l ttLt'n!: tnliln rung: - -r - + - Alvl B^l {-'}}
Dinh hu6ng khai th6c bdi to6n g6c : Hdy tim , :, ,,^
cac m6i fiAn h€ gitra d0 ddi cdc doqn thdng AO, BO, CO, AM, BN, CP.
Ldi gidi
Ta sir dgng phucrng ph6p tinh di6n tich tam
gi6c dC gi6i bdi ndy (hinh ve).
A
Btu____*- -*C
,AO BO CO
lu do co
-+-+-AM BN CP
* AM-OM +BN-ON +CP-OP
AM BN CP
" (Ottt , ON OP)_, ,_i-r
- J - r ) - | - L.tJ
\AM BN CP )
C6 hai hu'6ng khai th:[c h]ri to6n gOc dC tim
, ;, ..^
c6c mdi li6n hQ gita tlQ ddi c6c <lopn th[ng
AO, BO, CO, AM, BN, CP li sir dsng bien dOi dai s6 ho{c x6t tinh ch6t c6c hinh.
AO BO CO
. AM " BN' CP
cdc so ducrng virx * y - z : 2.
Trong c6ch su dung bi6n COi Aai s6 thi chi tim
tlugc c6c hQ thirc AOI vOi c6c ti s6 x, y, z, cdn
trong cSch xdt tinh chdt c6c hinh thi sE tim
dugc c6c hQ thric kh6ng chi d6i voi c6c ti s5
x, !, Z md cdn v6i c6c ti sti kh6c. Theo y6u
cAu cria dfnh hudng khai th6c bdi to6n g5c x6t
A^
cdc ti sd x, y, z ndr. du6i ddy ding phucrng
ph6p biOn AOi Oai si5 sE nhanh ggn hon.
1) Ap dung BDT Cauchy cho ba si5 ducrng
x, !, z v6'ix + !'+ v : 2 ta co 3W < x * ! + z : 2 n€n27xyz < 8. AOBOCO 8 AM BN CP-27 AM BN CP AO BO CO @d.c.*'trtt. -/ri sm*ffi'$Hteqil#"maiselr M
Ddt Sdac: t Sroc: St , Sco.t: Sr, Szoa: Sc
*r^; oM _suo, _se oN _sroc _su
Ltll - "^^- AM S.Euc ^t ' ,BN Sruc S ' oP =snou *sc cP snrc s ' X6t diQn tich cdc tam gi6c tr6n c6
S: S,n + Sr+ Sc^ OM ON OP_Sr+Sr+Sc_ ^ OM ON OP_Sr+Sr+Sc_ AMBNCPS T$f;N l"tSC 6+s&6+e&ese*'46s e f a&6e&sd&&*$'p * c[u#ib]e (1) 27 >_ 8
2) rt (1) vd f *1* I = 3.rf ,u.5 = 3.rf ,u.5 x y z \xyz AMBNCP9 -r-r- \ - AO BO CO-2
3) Tri (1) vd 6p dsng BDT Cauchy cho ba s5
dugc
dM trN EF lE 3J6
/-r ,-r l->'ljl l-: -
\eo'\ao'\co--!!B 2
Cht i ring trong tdt cil chc U6t ding thric n6u
tr6n thi tling thirc xiry rakhi x : y :, : ?
J
Khi d6 O li trgng tdm cira LABC.
4) Ta co 0 < (3x -Z)2 + (3y -2)2 + (32 -2)2
: g@2 + y2 + r21 - tz1* + y + z)+ 12
:9(.* + y'+ *1 -tz,
suyra e(*+f +*1>tzn€ni+f +*>1.
3
Ding thric xity rakhix : y :, :
1.
uu, '" ( !9\' *( !9\' *(gg\' , !
\AM ) [BNl \cP ) - 35) ra c6 fJi * ,,[i * J, )' 5) ra c6 fJi * ,,[i * J, )'
: x * ! * z -r 2,[*y + ZJi + 2 rli
<x * ! * z -l (x * y + x * z * y + z) : 6.Eingthfc xhy rakhix :, :, : Eingthfc xhy rakhix :, :, :
Ivav Ji+Ji +J) s Ja , vav Ji+Ji +J) s Ja ,
lIo m Eo r=hav ra hav ra
I nn-*r/r, *t/., < v6 .
