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7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
1/121
EF@OXNKREF\
EFTNBEFTOHX
NB\M@X CM\ERIF
O@T\MDRHTOM@Qb frb cfeoiofr wotn tnb snfpbs mc ef`y pf`b himsbd coaurbs suhn fs squfrbs, rbhtf`aibs, qufdroiftbrfis, roant
trof`aibs, bquoiftbrfi trof`aibs, osmshbibs trof`aibs, shfib`b trof`aibs, bth. Qb k`mw tnb ruibs tm co`d tnb pbroebtbrs
f`d frbf mc smeb mc tnbsb coaurbs. Cmr bxfepib, f rbhtf`aib wotn ib`atn 6< e f`d wotn > e nfs pbroebtbr bqufi tm
< (6 ) 5 4:e. Tnb frbf mc tnos rbhtf`aib os bqufi tm (6< >) e< 5 ?3 e
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
Cmr bxfepb, f roant f`aibs trof`aib nfvo`a twm sodbs mc ib`atn 8 e f`d 1 e (mtnbr tnf` tnb nypmtb`usb), nfs ots
frbf 5: e f`d h 5 2 62 5 12 e
Ot os aovb` tnft pbroebtbr 5 42:
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
5/121
EF@OXNKREF\
EFTNBEFTOHX
Bx.4 Tnb ib`atns mc tnb sodbs mc f trof`aib frb 2 he, 6< he f`d 68 he. Co`d tnb ib`atn mc pbrpb`dohuifr crme tnb
mppmsotb vbrtbx tm tnb sodb wnmsb ib`atn os 68 he.
Xmi. Nbrb, f 5 2, l 5 6< f d h 5 68.
s 5 he
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
Bx.1 O` coa. FLHD os f cobid o` tnb cmre mc f qufdroiftbrfi wnmsb sodbs frb o`dohftbd o` tnb coaurb.
OcDFL 5 ?::, co`d tnb frbf mc tnb cobid.Xmi. Hibfriy,DFL os f roant-f`aibd trof`aib. Tnbrbcmrb,
DL< 5 DF< + FL< [Rso`a Zytnfamrfs TnbmrbeU
DL< 5 ?< + 4:6
5 46e
O`DFL, wb nfvb
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
Xmi. Ibt FLHD lb tnb aovb` trfpbzoue o` wnohn FL 5
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
5 :::>:6>: ee
Nb`hb, bfhn mc tnb twm hnoidrb` woii abt f` frbf mc 4>:: e : de nfs tnb dofam`fi bqufi tm 31< de. Co`d ots frbf tm tnb `bfrbst squfrb ebtrb.
66. F trfpbzoue wotn ots pfrfiibi sodbs o` tnb rftom 63 7 62 os hut crme f rbhtf`aib wnmsb sodbs ebfsurb 38 e f`d 2 e
rbspbhtovbiy. Tnb frbf mc tnb trfpbzoue os62
4mc tnb frbf mc tnb rbhtf`aib. Co`d tnb ib`atns mc tnb pfrfiibi sodbs mc
tnb trfpbzoue.
6
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
MLGBHTOSB T_ZB VRBXTOM@X
6. Tnb lfsb mc f roant f`aibs trof`aib Os - 2 ebtrbs f`d nypmtb`usb os 68 ebtrbs. Ots frbf woii lb
(F) >.>: (L) \s. 182 (H) \s. 18:.>: (D) \s. 1
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
(F) )%( 6::
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
F`s. H L L F H D L L F L
Vub. 66 6< 68 64 62 63 61 6> 6?
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
26/121
EF@OXNKREF\
EFTNBEFTOHX
XR\CFHB F\BFX & SMIREX
O@T\MDRHTOM@X
Rp toii `mw wb nfvb lbb` dbfio`a wotn coaurbs tnft hf` lb drfw` m` tnb pfab mc mur `mtblmmk mr m` tnb
lifhklmfrd. Tnbsb frb hfiibd p i f ` b c o a u r b s . O` tnos hnfptbr, wb snfii study flmut smeb smiod coaurbs iokb hulmods,
hulb, hyio`dbr f`d spnbrb. Tnbsb coaurbs frb t n r b b d o e b ` s o m ` f i c o a u r b s . Qb snfii fism ibfr` tm co`d tnb surcfhb
frbfs f`d vmiuebs mc tnbsb coaurbs.
R@OTX MC EBFXR\BEB@T MC F\BF F@D SMIREB
Tnb o`tbr-rbiftom`snops lbtwbb` vfromus u`ots mc ebfsurbeb`t mc ib`atn, frb f`d - vmiueb frb iostbd lbimw cmr
rbfdy rbcbrb`hb 7
IB@ATN
6 Hb`toebtrb (he) 5 6: eoioeobtrb (ee)
6 Dbhoebtrb (de) 5 6: hb`toebtrb6 Ebtrb (e) 5 6: de 5 6:: he 5 6::: ee
6 Dbhfebtrb (dfe) 5 6: e 5 6::: he
6 Nbhtmebtrb (ne) 5 6: dfe 5 6:: e
6 Koimebtrb (ke) 5 6::: e 5 6:: dfe 5 6: ne
6 Eyrofebtrb 5 6: Koimebtrb
F\BF
6 he< 5 6 he 6 he 5 6: ee 6: ee 5 6:: ee> he> he>
< 1
>
r 564
r 5 6
< 5 he8 5 ?.2< he8
Nb`hb, tnb vmiueb mc tnb hm`b 5 ?.2< he 8.
