khaph tk ldrh gs 4) fdlh Li'kZ js[kk ds Li'kZ fcunq ij Mkyk x;k yc o`ùk ds dsunz gksdj xqtjrk gs 5) ;fn nks o`ùk,d&nwljs dks Li'kZ djsa rks muds Li'kZ fcunq muds dsunzksa dks tksm+us okyh js[kk ij flkr gksrk gs (i) = (ii) O O (iii) + O = (iv) O,oa O dk f}hkktd gs (v) O, dk yc lef}hkktd gs (vi) < 8) fdlh o`ùk dh Li'kZ js[kk }kjk cuk;k x;k,dkurj o`ùk[k.m dk dks.k leku gksrk gs S = S & = T T MTHMTIS Y- RHUL RJ 34 GOMTRY & MNSURTION

34 X Y 1) ;fn,d o`ùk f=khkqt ds Hkqtk dks fcunq ij Li'kZ djrk gs,oa Hkqtk rkk dks c<+kus ij rkk R ij Li'kZ djrk gs rks = 1 ( dk ifjeki) X = YX & Y = XY 9) ;fn,d thok,oa,d Li'kZ js[kk o`ùk ds ckgj,d nwljs dks izfrnsn djs rks thok ds [k.mksa (segments of chord) dk xq.kuiy Li'kZ js[kk ds oxz ds cjkcj gksrk gs R T R 13) ;fn fcunq T ls O dsunz okys,d o`ùk ds nks fcunqvksa rkk ij Li'kZ js[kk, T rkk T [khaph xbz gks rks T = O. R x R = TR 10) ;fn nks o`ùk,d nwljs dks ckg~;r% Li'kZ djs rks Li'kZ fcunq ls xqtjus okyh js[kk ds var fcunqvksa ij [khaph xbz Li'kZ js[kk, lekarj gksrh gs T O 14) O 11),d o`ùk ds ifjxr [khaps x;s prqhkqzt ds leq[k Hkqtkvksa dk ;ksx cjkcj gksrk gs ;fn rks O = 90 0 c b d a 15) o`ùk ds O;kl ds nksuksa var fcunq ij [khaph xbz Li'kZ js[kk lekarj gksrh gs R S + = + rea = s(s a)(s b)(s c)(s d) RS MTHMTIS Y- RHUL RJ 35 GOMTRY & MNSURTION

35 16) nks o`ùkksa dh mhk;fu" Li'kZ js[kk, (ommon tangent of two circles),d Hkh mhk;fu" Li'kZ js[kk ugha 18) mhk;fu" vuqli'kz js[kk dh yckbz (Length of direct com- mon tangent) = ( ) d = dsunzksa ds chp dh nwjh d r r, 1 19) nks o`ùkksa ds nks mhk;fu" vuqizlk Li'kZ js[kkvksa (Transverse common tangents) dh yckbz leku gksrh gs,d mhk;fu" Li'kZ js[kk = d r + r 0) mhk;fu" vuqizlk Li'kZ js[kk dh yckbz (Length of trans- verse common tangents) = ( ) 1 nks mhk;fu" Li'kZ js[kk 1) nks o`ùkksa ds mhk;fu" vuqizlk js[kkvksa dh izfrnsn fcunq muds dsunzksa dks tksm+us okyh js[kk ij flkr gksrk gs rhu mhk;fu" Li'kZ js[kk ) G pkj mhk;fu" Li'kZ js[kk 17) nks o`ùkksa ds nks mhk;fu" vuqli'kz js[kk, (irect common tangents) dh yckbz leku gksrh gs 3) (i) = = (ii) = = G = G = = = = 90 0 MTHMTIS Y- RHUL RJ 36 GOMTRY & MNSURTION