(Eing thirc xdy ra khi vd chi khi O ld treng
t6,m MBC).
Ta cdn chtmg minh dugc v6i c6c s5 ducrng
x, !, Zth6a mdn x * ! * z : 2 thl:
6) x3 + y3 + z3 ) 1 *t sir dung BET
9
, /l\3 /t\3 Ix'*l-l+l-l>-x. x'*l-l+l-l>-x.
\.3/ \.3i - 3
Tuong t.u khi xdt lfiy thria b6c b6n.
T trl-, *trli *al; .zf- khi su dpng BDr
22 w-
y*1 *:>3?11x.
3 3 Y3'
Hoan ngh€nh c6c b4n sau dd n6u dugc nhidu
h9 thirc gita c6c d0 dei dd cho:
Nguydn Ngpc Hdn, GV THCS Vinh Tulng,
Vinh Phrfic; Hodng Vdn Chung, GV THCS
CAm Vt, CAm Gidng, Hii Duong; Trdn Duy
Qudn, 11T1, THPT chuy6n Nguy6n Binh
Khi6m, Vinh Long.
X
XXXXX XXX
Mdi c6c bpn gui loi gi6i vcri hufng khai thSc
dd cho cria BAr roAN GOC 3 (THPT) sauddy vC Tda soan TH&TT tru6c ngey ddy vC Tda soan TH&TT tru6c ngey
3t/t2120t4.
BAr roAN GOC 3 (THPT)
Ddy sd @) vdi n bdng l, 2,3,.-. dtrqc xdc
dinh nhr sol{ uy: a td s6 thtrc dd cho vd
.2
un+l : u,* n' voi n bdng 1.2,3,...
Hdy tim tii hqng tiing qudt cfia ddy si| @).Dinh hudng khai th6c bii to6n gi5c : Hdy tim Dinh hudng khai th6c bii to6n gi5c : Hdy tim
.a , .:
s6 hqng tong qudt cda ddy s6 (u,) khi trong
ding thuc un+1: un* n2 ta thay ,6 n2 brr bi6,th*c khdc vd thay u" bdi kun vdi m)t vdi gid th*c khdc vd thay u" bdi kun vdi m)t vdi gid tr! nguyAn duong cila s6 k.
TRAN MINH DAT
(GV THPT Hu)nh Tiin Phtit, Binh Dqi, Bdn Tre)
l. Lli gi6i thiQu
Bdi to6n tinh thC tich cira ttrOi ea diQn li bdi
toSn img dung rQng rdi trong thUc t6 vd ludn
xu6t hiQn trong c6c eC tni tuy6n sinh Dai hqc
- Cao tllng cria c6c nlm.
Chinh vi vfly, chring t6i de cO gfing tim hi6u
^).
sdu vO c6c biri toan tinh the tich, v6i mong mu5n gi6i thiQu d6n c5c ban kinh nghiQm
ilugc dric rut qua nhi6u n[m cta chting t6i v6 v6n d6 ndy, d6 chring ta sE tg tin, chtt tlQng
trong vi€c tim loi giii cho bdi to6n ndy. D6 gi6i bdi toSn tinh th6 tich cira ttrOi Oa Aign
thi chring ta thucrng ding hai phuong ph6p:
phuong ph6p tqa dQ, phucrng ph6p t6ng hqp. + Phrong phdp t7a d0: ta chi cdn gfn tqa dQ mQt.cSch hqp li vdo bii toSn ld c6 th6 gi6i
quy6t dugc.
+ Phuong phdp t6ng hqp trong viQc tinh thrS
tich cria ttrOi ea diQn li phucrng ph6p tinh th€
tich th6ng qua viQc tinh diQn tich tl6y vdtludng cao mQt cSch tr.uc titip bing suy luQn. tludng cao mQt cSch tr.uc titip bing suy luQn.
