iap clii s6 19. lliang S-2016. Inrong Dai hpc Ta> Nguven [SSN 1859-461 P H L O X G P H A P G I A I M O T S O D A N G T O A N
H I N H H O C K H O N G G I A N T R O N G C H U O N G T I U N H D U B I D A I H O C Kieu Manh Hung' Ngay nhan bai 20/7/2016; Ngay phan bien ihong qua: 01/8/2016; Ngay duyel dang: 15/8/2016
T O M T A T
Trong bai bao nay, chiing toi irinh bay phuong phap (PP) giai mol .so dang loan c o ban ciia Hinh hpc khong gian (HHKG) Trong moi dang loan, ngoai vice neu Ien PP giai chung chung toi con dua ra cac chii >', eae bude giai cu ihe cung nhu cac vidu (VD) minh hoa. Qua cac VD nay. mgt ldn nfra giup hpc sinh (HS) ciing co PP. ren luyen kT nang de giai lung dang loan tuong img.
Tie khda. tTmb hoc khdng gian. dir bi dai hgc. kT nang, giai lodn.
1. .MO DAU
Nhiem vu day va hpc kien ihuc r non Toan noi chung cung nhu kien ihuc mon HHKG chi dat kk qua cao neu ngudi giao vien (GV) biel he thong chuang irinh. npi dung tirng chuong trinh va eac chuan kien ihirc, kT nang can co trong lirng khai mem (KN). ben eanh do dua ra duoc cac dang loan va PP giai cho liirng dang loan (Ngu\en Thanh Himg. 2009) Theo lai iieu chu^n kidn ihirc mon Toan do nhom lac gia Dao Tam, Chu Trpng Thanli va Nguven Chien Thdng bien soan nam 2011 ihi l i s sau khi hoc \ong mm chuong. moi phim cua mon Toan irong ehuang Irinh ph6 thiing phai niim dugc nun so chuan kien ihirc \ a kT nang lutTng img. Van de nay cang phai duac chu trgng dtii vai HS di^ hi dai htic (DBDH). Do do doi h6i nguoi giao vien khi giang day cho doi luong nay phai dua ra mpi so chuan kidn thiic va k? nang ckn dai duac Vi day la doi lugng chu>'cn hici. lliupe dimg bao dan lgc thieu so noi co di^u kien kinh le kho khan nen ngoai nang luc su pham cua nguai GV con can ihcm long nhicl husk cung nhu kinh nghiem iruc licp giang da\. De neu PP giai mgt s6 dangloan IIIIKG cho HS DBDH. la lim hi^u mol so van de sau.
2. PHUONG P H A P N G I H E N CUU Irong b;ii bao na>-. chimg i6i sir dung phuong phap nghicn cim li luan - (sach giao khoa. giao irinh, ) noi dung ciia cac mon HHKG va phuoiie phap giai mot so dang loan HHKG tronii ehuan«
irinh DIJDH
3. KKT QU.\ NCHIK.N ClfU VA TH.-\0 LUAN 3.1. Niii dung phin, HHKG irong chirang irhih
DBDH ^ Ciom 5 durtmg (Dai euong \ ^ duima ihana (dl)
\a mai phang (mp), Quan he song song (ss), Quan he Miong giie (vg). DiCm lich \ a liie ueh khfii da dicn. Mai eau) Kicn thirc phAn HHKG na\ iiron"
doi kho, mang tinh triru lirgng cao do do gay ra nhieu kho khan cho HS DBDH.
3.2. Phirang phdp gidi mat so dgng todn HHKG trong chuong trinh DBDH
Do mai bang kien thiic cua cac em HS DBDH khong cao, Eu tuong cita cac em co phSn tu ti, mac cam nen viec Iruyen dat kien thirc, giiip cac em giai dugc cac dang loan HHKG vo ciing kho khan.