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
Bx.
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
46/121
EF@OXNKREF\
EFTNBEFTOHX
Cmreuifb
Cmr f spnbrohfi snbii wotn bxtbr`fi rfdous 5 \ f`d o`tbr`fi rfdous 5 r, wb nfvb 7
Tnohk`bss m snbii 5 (\ - r) u`ots.
Mutbr surcfhb frbf 5
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
Smiueb mc tnb eftbrofi 5 8
Qnft ib`atn mc tfrpfuio` 8 e wodb woii lb rbquorbd tm efkb hm ohfi tb`t mc nboant > e f`d lfsb rfdous 3 e=
fssueb tnft tnb bxtrf ib`atn mc eftbrofi tnft woii lb rbquorbd cmr stothno`a efrao`s f`d wfstfab o` hutto`a
os fpprmxoeftbiy
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
56/121
EF@OXNKREF\
EFTNBEFTOHX
. F pimt mc if`d o` tnb cmre mc f rbhtf`aib nfs doeb`som` : e. F drfo` ibt 6: e wodb os dua fii frmu`d ot
(m` tnb mutsodb) f`d tnb bfrtn dua mut os bvb`iy sprbfd mvbr tnb pimt, o`hrbfso`a ots surcfhb ibvbi ly he 83 he. Crme bfhn m`b mc ots hmr`brs, f
squfrb mc > he os hutmcc. F` mpb` lmx os efdb mc tnb rbefo`o`a snbbt. Co`d tnb vmiueb mc tnb lmx.
6:. Qftbr o` f hf fi, 8: de wodb f`d 6< de dbpp, os cimwo a wotn f vbimhoty mc a.
6
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
6>. F wmmdb tmy os o` tnb cmre mc f hm`b suremu`tbd m` f nbeospnbrb. Tnb dofebtbr mc tnb lfsb mc tnb hm`b os 3 he
f`d ots nboant os 4 he. Co`d tnb hmst mc pfo`to`a tnb tmy ft tnb rftb mc \s. 2 pbr 6::: he
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
6?. Tnb crm t hmepmu d wfii mc f nmusb os dbhmrftbd ly wmmdb` spnbrbs mc dofebtbr
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
6
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
66/121
EF@OXNKREF\
EFTNBEFTOHX
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
67/121
EF@OXNKREF\
EFTNBEFTOHX
MLGBHTOSB T_ZB VRBXTOM@X
6. F rbhtf`auifr sf`d lmx os 2 e wodb f`d < e im`a. Nmw ef y huloh ebtrbs mc sf`d frb `bbdbd tm coii tnb lmx uptm
dbptn mc 6: he =
(f) 6 (l) 6: (h) 6:: (d) 6:::
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
(f) \s. 8?3 (l) \s. 4:: (h) \s. 46: (d) \s. 4
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
6
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
70/121
EF@OXNKREF\
EFTNBEFTOHX
(F) 3 he (L) 1 he (H) > he (D) ? he
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
71/121
EF@OXNKREF\
EFTNBEFTOHX
XR\CFHB F\BFX & SMIREBX F@XQB\ KB_ B^B\HOXB
(F) SB\_ XNM\T F@XQB\ T_ZB VRBXTOM@X 7
6.< ( + l)n 8
26 e8
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
72/121
EF@OXNKREF\
EFTNBEFTOHX
Vub
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
73/121
EF@OXNKREF\
EFTNBEFTOHX
trof`aibs frb bqufi o.b. sox bqufiotobs nmid ammd, tnrbb mc tnb hmrrbspm`do`a sodbs f`d tnrbb mc tnb hmrrbspm`do`a
f`aibs.
O` tnos sbhtom`, wb snfii prmvb tnft oc tnrbb prmpbriy hnmmsbs hm`dotom`s mut mc tnb sox hm`dotom`s frb sftoscobd, tnb`
tnb mtnbr tnrbb frb futmeftohfiiy sftoscobd. Ibt us `mw doshuss tnmsb hm`dotom`s wnohn b`surb tnb hm`arub`hb mc
twm f`aibs.
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
O. XODB-F@AIBX(XFX)HM@A\RB@HB H\OTB\OM@
F^OME 7 Twm trof`taibs frb hm`arub`t oc twm sodbs f`d o`hiudbd f`aib mc m`b trof`aib frb bqufi tm tnb
sodbs f`d tnb o`hiudbd f`aib mc tnb mtnbr trof`aib.
O` tnb aovb` coaurb, o` FLH f`d DBC, wb nfvb 7
FL 5 DB, FH 5 DC f`d .DF
DBCFLH [Ly XFX-hrotbrofU
\BEF\K 7 XFX-Hm`arub`hb ruib nmids lut FXX mr XXF mr XXF ruib dmbs `mt nmid.
OO. F@AIB (FXF) HM@A\RB@HB H\OTB\OM@
TNBM\BE-6 7 Twm trof`aibs frb hm`arub`t, oc twm f`aibs f`d tnb o`hiudbd sodb mc tnb m`b trofaib frb bqufitm twm f`aibs f`d tnb o`hiudbd sodb mc tnb mtnbr trof`aib.
Aovb` 7Twm trof`aibs FLH f`d DBC suhn tnft CHBL , f`d LH 5 BC.
Tm prmvb 7 .DBCFLH
Zrmmc 7 Tm prmvb .DBCFLH wb `bbd tm hm`sodbr tnrbb pmssolib sotuftom`s 7
Hfsb 6. Ibt FL 5 DB.