?Oi vOi hai phucrng phSp tr6n, phucrng ph6ptdng hqp n6i ri6ng, tli6u quy6t tlinh ctra tdng hqp n6i ri6ng, tli6u quy6t tlinh ctra
phuong phrip ndv ld ta phii tim dugc tludng cao cira kh6i tla diQn. Nhrmg viQc ldm d6 sC
tuong OOi pntlc tap trong nhi6u bdi to6n. Sau ddy, chring t6i xin gi6i thiQu.mQt phuong ph6p tim cluong cao trong c6c kh6i cla diqn.
Il. NQi dung
A. Bii to6n t6ng quit
Cho hinh ch6p 5.A1A2...4 c6 dinh la S va
ddy ld da gidc n cqnh AtAz...A,. Tinh thii tich
cia hinh chdp S.$,47...4".
Ldi gidi
1
Phdn tlch: Ta c6 cdns thuc V" n,. ,.. = 1.Sh .
3
Trong d6 S: diQn tich d6y (thucmg d6 thuc
hiQn); h: dO ddi duong cao ctra ttrOi ctrOp
(chtng ta tim hi6u sdu vO v0n dO ndy).
ilfi PHUUITG PrilP ilu orm{G clo
*
IRO]IG KHfi OA DIEil
HOANG THE TOI
(GV THPT Tdn Dd, Thanh Thly, Phrt Thg)
Cdc brbc thwc hiQn:
+) Badc 1 : gii str ta tinh tlugc diQn tich <16y li S. +) Budc 2: xhc rlinh dudng cao vi tinh d0 ddi
dulng cao (h.1).
oXdt LSA$ : x6c dinh duong cao SM (SM L Ah tai M.
.Xdt LS44: x6c dinh tludng cao SN
(^SN -i- 4k tai N.
oTrong mp(44...4) : Qua M k6 dudng
thing (d) sao cho(d) L 4A, tqi M. Qua Nk6
tludng thing (d1) sao cho (d1) L 4,42 t1i N. . Gsi (d) O(dr) = {n\ thi ,SH chinh ld ttudng
cao cira hinh ch6p 5.44...A,vi gi6 str ta tinh
dvgc SH: fu.
+) Budc3: Tinh th6 tich Vs.an...n, :!.tr. 3 Ch*ng minh budc 2 (h.1) .t (th ,(! -t -L (s r). L 44' L Arh nHM = ,HM c. L(SMH ,. LSH M. IM M( M,, bL 414 {fi 4,4, 'a1 4 ra Vi nCn suy fieo c (theo {er\ SMH .Me /fi{) Met Az c5ch fimg) r c6ch dUne) kh6c SFl c. (SMH) (1)
(3)
vi l\,qr r sN (theo cdch dr,mg)
) *o, L HN (theo crich drng)
lsn0uN:{N}
I
[.^SN,FIN c (SIVH)
n€n 44 r (SNH) . Mat kh6c S11 c (sNH)
suy ra 44 L SH (2)
Ta c6: f AtAr, AtA", c. (4Ar...4,)
) ' ''-' ' "": "": (3) It"q.,f1 A'A': \'q'\ Tt (1), (2) vir (3) suy ra SH I (4Ar...A") (<Ipcm).
B. MOt s6 ttri dg minh ho4
*rni dg t. Cho hinh ch6p S.ABC co AB :
Scm, AC : '7cm, SA : Zcm, GA = l50o .
Goc tqto boi hai mp (SAB) vd (ABC) biing 450.
Gdc tao boi hei nrp (SAC) vd (ABQ bdng
3oo. Tinh vs .ror . s
Tam grilc HMS vu6ng cAn tai H nOn
SH = MH (2)
X6t A1INS vu6ng t1i H c6:
tun.ffi =#:;;= su:ftmr
Hinh 2 A
Ke SM L AB t1i M, trong mp(ABC) k6rluong thing (d) L AB tai M. rluong thing (d) L AB tai M.
Ke SN L AC t4i lI, trong mp(ABC) k6 <lulngthing (dr) LAC taiN. thing (dr) LAC taiN.
Gqi 11 = (d)a(d,). Khi d6 SH ld cludng caocua hinh ch6p S.ABC. cua hinh ch6p S.ABC.