Tuy nhicn, neu nguoi giao vien (GV) biet dira ra cac dang loan co ban, cac chuan kiin ihiic, kTnang va cdc PP giai cho limg dang loan. K6t hpp vdi vice kicn tri huang dan HS ihuc hien theo dung PP do. thuong xuyen luyen tap cac bai loan luong lu Ihi vice giang day ciia GV ciing nhu vice hpc va giai loan HHKG cua HS se d o kho hon. Qua do giilp HS CO the hgc va giai dugc cac bai loan HHKG cung nhu cae ki^n thirc v6 Hinh hpc. Tao lien de de cac em buoc vao li6p can cac npi dung kien ihirc ciia chirong trinh dai hgc sau nay Sau day la tim hieu mgt so dang loan sau:
3.2.1. Dgng lodn I: rim giao tiiySi (gl) cua hai mgl phang
Yeu cau. Biel dugc kn gl, kn didm chung ciia hai mp. Cac djnh ly vc gt ciia hai mp (Ddo Tam vd cs. 2(110).
^ a) Pl> Cdch T. Tim hai diem chung ciia hai mp do (Dicm chung thir nhat thuang d l thay; Di^m chung Ihu hai la giao di^m (gd) ciia hai duong thang (dl) con Iai, khong qua dilm chung thir nhat) Cdch 2. N5u Irong hai mp co chira hai dl SS Ihi chi can lim mot didm ehung. gl la dl di qua diem chung va ss voi hai dt nay.
b) VD hinh 3 2 1: C h o hinh thang ABCD va ABEF co chung canh 1cm AB va khong cimg nam trong mgl mp. Xac dinh gl ciia cac cap mp sau (AEC) va iBFO), (BCE) va {ADF) (Van Nhu Cuong. 2011).
/ //A- kh-iaihrhiwiT
" irong Dai hue Tm .\gin lanlillii,!^, DT 0<-J4'ir)^95g Email
ic'diiaigdhriii'a-f^iiiail tai
l £ E £ l l L i 2 J ? i l ' ' ' a n g 8-2016. Irutms Dai hpe Tav \ s u '
a) Xci hai mplAEO va iBFD). (Hinh 3 2.1).
v.'^-.c^.««=,|'-^^--(^^^-)^„i
[l = BD::{BFD) diem ehung cua (AEC) xi, (BFD) Goi J-.4Er,FB
_^U = .-IEa{AEC)
\j = FB^{BFD)^
/ l i diem ehung cua {AEC) vi, (BFD)^ (AEO nl.BFD)=U
Hinh 3,2,1
oH li diem chung ciia
> K li diem ehung b) Xet hai mp I.BCE), (.ADF). H = FDr\ EC
\HSFD^(ADF)
\H<:ECCZ(BCE) ~ {BCE)vi,(ADF). K=ADr-,BC
lKt:ADc:{ADF) [HeBCciBCE) '
cua (BCE) va i.-lDF). Vay (BCE)n(ADF)' HK.
3.2.2. Dgug tadn 2: Taa gd c„a d, a vd a,p {P).
Yen eiu: Biet cich nhin ra eic dt d6ng phing.
eac dt khong dong phang,
a) PP: Tim gd ciia a vol mot dt b nio do n5ni trong (P). Trong truong hgp dt b khong the hicn ro ta tiiin hanh tim dt b theo cac buoc sau, Tim mp (Q) chira a. Tim gl b cua (P) v i (Q): Tim .4 = a n b K h i d 6 A = a n ( P )
Ifinh 3.2.2
b) VD lanh 3 2 2: Cho hinh chop dlnh 5. day li hinh thang ABCD vo, AB la day loii. Goi M, N lan lugl li laing dii3m cua cic canh SB, SC Tim gd eua dt SD vgi mp t.AMN) (Van Nhll Cuong, 2 0 1 0 .
G,a,
Xet hai mp (SAD) v i (AMN) eo A la didm chung X M hai mp (SAD) xi, (.SBC) co .?£ li gt (Hinh 3.2,2) Ggi F li, gd eUa AW va ,?£
f is,V£\'c:(/l,VCV) F e 5 £ c ( , S X D )
Fe{AMN) F e {.UD)
, Viy do X F li gt eua (,-l.\i\') va
(SAD). Goi G li gd c i a .SD va AF
idSD iGeSD
^ [G e AF c (X,'l/A') ^ [G e{AMN) G-SDr-(A.m').