XTFTBEB@T \BFXM@
6.
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
XTFTBEB@T \BFXM@6.
:
Xoeoifriy. CBD :6>:
Aovb`Aovb`Crme(8)Ly FXF hrotbrof
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
OS XODB-XODB-XODB (XXX) HM@A\RB@HB H\OTB\OM@
Twm t r o f ` a i b s f r b h m ` a r u b ` t , o c t n b t n r b b s o d b s m c m ` b t r o f ` a i b f r b b q u f i t m t n b h m r r b s p m ` d o ` a t n r b b s o d b s
m c tn b m tn b r tr o f ` a i b .
Oc tnb aovb` coaurb, o`FLH f`dDBC, wb nfvb 7
FL 5 DB, LH 5 BC f`d FH 5 DC.
FLHDBC [Ly XXX-hrotbrofU
S \OANT F@AIB-N_ZMTB@RXB-XODB (\NX) HM@A\RB@HB H\OTB\OM@
T w m r o a n t f ` a i b s f r b h m ` a r u b ` t , o c t n b n y p m t b ` u s b f ` d m ` b s o d b m c m ` b t r o f ` a i b f r b r b s p b h t o v b i y b q u f i t m
t n b n y p m t b ` u s b f ` d m ` b s o d b m c t n b m t n b r t r o f ` a i b .
O` tnb aovb` coaurb,FLH f`dDBC frb roant - f`aibdtrof`aibs o` wnohn Nyp. FH 5 Nyp. DC f`d LH 5 BC.
FLHDBC [Ly \NX-hrotbrofU
Bx.6 O` coa. MF 5 ML f`d MD 5 MH. Xnmw tnft [@HB\TU
(o) FMD LMH f`d (oo) FD || LH.Xmi. Aovb` 7 O` coa. MF 5 ML f`d MD 5 MH.
Tm prmvb 7(o)FMD LMH f`d (oo) FD || LH.
Zrmmc 7
Nb`hb, prmvbd.
Bx.< O@ FLH, FL 5 FH. Oc Z os f pmo`t mc FL f`d V osf pmo`t m` FH suhn tnft FZ 5 FV. Zrmvb tnft
(o) FZH FVL (oo) LZH HVL.Xmi. Aovb`7 O`FLH, FL 5 FH. Z os f pmo`t m` FL f`d V os
f pmo`t m` FH suhn tnft FZ 5 FV.
XTFTBEB@T \BFXM@
6.
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
77/121
EF@OXNKREF\
EFTNBEFTOHX
Tm prmvb 7 (o)FZH FVL (oo)LZH HVL.
Zrmmc 7
Nb`hb, prmvbd.
Bx.8 O` coaurb, FH 5 FB, FL 5 FD f`dLFD 5BFH, prmvb tnft LH 5 DB. [@HB\TUXmi. Aovb` 7 O` coaurb, FH 5 FB, FL 5 FD f`dLFD 5BFH.
Tm prmvb 7LB 5 DB.
Hm`struhtom` 7 Gmo`t DB.
XTFTBEB@ \BFXM@
6.
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
78/121
EF@OXNKREF\
EFTNBEFTOHX
Nb`hb, prmvbd.
Bx.4 O` coaurb, dofam`fi FH mc f qufdroiftbrfi FLHD losbhts tnb f`aibs F f`d H. Zrmvb tnft FL 5 FD f`d HL 5
HD/
Xmi. Aovb` 7 O` coaurb, dofam`fi FH mc f qufdroiftbrfi FLHD
losbht tnb f`aibs F f`d H.
Tm prmvb 7 FL 5 FD f`d HL 5 HD
Nb`hb, prmvbd.
Bx.2 FL os f io`b sbaeb`t f`d Z os ots eod-pmo t. D f`d B frb pmo ts m` tnb sfeb sodb mc FL suhn tnftLFD 5FLB f`dBZF 5DZL. Xnmw tnft (o) DFZ BLZ (oo) FD 5 LB. [@HB\TU
Xmi. Aovb` 7 FL os f io`b sbaeb`t f`d Z os ots eod-pmo`t. D f`d B frb pmo`ts mc tnb sfeb sodb mc FL suhn tnftLFD 5
FLB f`dBZF 5DZL.
XTFTBEB@ \BFXM@
6.
8.
8.
O`FLH f`dFDB
(o) FL 5 FD
(oo) FH 5 FB
(ooo)LFD 5BFH
LFD +DFH 5DFH +BFH
LFH DFB
FLH FDB
LH 5 DB
Aovb`
Aovb`
Aovb`
Fddo`aDFH tm lmtn sodbs
Ly XFX hrotbrof
H.Z.H.T.
XTFTBEB@ \BFXM@
6.
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
Tm prmvb 7(o)DFZ BLZ(oo) FD 5 LB
Zrmmc 7
Nb`hb, prmvbd
Bx.3 FL os f io`b-sbaeb t. Fs f`d Ly frb twm bqufi io`b - sbaeb ts drfw` m` mppmsotb sodbs m` io`b FL suhn
tnft F^ || L_. oc FL f`d ^_ o`tbrsbht bfhn mtnbr ft Z. Zrmvb tnft 7
(o) FZ^ LZ_ (oo) FL f`d ^_ losbht bfhn mtnbr ft Z.
Xmi. Aovb` 7 FL os f io`b-sbaeb`t. F^ f`d L_ frb tm bqufi io`b-sbaeb`ts drfw` m` mppmsotb sodbs mc io`b FL suhn tnft
F^ || L_.