Ddt h : SH, ta co l^l Szac : =,lA.lC.s;nCAB :1 AB.Ac.sin1500 22 1__t 3s : :_.5.'l 224.:_ - _1cm, ). Ta c6 SMH = 450 vd SNH = 300 (theo gii thico (1)
Ta th6y ili6mHnim trong g6c dZE $heo (l))
Khi d6 AE*frHM:1800 (v\ AMHN ld ttr
gi6c nQi ti6p) = NHM:300.
Ap dpng dinh li c6sin vdo LMHN ta dugc:
IVfr,{z : HMz + HNz -2.HM.HN.cotffiN e MNz = h2 + (Jllr1' - z.tr.djD.cos 3oo (theo (2) va (3)) ^6 e MN2 = h2 +3h' _2.J3h2."" - h2 2 eMN:hhay MN=SH (4) Tt (2), (4) suy ra MN = MH = A,MHN cdn
tqi M = ffiE :ffifi =3oo
-ffii:ffiE =3oo (hai g6c nOi titip
ctng ch5n cr;r:.g MH ). Xet LMAH vudng t4i M co:
sinffiE - w
<> sin3oo - lnlrl
AH AH
o1: Yo AH =2.MH:2h (theo(2)).
2AH
Xet LHAS vudng tai H c6 SAz : SH2 + AHz
(dinh li Pythagore)e32 _h2 +(2h)2 og=5h2 e32 _h2 +(2h)2 og=5h2 93 eh2--eh: sJs -.VQy 1- l3s 3 7Ji. Vrnr, : :.5 ABC.SH = -.::-.-:- : -::-(cm3). 3 3 4 JS 4
*rni dq z. cho hinh chrip s.ABC cd
AB :9 cm, BC : ll cm, AC : 6 cm, SA :3
cru, SB :7 cm, SC : 5 cm. Tinh th| fich khdi
ch(tp S.ABC.
Ldi gi,fiL Ke SM L AB tpi M, trong mp(ABC), k6 dudng thing (d) L AB t4i M. (ABC), k6 dudng thing (d) L AB t4i M.
Ke,SN L AC tpi N, trong mp(ABC), ke duongth[ng (dr) L AC t?i N. Goi H = (d)fr(dr) . th[ng (dr) L AC t?i N. Goi H = (d)fr(dr) .
Khi d6 SHld dudng cao ctra hinh ch6p S.ABC
(h.3).
Iff[rfi?a .osr.'o.srr...{..)o'c'.e 16 ..rrocr..ro{r...er.i'.o.rr ", @ric'ryvt' -/6
Hinh 3Ddth: SH; Ddth: SH;
AB+BC+CA 9+11+6
p= -- ^^ --l?
2
Khi d6 theo c6ng thric H6ron cho LABC tac6
S nr" p(p-AB)(p-BC)(p-cA)
= sN :''t-:1" ='''y: "E.1"* ) (2)
AC63
Ap dpng dinh li c6sin vio LABC ta dugc:
BCz = AB2 + ACz -2.AB.AC.corilZ
= "o"6ii _ AB2 + AC2 - BCz AB2 + AC2 - BCz 2,AB.AC 92 +62 -IP -1 =-=*2.9.6 27 i----------------
+ sinBAC = .i 1- cosz BAC
f I 2Jrn
=n/1__ =_ (4)
\ 729 27
Vi kong LSAB c6 AB > ,SB > &4 n6n M nim
gita tlopn thing AB; trong LSAC c6
AC > SC > SA n6n N nim gita dopn thingAC. Ygy tti6m Ilnim trong g6c 6Za . AC. Ygy tti6m Ilnim trong g6c 6Za .
Khi d6 6Zc + Nfu = l80o (vl AMHN ld tft^...r . ^...r .
gi6c nQi tiep) = sinNHM = sinBAC (5)
Xet LMSAvuOng tqi M co:
SAz - SM2 + AM2 (dinh li Pythagore)
/ -';2
e AMz : sAz - sM2 =32 '\. -l '11235 18 ) I = tostzz+
A1o AM =--l(cm). o AM =--l(cm).