36 4) ;fn nks o`ùk dh mhk;fu" Li'kZ js[kk,oa muds dsunzksa dks tksm+us okyh js[kk,d nwljs dks fdlh fcunq ij izfrnsn djs rks izfrnsn fcunq muds dsunzksa dks tksm+us okyh js[kk dks ckg~; #i ls muds f=kt;kvksa ds vuqikr esa fohkkftr djrh gs {ks=kfefr ( MNSURTION- 3) UOI (arallelepiped)?kukhk (lekarj "kv~iyd) r 1 r o o' h l O r1 = O ' r o`ùk ds {ks=kiy,oa ifjeki (rea and erimeter of ircle) b 1) (vk;ru) Volume = vk/kj dk {ks=kiy x m pkbz = rea of base x height ) (vk;ru) Volume = yackbz x pksm+kbz x m pkbz (l x b x h) 1. o`ùk dk {ks=kiy = π r. o`ùk dk ifjeki = π r 3. v¼zo`ùk dk {ks=kiy = 1 r π 4. v¼zo`ùk dk ifjeki = ( π +)r 5. prqkkzal dk {ks=kiy = 1 r 4 π π 6. prqkkzal dk ifjeki = + r v 7. =kt;[k.m dk {ks=kiy = θ r π 360 θ 8. pki dh yckbz = r π 360 MTHMTIS Y- RHUL RJ 37 3) (vk;ru) Volume = 1 3 tgk 1, rkk 3 Øe'k% rhu layxu lrgksa dk {ks=kiy gs 4) (fod.kz) iagonal = y 0 + p k S 0 + m 0 5) ( ik'ohz; lrg dk {ks=kiy ;k pkjksa nhokjksa dk {ks=kiy ) Lateral surface rea or rea of four walls = vk/kj dk ifjeki x m pkbz 6) ik'ohz; lrg dk {ks=kiy (Lateral surface rea) = ( ya0 + pks0) x m 0 7) liw.kz lrg dk {ks=kiy (Total surface area) = ( ya0 x pks0 + pks0 x m 0 + ya0 x m 0) 8) laiw.kz lrg dk {ks=kiy (Total surface rea) = ( ya0 + pks0 + m ) - ( fod.kz ) 9) <dunkj ckwl ds fy, ( or a box having closed top ) (i) Hkhrjh yckbz = ckgjh yckbz x eksvkbz (ii) ckgjh yckbz = Hkhrjh yckbz + x eksvkbz (iii) Hkhrjh pksm+kbz = ckgjh pksm+kbz x eksvkbz (iv) ckgjh pksm+kbz = Hkhrjh pksm+kbz + x eksvkbz (v) Hkhrjh m pkbz = ckgjh m pkbz x eksvkbz (vi) ckgjh m pkbz = Hkhrjh m pkbz + x eksvkbz GOMTRY & MNSURTION

37 10) fcuk <du dk ckwl ds fy, ( box having open top) (i) Hkhrjh yckbz = ckgjh yckbz x eksvkbz (ii) ckgjh yckbz = Hkhrjh yckbz + x eksvkbz (iii) Hkhrjh pksm+kbz = ckgjh pksm+kbz x eksvkbz (iv) ckgjh pksm+kbz = Hkhrjh pksm+kbz + x eksvkbz (v) Hkhrjh m pkbz = ckgjh m pkbz eksvkbz (vi) ckgjh m pkbz = Hkhrjh m pkbz + eksvkbz U le"kv~iyd ) 3) oø i`" {ks=kiy (urved surface rea) = vk/kj dk ifjeki x m pkbz 4) oø i`" dk {ks=kiy (urved surface rea) = πrh 5) laiw.kz lrg dk {ks=kiy (Total surface rea) = πrh + πr = πr ( h + r) Hollow ylinder ( [kks[kyk csyu ) R h h l r b 1) vk;ru = Hkqtk 3 ) ik'ohz; lrg dk {ks=kiy (Lateral surface rea) = 4 x Hkqtk 3) laiw.kz lrg dk {ks=kiy (Total surface rea) = 6 x Hkqtk 4) fod.kz (iagonal) = 3 x Hkqtk Right ircular cylinder ( yc o`ùkh; csyu ) 1) olrq dh eksvkbz (Thickness of material) = R r t ) mijh ;k fupyh lrg dk {ks=kiy (rea of each end) = π( R r ) 3) ckg~; lrg dk {ks=ky (xternal surface rea) = πrh 4) van:uh lrg dk {ks=kiy (Internal surface rea) = πrh 5) oø i`" {ks=kiy (urved surface rea) = πrh + πrh = π (R + r) h h r 6) laiw.kz i`" {ks=kiy (Total surface rea) = πrh + πrh + ( πr - πr ) = π (R + r) ( R r + h) 1) vk;ru = vk/kj dk {ks=kiy x m pkbz ) vk;ru = π r h MTHMTIS Y- RHUL RJ 38 7) /krq dk vk;ru (Volume of material) = kkgjh vk;ru Hkhrjh vk;ru = πr h πr h = π (R r ) h GOMTRY & MNSURTION