Tm prmvb 7(o)FZ^ LZ_ (oo) FL f`d ^_ losbht bfhn mtnbr ft Z.
XTFTBEB@ \BFXM@
6.
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
Nb`hb, prmvbd.
Bx.1 O` coa., FD os f ebdof` f`d LI, HE frb pbrpb`dohuifrs drfw` crme L f`d H rbspbhtovbiy m` FD f`d FD
prmduhbd. Zrmvb tnft LI 5 HE.
XTFTBEB@ \BFXM@
6.
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
81/121
EF@OXNKREF\
EFTNBEFTOHX
Xmi. Aovb` 7 O` coa., FD os f ebdof` f`d LI, HE frb pbrpb`dohuifrs drfw` crme L f`d H rbspbhtovbiy m` FD f`d FD
prmduhbd.
Tm prmvb 7 LI 5 HE
Nb`hb, prmvbd.
Bx.> FD f`d LH frb bqufi pbrpb`dohuifrs tm f io`b sbaeb`t FL. Xnmw tnft HD losbhts FL. [ @HB\TU
Xmi. Aovb` 7 FD f`d LH frb bqufi pbrpb`dohuifrs tm f io`b sbaeb`t FL.
Tm Zrmvb 7HD losbhts FL.
Nb`hb, prmvbd.
Bx.? Oc coa.LHD 5 FDH f`dFHL 5LDF. Zrmvb tnft FD 5 LH f`dF 5L/
XTFTBEB@ \BFXM@
6.
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
Xmi. Aovb` 7 O` coaLHD 5FDH f`dFHL 5LDF
Tm prmvb 7(o) FD 5 LH (oo)F 5L
Nb`hb, prmvbd.
Bx.6: O` tnb coa. FLHD os f qufdroiftbrfi o` wnohn FL 5 FD f`d LH 5 DH. Zrmvb tnft 7
(o) FH losbhts bfhn mc tnb f`aibs F f`d H.
(oo) LB 5 BD.
(ooo)FLH 5FDH. Hf` wb sfy tnft FB 5 BH =Xmi. Aovb` 7 O` tnb coa. FLHD os f qufdroiftbrfi o` wnohn
FL 5 FD f`d LH 5 DH.
Tm prmvb 7 (o) FH losbhts bfhn mc tnb f`aibs F f`d H.
(oo) LB 5 BD.
(ooo)FLH 5FDH. Hf` wb sfy tnft FB 5 BH =
XTFTBEB@T \BFXM@
6.
::
Hmeem`
Crme (6)
Aovb`
Aovb`
Ly FFX hrotbrof
H.Z.H.T.
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
@m. FB BH. Crme FB tm lb bqufi tm BH, ot os `bhbssfry tnftFBD HBD cmr wnohn< eust lb bqufi tm4.
Nb`hb, prmvbd.
Bx.66 FL os f io`b sbaeb t, Z f`d V frb pmo ts m` mppmsotb sodbs mc FL suhn tnft bfhn mc tnbe os bquodostf t
crme tnb pmo`ts F f`d L. Xnmw tnft tnb io`b ZV os tnb pbrpb`dohuifr losbhtmr mc FL. [@HB\TU
Xmi. Aovb` 7 FL os io`b sbaeb`t. Z f`d V frb pmo`ts m` mppmsotb sodbs mc FL suhn tnft bfhn mc tnbe os bquodostf`t crme
tnb pmo`ts F f`d L.
Tm prmvb 7ZV os tnb pbrpb`dohuifr losbhtmr mc FL.
XTFTBEB@T \BFXM@
6.
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
Crme (3) f`d (1) wb hm`hiudb tnft ZV os tnb pbrpb`dohuifr losbhtmr mc FL.
Nb`hb, prmvbd.
XTFTBEB@T \BFXM@
6.
::
::
FHZ 5 ?::
Aovb`
Aovb`
Hmeem`
Ly XXX hrotbrof
H.Z.H.T.
Aovb`
FZV 5LZV, crme (8)
Hmeem`
Ly XFX hrotbrof
H.Z.H.T.
Io`bfr pfor mc f`aibs
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
Bx.6
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
Nb`hb, prmvbd.
Bx.64 FD, LB f`d HC, tnb fitotudbs mc FLH frb bqufi. Zrmvb tnft FLH os f` bquoiftbrfi trof`aib.
Xmi. Aovb` 7 FD, LB f`d HC frb tnb fitotudbs mcFLH f`d frb bqufi.
Tm prmvb 7FLH os f` bquoiftbrfi trof`aib
Nb`hbFLH os f` bquoiftbrfi trof`aib.
Bx.62 O` f FLH, tnb o`tbr`fi losbhtmrs mcL f`d H ebbt ft M. Zrmvb tnft MF os tnb o`tbr`fi losbhtmr mcF.
XTFTBEB@T \BFXM@
6.
7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles
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EF@OXNKREF\
EFTNBEFTOHX
Xmi. Aovb` 7 O` fFLH, tnb o`tbr`fi losbhtmrs mcL f`dH ebbt ft M.
Tm prmvb 7MF losbhtsF.
Hm`struhtom` 7 Drfw MD LH, MB, HF f`d MC FL.
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EF@OXNKREF\
EFTNBEFTOHX
Zrmmc 7
Nb`hb, prmvbd.