3.2.3. Diing tadn 3: Chin,g „„„h (ca) ba d,ea, ,l,d„g bang
Yeu eau: Xae dinh duge vi tri cua ba diem vgi cac dt \ a cic mp trong bai toin.
a) PP: De em ba dicm thing hang ta em ba dicm niy thuoc hai mp phan biet (vi khi do chung thugc gt eua hai mp nay )
Tap chi s6 19. thing 8-2016. iruong Dai hpc Tav Nguyen
Hinh 3.2.3
b) VD limh 3.2.3. Cho b6n diam O, A. B, C khong dong phang Tren eac dl OA. OB, OC lan luot lay cac diem A'. B\ C khac O sao cho cac dl sau day c^i nhau BC va B'C\ CA va C'A\ AB va A 'B • Gpi //. /,,/ lan luoI la gd ciia A 'B'. B 'C \ CA' va\ mp {.'[BC] Chimg mmh rSng H. I. ./
ihang hang.
Giai:
Gia su A 'B' cai AB lai dicm H Khi do diC-m / / thupc ca hai di A'B' va AB Mai khac. AB nSm irong mp {ABC) nen / / la gd ciia A'B' voi mp {ABC). Ggi / , . / lan luoi la gd cua cac dt B 'C' va CB. C 'A ' va CA ihi /,./ Iheo thir lu la gd ciia B T ' , CA • vai mp {ABC) (Hinh 3.2.3) Ta ctS li. I../ ]k\
lutn ihupc cac dl A 'B '.B'C'. C 'A' nen chiing cimg ihugc mp {A'BC). Mat khae H. 1. J ciing ihuoc mp (.-IflC) Do do ba diem //. /. y ihuoc gt •i ciia hai mp phan bicl {ABC) \ a lABC) nen chimg liiang hang.
3.2.4. Dgng lodn 4: Dimg ihit'i dien cua mp il') vd mdi kbdi da diOn T
Ytiu eau: Hieu dugc kn ihicl dien. PP giai bai itian tim gd ciia dl \ ii mp. iim at cua hai mp (3 2 I 3 2 2)
a) PP- Tim thici dien cua mp (/>) va kh6i da dicn T. la lim dtian gl eua mp [P] vai cac mai eua khoi da dicn T Cu ihc la ihue hicn cac buac sau, Bin'K I Tir cac diem chung cil s5n \ a c d m h g l d a u lien cua (P) vai mol mai cua 7". Buac 2 Keo dai gl da CO. lim gd vtVi eac canh cua mat nav tir do lam tirtTiig tir la lim dutTc cac gt con lai, cho lai khi eac doan gl khep km Thiel dien can dung la da eiac gom eac doan gl khep km nav
Hinh 3.2.4
b) VD hinh 3.2 4: Cho hinh chop tir giac SABCD. Ba diem A\ B'. C lan lupt nam tren ba canh SA. SB. SC nhung khong trimg voi S. A. B. C.
Xac dinh thiet dien ciia hinh chop khi cat bai mp {A B C •) (Van Nhu Cuong, 2011).
Giai.
Ki hicu O la gd ciia hai duong cheo AC va BD.
Goi O • la gd cua /) C' va S0\ D ' ta gd ciia hai dl B ' O ' v a ^ D (Hinh 3,2.4).
- Neu D' ihuoc doan SD ihi ihiel dien la lir giac A-BVD:
- Neu D' nam Iren phan keo dai ciia canh SD, ggi E la gd ciia CD va C 'D \ F la gd ciia AD va // 'D \ Khi do Ihict dien la ngu giac A 'B C 'EF.
3.2.5. Dgug toiin 5: Cm hai dl a. b ss Yeu cau. Hi^u dugc djnh nghTa hai dl ss. Cac dinh ly de cm hai dl ss trong hinh hgc phdng da hgc a lap duoi.