18'
){.et LNSAvu6ng t4i N c6: SAz = SN2 + ANz
(dinh li Pythagore)
- _- f
a ANz = sA?* sN2 :32 -( ?I]!]' = ,,\:.) g \:.) g
5
+ AN:: (cm).
a\
J
Ap dqng dinh li c6sin vio LAMN k6t trqrpv6i (1), (2) vd (3) ta duoc: v6i (1), (2) vd (3) ta duoc:
Ivff,{z : AM2 + AN2 -2.AM.AN.co"6Fi
e MN2 - 168l *25 -2.41 .s .(-t\
324 9 183\27)
-2404e (6)
29r6
Ap dpng dinh li sin vdo LHMN ta dugc:
MN HN
sinNHM sinNMH
(Xem ti€p trang 46) (3)
@
2J182(cm,).
Khi d6 theo c6ng thric H6ron cho A&4,8 ta c6
Ss;a : : Jus . -, _ _1cm. ). I Mdt khric: Ssza : ,.SM.AB .r.,hn5 > SM:2.5t* AB -'' ^ 't'- = ,, J1235(cm)' (1) Df;t p, = SA+AB+SB 3+9+7 19 Dit pr: SA+AC+SC 3+6+5
Khi d6 theo c6ng thric H6ron cho A,SAC ta cb
Sszc =
=@=2Ji7cm2).I I
M[t khilc: Sszc = :.SN.AC
2
TOfiN HOC *of er'*a.oeN{rrio&.cs&6r"s6.
CACDI,/CrNA zU,,{,xl G GJ Ac> NHAII
TRONG Trt}IEN
Theo ei6 thi6t c6 AQ : BP
- k. voi k> o.
QD PC
cHo nn0r sAr roAru
{3 EAI fOAx I5 l\CG. Cho ru'cti€n ABCD.
Gr2i fu{, ll' theo thti' try la trtrng dii:nt cctc canh
.lR. CD. !ri.r dirin; l) lren carth BL'vu J.;i,n Q
tritt t:anrt .4D sa, c.rt, +: + (-,h*ng
QD PC
tttir,'t t'r:ttt,: dtrdtry tlfi.ng l,4N ctit PQ tai trtng didttr G t'ia doan PQ,
DINI{ HUCING 1. Str dsng phuong ph6pvecto d6 chimg minh hai vecto MG vd lE vecto d6 chimg minh hai vecto MG vd lE
cr)ng phucrng, suy ra ba di6m M, N, G thing
hdng (h.1).
A
Nhu vfly AQ: - kDQ vit BP: - kCP .
Tri d6 vd (1), (2) c6 MG: - kNG, suy ra ba
di6m M, N, G thing hdng, tric liL MN di qua
trung cliiim G ci,' doqn PQ .J
EIhIH HLTONG 2. GQi H, K theo tht r.u ld
trung cliiim ctra cdc do4n thing PA, PD. Ta sE
chimg mir$ H, K, G thing hing vd hai tam
gi6c GMH vd GNK d6ng d4ng, tu d6 suy ra
ffiE : frGK ndn M, N, G thinshdng (h.1).
Cach giai 2. Ta thdy GH, GK theo thir t.u li
dulng trung binh cira tam gi6c PAQ vi tam gi6c PDQ, do tt6 GH // AD // GK ndn H, K, G
thdng hdng
"d ry = AQ
(3)
KG QD
Ta th6,y MH, NK theo thir t.u ld dulng trung
binh cria tam gi6c ABP vd tam giSc DCP, do d6 MH // BC // NKvir \ry^ :1 : g g) BP2PC Tn (3), ( ) ve gi6 thi6t c6 HM BP_ _ AQ HG KN PC QD KG ^HM KN n6n _: __- (5) HG KG Do ba di6mH, K, G thinghdng vd MH // NK
md M vdN nim vC trai phia cira ducrng thingHK th\ GHM: GKN. Tn d6 vd (5) suy ra HK th\ GHM: GKN. Tn d6 vd (5) suy ra
C
Ilinh I
Cach giai 1. Tn gi6 thi6t M,ld G theo
li trung dr6m cilr- AB, CD, PQ ta c6 2MG : @*ttp :u,q *VA + ffi