Z\MZB\TOBX MC F@ OXMXHBIBX T\FO@AIB
O` tnos sbhtom`, wb woii ibfr` smeb prmpbrtobs rbiftbd tm f trof`aib wnmsb twm sodbs frb bqufi. Qb k`mw tnft f
trof`aib wnmsb twm sodbs frb bqufi os hfiibd f` osmshbibs trof`aib. Nbrb, wb woii fppiy XFX hm`arub`hb hrotbrof f`d
FXF (mr (FFX) hm`arub`hb hrotbrof tm study smeb prmpbrtobs mc f` osmshbibs trof`aib.
TNBM\E - 8 7 F`aibs mppmsotb tm bqufi sodbs mc f` osmshbibs trof`aib frb bqufi.
Aovb` 7FLH os f` osmshbibs trof`aib f`d FL 5 FH.
Tm prmvb 7 L 5H.Hm`struhtom` 7 Drfw FD tnb losbhtmr mcF. FD ebbts LH ft D.Zrmmc 7
XTFTBEB@T \BFXM@
6.
.
O`MDH f`dMBH(o)MHD 5MHB
(oo)MDH 5MBH
(ooo) MH 5 MH
MDH MBH
MD 5 MB
MD 5 MC
MB 5 MC
O` roantMBF f`dMCF
(o) MB 5 MC
(oo) Nyp. MF 5 Nyp, MFMBF MCF
MFB 5MFC
MF losbhtsF.
MH losbhtsH.
Bfhn 5 ?:: (ly hm`struhtom`).
Hmeem`
Ly FFX hrotbrof
H.Z.H.T.
Xoeoifriy, (MDL MCL).
Crme (8) f`d (4)
Crme (2)
Hmeem`
Ly \NX hrotbrof
H.Z.H.T .
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EF@OXNKREF\
EFTNBEFTOHX
Nb`hb, prmvbd.
HM\MIIF\_ bfhn f`aib mc f` bquoiftbrfi trof`aib os mc 3::.
Aovb` 7FLH os f` bquoiftbrfi trof`aib.
Tm prmvb 7 F 5L 5H 5 3::.Zrmmc 7
Nb`hb, prmvbd.
TNBM\BE-4 7 Tn b so d b s m p p m so tb tm b q u f i f ` a i b s m c f tr o f ` a i b b q u f i .
Aovb` 7FLH o` wnohnL 5H.
XTFTBEB@T \BFXM@
6.
:: 5 3::
FLH os f` bquoiftbrfi trof`aib.
FL 5 FH
FL 5 LH
Crme (
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EF@OXNKREF\
EFTNBEFTOHX
Tm prmvb 7FL 5 FH.
Hm`struhtom` 7 Drfw FD LH. FD ebbts LH o` D.
Zrmmc 7
Nb`hb, prmvbd.
Bx.63 O` tnb fdgmo`o`a coa, co d tnb vfiub mc x.
Xmi. Qb nfvb,
HFD +FDH +DHF 5 6>:: [F`aib sue prmpbrtyU
HFD +FDH + 34: 5 6>::
HFD +FDH 5 (6>:: - 34:) 5 663:
Lut HD 5 HFHFD 5FDH [s mppmsotb tm bqufi sodbs mc ffrb bqufiU
Xm, HFD 5FDH 5:
@mw,FDH 5FLD +DFL [Bxt.mc f5 sue mc o`t. mpp.sU
Lut, FD 5 LDFLD 5DFL.
Xm, FDH 5
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EF@OXNKREF\
EFTNBEFTOHX
Bx.61 FLH os f osmshbibs trof`aib os wnohn FL 5 FH. Xodb LF os prmduhbd tm D suhn tnft FD 5 FL. Xnmw tnftLHD os f roant f`aib.
Xmi. Aovb` 7FLH os f` osmshbibs trof`aib os wnohn FL 5 FH. Xodb LF os prmduhbd tm D suhn tnft FD 5 FL
Tm prmvb 7LHD 5 ?::
Zrmmc 7
Nb`hb, prmvbd.
Bx.6> O` f roant f`aibd trof`aib, m`b fhutb f`aib os dmulib tnb mtnbr. Zrmvb tnft tnb nypmtb`usb os dmulib tnb
sefiibst sodb.
Xmi. Aovb` 7 FFLH o` wnohnL 5 ?:: f`dFHL 5 ::
::
LHD 5 ?::
LHD os f roant f`aib.
Aovb`
s mppmsotb tm bqufi sodbs mc ffrb bqufi
Aovb`Crme (6) & (
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EF@OXNKREF\
EFTNBEFTOHX
Zrmmc 7
Nb`hb, prmvbd.
O@BVRFIOTOBX O@ F T\OF@AIB
TNBM\BE-2 7 O c tw m so d b s m c f tr o f ` a i b f rb u ` b q u f i , tn b ` tn b a rb f tb r so d b n f s a rb f tb r f ` a i b m p p m so tb tm o t .
Aovb` 7FFLH o` wnohn FH 0 FL.
Tm prmvb 7FLH 0FHL.
Hm`struhtom` 7 Efrk f pmo`t D m` FH suhn tnft FD 5 FL. Gmo` LD.
Zrmmc 7
Nb`hb, prmvbd.
XTFTBEB@T \BFXM@
6.
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EF@OXNKREF\
EFTNBEFTOHX
TNBM\BE-3 (Hm`vbrsb mc Tnbmrbe-2) 7 Oc twm f`aibs mc f trof`aibs frb u`bqufi, tnb` tnb arbftbr f`aib nfs
arbftbr sodb mppmsotb tm ot.
Aovb` 7FFLH o` wnohnFLH 0FHL.Tm prmvb 7 FH 0FL.