a) PP. Cdch 1. Ta cm a. h d6ng piling r6i ap dung cac PP cm hai dl ss Irong hinh hoc phdng, chang han ap dung dinh Ii Ta lei, duong trung binh, Cdch 2 Cm jiai dl a. b ciing ss voi mpt dt ihir ba t; Cdch 3: Ap dung dinh li v^ gl "Nau hai mp cdl nhau va ldn lugl chira hai dl ,ss cho Iruoc ihi gl eiia chiing cung ss vai hai ch dy"; Cdch 4: Ap dung dmh li ve gt cua ba mp '"Nau ba mp cdl nhau Iheo ba gl ihi ba gt nay hoac d6ng qui hoac ss. Sau do la cm them co hai gl trong ba gl do ss,
b) VD limh 3,2 5 Cho hinh chop SABCD co day/JflCD la hinh binh hanh Goi M N. P (^ la cac dicm lan lugt ndm Iren cac canh BC. SC. SD. AD sao cho MN//BS NP//CD. MQ//CD a) Cm POfi SA, b) Gpi K la gd cua MN va PO. Cm rdng SKJI AD//BC
IgPihi_so 19. thang 8-20i6. truang Dai hpc Tay Nguven ISSN ]S59--16il
5 \ ' SP a) Trong tam giac (ta) SCO co ^ =
~ SC SD S.\ BA/
SC ~ BC ' POI/ASiHlnhl
BM _ AO BC " .AD trong lg SCB co
SP AO
^ SD~ AD^ PQI/AS iHlnh 3,2.5);
b) Xat mp iSAD) va mp (SBC) co .SK la gl v a ldn lugt chua hai dt ss AD va BC => SK /AD BC.
3.2.6. Dgng lodn 6: Tim goe giira hai dl cheo nhau a, b.
Yeu cau: Biat duoc goe giua hai dt a. b !a goe giira hai dt cimg di qua mot diem va ss voi hai dl a, b. Tire la viec xac dinh goe giTra hai dt trong khong gian qui ve xac dmh goe giira hai dt trong cimg mot mp va khong phu ihuoc vao diem ehgn.
a) PP: Budc 1: La> mol diem O lii> \ , Budc 2:
Qua O dung a'Ila. b'llb:Budc 3' Goe nhgn tao boi o • va A" la goe giira hai dl a b.
b) Chli y: Nen chgn 0 ihuoc a hoac b khi do chi can ve them dt ss voi dl con lai
c) VD limh 3 2 6. Cho lang tru ABCA B 'C' cb tat ca cac canh deu bang a. Biei goe tao ihanh boi . canh ben va mat dav la 60" va hinh chiau H ciia dinh A ien mp (.-i B C') irung v 61 trung dicm cua canh B'C Tinh tan cua goe giCra hai dl BC va AC.
Giai:
Goi H I hinh ehicu cua xuong mp {.A'B'C ) . i.u> ra / / la irung dicm ciia BC (Hmh 3 2,6) Vi BC B'C=>
iBCAl"] = [B~X\AC']- XCI lg vuong ,-IC7/e6 aJi
. -rr^r -iH 2 n rr^i. ,„•.
Hinh 3.2.6 3.2.7. Dgng lodn 7: Cm dt a ss vai mp iP).
Yeu cau: Ve hinh va xae dinh duoc dl a va mp (P).
a) PP: Cdch / : Ta cm « ss voi mgl dl /> c (P).
Khi dl b khong the hien ro ta lam iheo cac buoc sau Birac I: Tim m p i m p ( O ) chira a: Bir&c 2- Tim h = {P)r'A,0): Birdc 3. Cm blla. Cdch 2. Cm a c
( 0 m a ( 0 (/•),
b) VD hinh 3-2-": Cho hinh chop S.ABCD. ddv .ABCD la hinh thang {AB CD) Mat phang (a) chira AB. cat cac canh SC. SD theo thir tu tai cac diam P O. Mat phdng ({J) chtia CD cdt cac canh SA. S S theo thir tu tai cac diam .\f. .V(Hinh 3 2 7 ) .
D C Hinh 3.2.7
i ) C m . l A {ABCDj.PO lABCD).
II) Goi / la gd eua .40 \ a D.\l. 71a gd cua BP v CV Cm rang U i.ABCDi. II {XfXPO)
Giur-
il Gia su ia } ~ \SCD}= PO. M CD AB m AB:zia)-pu CD~PO (ABCD) Tuon- n ...\B'C. Man I B'C .AC ] - . " ' gia su ( / ? ! - ( SAB\ = .\r\ . AB (_ D ma ( D
Tap ehi so I ^. thang 8-2016. tririmg Dai hoe Tay Nguyen ISSN 1859-4611
c ( / ; ) =..\D.- AB =. -in- (.IBCD).
ll) Ta CO (OABP) r:{i\.fDCK) = IJ m i CO;,(Q.-lBP) ^U-CD='a.-(ABCD) Tuong tu ta CO CD'-'PO ^IJ//PQ '^JJ--(.\P,'PO).