38 Right ircular one ( yac o`ùkh; 'kadq ) one I θ h l H L h = dks.k dh m pkbz (height of cone) l = fr;zd m pkbz (slant height of cone) r = dks.k ds vk/kj dh f=kt;k (radius of cone) 1) frjnh m pkbz (Slant height) = ) vk;ru (Volume) = r h + r 1 rea of base x height 3 3) vk;ru (Volume) = 1 3 πr h 4) oø i`" dk {ks=kiy (urved surface rea) = 1 vk/kj dk ifjeku x m pkbz = πrl 5) laiw.kz lrg dk {ks=kiy (Total surface rea) = πrl + πr = π r ( l + r ) 6) ;fn fdlh f=kt;[k.m ls,d dks.k cuk gks rks (If a cone is formed by sector of a circle then) (i) dks.k dh frjnh m pkbz (Slant height of cone) = f=kt;[k.m dh f=kt;k (Radius of sector) (ii) dks.k ds vk/kj dk ifjf/ (ircumference of base of cone) = f=kt;[k.m ds pki dh yckbz (length of arc of sector) 7) ;fn nks 'kadq (cones) dk 'kh"kzdks.k leku gks rks (Two cones having equal vertex angle ) (i) (ii) (iii) 'kadq -I dk vk;ru = 'kadq -I dk oøi`" dk {ks=kiy = one II h 'kadq -II dk vk;ru = a 'kadq -II dk oøi`" dk {ks=kiy = b H L R = = h l r θ H L R = = = a h l r r H L R = = = b h l r rustum of one ( fnuud ) h r R R l l 1) fnuud ds fr;zd m pkbz (Slant height of frustum) = h + ( R r) MTHMTIS Y- RHUL RJ 39 GOMTRY & MNSURTION

39 ) vk;ru (Volume) = 1 3 π(r + r + R r) h 1) vk;ru (Volume) = 4 3 π r3 h 3) vk;ru (Volume) = ( ) tg k 1,oa vk/kj vksj 'kh"kz dk {ks=kiy gs 4) oø i`" dk {ks=kiy (urved surface rea) = π (R + r) l ) laiw.kz i`" dk {ks=kiy (Surface rea) = 4 πr HMISHR ( v¼xksyk ) r r 5) laiw.kz lrg dk {ks=kiy (Total surface rea) = π (R + r) l + πr + πr = π [ (R+r) l + R + r ] 6) ml 'kadq dh m pkbz ftls dkvdj fnuud cuk;k x;k gs (Height of cone of which frustum is a part) = hr R-r 7) ml 'kadq dh frjnh m pkbz ftls dkvdj fnuud cuk;k x;k gs (Slant height of cone of which frustum is a part) = lr R- 8) fnuud ds mijh Hkkx ds 'kadq dk m pkbz ( Height of cone of upper part of frustum) = hr R-r 9) fnuud ds mijh Hkkx ds 'kadq dk fr;zd m pkbz (Slant height of cone of upper part of lr frustum ) = R-r r SHR ( xksyk ) 1) vk;ru (Volume) = 3 πr3 ) oøi`" dk {ks=kiy (urved surface rea) = πr 3) laiw.kz i`" dk {ks=kiy (Total surface rea) = 3 πr SHRIL SHLL ( xksykdkj [kksy ) 1) olrq dk vk;ru (Volume of material) = 4 3 π (R3 r 3 ) ) ckg~; i`" dk {ks=kiy (Outer surface rea) = 4 πr r TORUS R r R r MTHMTIS Y- RHUL RJ 40 GOMTRY & MNSURTION

40 1) vk;ru (Volume) = x π x R x r ) i`"h; {ks=kiy (Surface rea) = 4 x π x R x r RISM ( fizte ) 3) laiw.kz lrg dk {ks=kiy (Total surface rea) = ik'ohz; lrg dk {ks=kiy + vk/kj dk {ks=kiy TTRHRON ( leprq"iyd ) Hexagonal base Hexagonal face Hexagonal base 1) vk;ru (Volume) = vk/kj dk {ks=kiy x m pkbz ) ik'ohz; lrg dk {ks=kiy (Lateral surface rea) = vk/kj dk ifjeki x m pkbz 3) laiw.kz lrg dk {ks=kiy (Total surface rea) = ik'ohz; lrg dk {ks=kiy + x vk/kj dk {ks=kiy 1) vk;ru (Volume) = 1 a ) laiw.kz lrg dk {ks=kiy (Total surface rea) = 3 a OTHRON ( lev"viyd ) 3 YRMI 1) vk;ru (Volume) = 3 a3 1) vk;ru (Volume) = 1 vk/kj dk {ks=kiy x m pkbz 3 ) laiw.kz lrg dk {ks=kiy (Total surface rea) = 3a ) ik'ohz; lrg dk {ks=kiy (Lateral surface rea) = 1 vk/kj dk ifjeki x fr;d m pkbz MTHMTIS Y- RHUL RJ 41 GOMTRY & MNSURTION