Zrmmc 7
Nb`hb. prmvbd.
TNBM\BE-1 7 Tnb sue mc f`y twm sodbs mc f trof`aib os arbftbr tnf` ots tnord sodb.
Aovb` 7FFLH.
Tm prmvb 7 (o) FL + FH 0 LH
(oo) FL + LH 0 FH
(ooo) LH + FH 0 FL.
Hm`struhtom` 7 Zrmduhb LF tm D suhn tnft FD 5 FH. Gmo` HD.
Zrmmc 7
XTFTBEB@T \BFXM@6. Qb efy nfvb tnrbb pmssoloiotobs m`iy
(o) FH 5 L(oo) FH 9 FL(ooo) FH 0FL
Mut mc tnbsb bxfhtiy m`b eust lb trub.Hfsb-O. FH 5 FL.
FLH 5FHL.Tnos os hm`trfry tm wnft os aovb`.
FH 5 FL os `mt trubHfsb -OO FH 9 FL
FL 0 FHFHL 0FLHTnos os hm`trfry tm wnft os aovb`FH 9 FL os `mt trub.Tnus, wb frb ibct wotn tnb m`iy pmssoloioty
FH 0 FL wnohn eust lb trub.
s mppmsotb tm bqufi sodbs mc ffrb bqufi
Arbftbr sodb nfs arbftbr f`aib mpp. tm ot.
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EF@OXNKREF\
EFTNBEFTOHX
Nb`hb, prmvbd.
\BEF\K 7 (o) Tnb ifrabst sodb mc f trof`aib nfs tnb arbftbst f`aib mppmsotb tm ot f`d hm`vbrsb os fism trub(oo) Tnb sefiibst sodb mc trof`aib nfs tnb sefiibst f`aib mppmsotb tm ot f`d hm`vbrsb o` fism trub
Bx.6? O` coa, snmw tnft 7 (o) FL 0 FH (oo) FL 0 LH f`d (ooo) LH 0 FH.
Xmi. Aovb` 7 FFLH o` wnohnL 5 4:: f`dFHD 5 6:::.
Tm prmvb 7 (o) FL 0 FH
(oo) FL 0 LH
(ooo) LH 0 FH.
Zrmmc 7
Nb`hb, prmvbd.
Bx.:: f`dF 5 3::
Arbftbr f`aib nfs arbftbr sodb mpp. tm ot.
F 5 3:: f`dL 5 4::
Arbftbr f`aib nfs arbftbr sodb mpp. tm ot.
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EF@OXNKREF\
EFTNBEFTOHX
Zrmmc 7
ZE os tnb snmrtbst mc tnb io b sbaeb`ts crme Z tm FL.
Nb`hb, prmvbd.
Bx.
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EF@OXNKREF\
EFTNBEFTOHX
TNO@AX TM \BEBLB\
6. Twm coaurbs frb hm`arub`t , oc tnby frb mc tnb sfeb snfpb f`d mc tnb sfeb sozb.. F`aib mppmsotb tm bqufi sodbs mc f trof`aib frb bqufi.
?. Xodbs mppmsotb tm bqufi f`aibs mc f trof`aib frb bqufi.
6:. Bfhn f`aib mc f` bquoiftbrfi trof`aib os mc 3::.
66. Oc tnrbb sodbs mc m`b trof`aib frb bqufi tm tnrbb sodbs mc tnb mtnbr trof`aib, tnb` tnb twm trof`aibs frb
hm`arub`t (XXX hm`arub`hb ruib).
6
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EF@OXNKREF\
EFTNBEFTOHX
HLXB LFXBD XMEB OEZM\TF@T VRBXTOM@X
V.6 O` coa, DZ 5 LV,DZL 5 LVD f`dFDZ 5HLV, Xnmw tnft FDZ HLV.
[No`t 7DZL 5LVD
6>:: -DZL 5 6>:: -LVDFZD 5HVLU
V.< f`d e frb twm pfrfiibi io`bs o`tbrsbhtbd ly f`mtnbr pfor mc pfrfiibi io`bs p f`d q. Xnmw tnft
FLH HDF. [@HB\TU
[No t7 (o)LFH 5DHF
(oo)FHL 5HFD
(ooo) FH 5 HFUV.8 \fe wosnbs tm dbtbreo`b tnb dostf`hb lbtwbb` twm mlgbhts F f`d L, lut tnbrb os f` mlstfhib lbtwbb`
tnbsb twm mlgbhts fs snmw` o` coa, wnohn prbvb`ts nos crme efko`a f dorbht ebfsurbeb`t. Nb dbvosbs f`
o`ab`omus wfy tm mvbrhmeb tnos doccohuitiy. Corst, nb coxbs f pmib ft f hm`vb`ob`t pmo`t M sm tnft crme M,
lmtn F f`d L frb vosolib. Tnb` nb coxbs f`mtnbr pmib ft tnb pmo`t D m` tnb io`b FM (prmduhbd) suhn tnft FM
5 DM. O` f soeoifr wfy, nb coxbs f tnord pmib ft tnb pmo`t H m` tnb io`b LM (prmduhbd) suhn tnft LM 5 HM.
Tnb` nb ebfsurbs HD f`d co`ds tnft HD 5 61: e. Zrmvb tnft tnb dostf`hb lbtwbb` tnb mlgbht F f`d L os
fism 61: e.
V.4. O` roant trof`aib FLH, roant f`aibd ft H, E os tnb eod-pmo t mc nypmtb`usb FL. H os gmo bd tm E f`d
prmduhbd tm f pmo`t D suhn tnft DE 5 HE. Zmo`t D os gmo`bd tm pmo`t L. Xnmw tnft 7
(o) FEH LED(oo) DLH os f roant f`aib.(ooo) DLH FHL
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EF@OXNKREF\
EFTNBEFTOHX
(ov) HE 5::
DLH 5 ?::
(ooo) Ly XFX hm`arub`hb ruib,DLH FHL
(ov)DLH FHLHD 5 FL
FEH LED HE 5 DEHE 5
. O` tnb fdgmo`o`a coaurb, prmvb tnft 7
(o) FL + LH + HD 0 DF
(oo) FL + LH + HD + DF 0
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EF@OXNKREF\
EFTNBEFTOHX
?. O` tnb fdgmo`o a coaurb, M os tnb hb trb mc f horhib, ^_ os f dofebtbr f`d ^W os f hnmrd. Zrmvb tnft ^_ 0 ^W.
6:. O` tnb aovb` coaurb, FD 5 FL f`d FB losbhtsF. Zrmvb tnft 7
(o) LB 5 BD
(oo) FLD 0LHF.
(H) @HB\T VRBXTOM@X 7
6. O` qufdroiftbrfi FHLD, FH 5 FD f`d FL losbhtsF. Xnmw tnftFLH FLD. Qnft hf` ymu sfy flmut LH f`d
LD =
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EF@OXNKREF\
EFTNBEFTOHX
(o) FZL FVL
(oo) LZ 5 LV mr L os bquodostf t crme tnb fres mcF.
4. O`FLH, FD os tnb pbrpb`dohuifr losbhtmr mc LH. Xnmw tnftFLH os f` osmshbibs trof`aib o` wnohn FL 5 FH.
2. FLH os f` osmshbibs trof`aib o` wnohn fitotudbs LB f`d HC frb drfw` tm bqufi sodbs FH f d FL rbspbhtovbiy. Xnmw
tnft tnbsb fitotudbs frb bqufi.
3. FLH os f roant f`aibd trof`aib o` wnohnF 5 ?:: f`d FL 5 FH. Co`dL f`dH.
1. FD os f` fitotudb mc f` osmshbibs trof`aib FLH o` wnohn FL 5 FH. Xnmw tnft 7
(o) FD losbhts LH
(o) FD losbhtsF.
>. LB f`d HC frb twm bqufi fitotudbs mc f trof`aib FLH. Rso`a \NX hm`arub`hb ruib, prmvb tnft tnb trof`aib FLH os
osmshbibs.
?. FLH os f` osmshbibs trof`aib wotn FL 5 FH. Drfw FZ LH tm snmw tnftL 5H.
6:. FL f`d HD frb rbspbhtovbiy tnb sefiibst f`d im`abst sodbs mc f qufdroiftbrfi FLHD. Xnmw tnftF 0H f`dL
0D.
66. D os f pmo t m` sodb LH f`dFLH suhn tnft FD 5 FH. Xnmw tnft FL 0 FD.
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EF@OXNKREF\
EFTNBEFTOHX
(D) QNOHN MC TNB CMIIMQO@A XTFTBEB@T FB T\RB (T) F@D QNOHN F\B CFIXB (C) 7
(f) F`aibs mppmsotb tnb bqufi sodbs mc f trof`aib frb bqufi.
(l) Bfhn f`aib mc f` bquoiftbrfi trof`aib os 3::
(h) Xodb mppmsotb tnb bqufi f`aibs mc f trof`aib efy lb u`bqufi.
(d) Tnb losbhtmrs mc tnb twm bqufi f`aibs mc f trof`aib frb bqufi.
(b) Twm roant trof`aibs frb hm`arub t, oc nypmtb`usb f`d f sodb mc m`b trof`aib frb rbspbhtovbiy bqufi tm tnb
nypmtb`usb f`d f sodb mc tnb mtnbr trof`aib.
(c) Oc f y twm sodbs mc roant trof`aib frb rbspbhtovbiy bqufi tm twm sodbs mc mtnbr roant trof`aib, tnb` tnb twm
trof`aibs frb hm`arub`t.
(a) Tnb twm fitotudbs hmrrbspm`do`a tm twm bqufi sodbs mc f trof`aib `bbd `mt lb bqufi.
(n) Oc twm sodbs f`d f` f`aib mc f trof`aib frb rbspbhtovbiy bqufi tm twm sodbs f`d f` f`aib mc mtnbr trof`aib,
tnb twm trof`aibs frb hm`arub`t.
(o) Oc twm sodbs f`d tnb o hiudbd f`aib mc f trof`aib frb rbspbhtovbiy bqufi tm twm sodbs f d tnb o hiudbd f`aib
mc tnb mtnbr trof`aib, tnb twm trof`aibs frb hm`arub`t.
(g) Oc twm f`aibs f`d tnb o hiudbd sodb mc m`b trof`aib frb rbspbhtovbiy bqufi tm tnb twm f`aibs f`d tnb
o`hiudbd sodb mc tnb mtnbr trof`aib, tnb twm trof`aibs frb hm`arub`t.
(k) Oc FLH Z\V, tnb` FL 5 ZV.(i) Oc twm f`aibs f`d f sodb mc f trof`aibs frb rbspbhtovbiy bqufi tm twm f`aibs f`d f sodb mc tnb mtnbr trof`aib,
tnb` tnb twm trof`aibs frb hm`arub`t.
(e) Oc DBC \ZV, tnb`D 5V.
(`) Oc ZV\ HFL, tnb` ZV 5 HF.(m) Oc twm sodbs mc f trof`aib frb u`bqufi, tnb` ifrabr sodb nfs tnb sefiibr f`aib mppmsotb tm ot.
(p) Oc twm f`aibs f trof`aib frb u`bqufi, tnb` tnb arftbr f`aib nfs tnb ifrabr sodb mppmsotb tm ot.
(q) Xue mc f`y twm sodbs mc f trof`aib os arbftbr tnf` tnb tnord sodb.
(r) Doccbrb`hb mc f`y twm sodbs mc f trof`aib os bqufi tm tnb tnord sodb.
(s) Mc fii tnb io`b sbaeb ts tnft hf` lb drfw` crme f pmo t tm f io`b `mt hm`tfo`o a ot, tnb pbrpb`dohuifr io b
sbaeb`t os tnb snmrtbst.(t) Xue mc tnb tnrbb sodbs mc f trof`aib os ibss tnf` tnb sue mc ots tnrbb fitotudbs.
(B) COII O@ TNB LIF@KX 7
(f) Xodbs mppmsotb tnb bqufi f`aibs mc f trof`aibs frb ]]]]].
(l) F`aibs Mppmsotb tnb bqufi sodbs mc f trof`aibs frb ]]].
(h) O` f` osmshbibs trof`aib FLH wotn FL 5 FH, oc LD f`d HB frb ots fitotudbs, tnb` LD os ]]]] HB.
(d) Oc fitotudbs HB f`d LC mc f trof`aib FLH frb bqufi, tnb` FL 5 ]]]].
(b) O` roant trof`aibs ZV\ f`d DBC, oc nypmtb`usb ZV 5 nypmtb`usb BC f`d sodb Z\ 5 DB, tnb`
ZV\ ]]]].
(c) O` f trof aib FLH, oc LH 5 FL f`dH 5 >::, tnb`L 5 ]]]].
(a) O` f trof`aib ZV\, oc Z 5\, tnb` ZV 5 ]]]].
(n) Oc twm sodbs f`d tnb ]]] f`aib mc m`b trof aib frb rbspbhtovbiy bqufi tm twm sodbs f`d tnb o hiudbd f`aib mctnb mtnbr trof`aib, tnb` tnb trof`aibs frb hm`arub`t.
(o) Oc ]]]] sodbs mc f trof`aib frb rbspbhtovbiy bqufi tm tnb tnrbb sodbs mc tnb mtnbr trof`aib, tnb` tnb trof`aibs
frb hm`arub`t.
(g) Oc o` twm trof`aibs FLH f`d ZV\, FL 5 V\, F V f`dL 5\, tnb`FLH ]]]].
(k) Oc o` twm trof`aibs FLH f`d DBC, FL 5 DC, LH 5 DB f`dL 5D, tnb`FLH ]]]].
(i) Oc o` twm trof aibs ZV\ f`d DBC, Z\ 5 BC, V\ 5 DB f`d ZV 5 CD, tnb`ZV\ ]]]]].(e) Xue mc f`y twm sodbs mc f trof`aib os ]]]] tnf` tnb tnord sodb.(`) Oc twm f`aibs mc f trof`aib frb u bqufi. tnb tnb sefiibr f`aib nfs tnb ]]] sodb mppmsotb tm ot.(m) Mc fii tnb io`b sbaeb ts drfw` crme f pmo t tm f io`b `mt hm`tfo`o`a ot, tnb ]]] io`b sbaeb`t o` tnb snmrtbst.(p) Doccbrb`hb mc f`y twm sodbs mc f trof`aib os ]]]] tnf` tnb tnord sodb.
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EF@OXNKREF\
EFTNBEFTOHX
(q) Oc f`y twm sodbs mc f trof`aib frb u`bqufi, tnb` tnb ifrabr sodb nfs tnb ]]] f`aib mppmsotb tm ot.(r) Tnb sue mc tnb tnrbb fitotudbs mc f trof`aib os ]]]] tnf` ots pbroebtbr.(s) O` f roant trof`aib, tnb nypmtb usb os tnb ]]] sodb.
(t) Tnb pbroebtbr mc f trof`aib os ]]] tnf` tnb sue mc ots ebdof`s.MLGBHTOSB T_ZB VRBXTOM@X
6. Cmr f trof`aib FLH, tnb trub stftbeb`t os 7
(F) FH< 5 FL< + LH< (L) FH 5 FL + LH (H) FH 0 FL + LH (D) FH 9 FL + LH
::
(L) 83::
(H) ?::
(D)
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EF@OXNKREF\
EFTNBEFTOHX
(D) FL FZ
6:. Tnb ebfsurbs mc tnrbb f`aibs mc f trof`aib frb o` tnb rftom 6 7 < 7 8 Tnb` tnb trof`aib os 7
(F) \oant f`aibd (L) Bquoiftbrfi (H) Osmshbibs (D) Mltusb f`aibd
66. Tnb nypmtb`usb mc f roant f`aibd trof`aib os
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EF@OXNKREF\
EFTNBEFTOHX
6>. M`b sodb mtnbr tnf` tnb nypmtb`usb mc f roant f`aibd osmshbibs trof`aib os 4 he. Tnb ib`atn mc tnb pbrpb`dohuifr m`
tnb nypmtb`usb crme tnb mppmsotb vbrtbx os 7
(F) > he (L)
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EF@OXNKREF\
EFTNBEFTOHX
Oepmrtf`t @mtbs