3.2.8. Dfing lodn 8: D,n,g ,l,iel d,e„ ss vdi „,6, d, a cho ,rudc-
Yeu cau : Ve duge hinh, nam chac eae dinh li trong chuong quan he ss va biet PP giii bai toan 3 2 1,3,2.2."'
a) PP : Su dung tinh chit "Cho dt a ss vol mp (P). neu cat (O) chila u va cat (P) khi do gt b cila (P) v i (Q) ss vol a~.
S
Hinh 3.2.8
b) VD hmh 3 2 ,V. Chg liinh ch6p SABCD. c6 diy .lliCD la miit tir giac [dv Goi O li, gd eua hai dutmg eheo .IC va BD Xic dinh Ihiei" dien ciia hinh chop cil boi mp (u) dl qua O. ss vgi AB \ a SC Thii't dicn dd la hinh gi?
GIOI.
Vi mp [o) q u a i ^ v a s s v o i . - l f i => gt cua {a) vol mp (ABCD) li dt d qua <). ss vgi .-IS. cil AD 1.11 E. BC lai F Mat khac ( a ) qua F. ss viii ,?C =>
gl cUa [a] vcri mp (SIIC) li d i i / ' q u a F. ss voi .W c a l , V » l a i G ( l l i n h 3 2,S) Hon nua ( a ) q i i a & s s liii AB suy ra gl eua (cr) viii mp (.4.SB) li, dt l i "
qua C, ss vol AB. cat .?..] tai / / Suv ra thift dicn la lu giac HGEF Hon nua EF AB Gll'/.IB ,K„
IIGEF la hinh thang
.1.2.9. Daag loan 9: Diaig duii dap, ci, bm
„„), mp (/•)>! vd, mp (O),ho „•„,-„.
Hinh 3.2.9
a) PP: Sli dung dinh ly "N^u hai mp ss bi eit boi mgt mp thir ba thi hai gt ss vol nhau".
b) VD hiah 3.2 9: Cho hinh hop ABCD.A'BVD: Tren ba eanh AB. DD; CB' lan luot lay ba diem M. N. P khong triing voi cac
, AM D'lV B'P dmh sao cho - — = - — - = -—— . Xac dinh thiet
AB D'D BC dicn cua hinh hop khi cit boi mp (MNP) (Van Nhir Cuong, 2011)
G,a, -.
Kc ME//AB; EEEB. Ta co , ir_£_Mf _BV£_ B'P
B'B ^ AB^ g.jj' g.^.,^EP//BC:3 EP//AD-^(MEP)ll(ABD-). Vl D'N - BE nen EN//BD- ma B'lycz{AB'D') v i E€{MEP) nen EAic (MEP), luc (IvlNP) triing vol (MEP).
Suy ra (MNPyi(AB 'D'). Tir M ke ME//AB', tir P kd PF//B -D: Tir N kc NK//AD' cil /40 tai K. Do do Ihiel dien can tim li luc giae MEPFNK co cac canh doi ss (Hinh 3.2.9),
3.2. HI. D,„,g raau Ilh Ca hen mp ,v,v.
Yen eau: Vc diroc hinh v i .vie djnh duge hai mp. Hieu va nim chic c i c dinh li lien quan iin hai mp ss (Dao Tam v i cs, 2010)
a) PP Ta em mp niy chua hai dt cit nhau lan lirol ss vol mp kia.
.52
l i i L ^ si' 19. thing 8-2016. trucmg Dai hgc T i \ Nguven ISSN 1859-461
> EP 11 BC •^EPII Hinh 3.2.10
b) VD hinh 3 2 / 0 ' C h o hinh hop .tBCD.yB'CO' . Tren ba eanh AB. DD'. CB' lin luot lay ba diem M. N. P khong trung voi cac dlnh sao cho
AM D'N B'P
. Cm ring mp (KINP) vi mp
AB D'D B'C '^
(AB 'D') ss vol nhau, Gia,.
Ke ME.'/AB'. (E'^EB) (Hinh 3 2 10). Ta co . irE__AM^ B'E _ B'P
B'B~ AB ITB^ B'C
AD '^ (MFP)II(AB D'). Vi D 'iV -B'E nen ENII B'D'mz B'D'^(AB'D'] v i EG(MEP) nen tir
=> EN,^{,),1EP). tire (A-INP) (rung vol (,WEP).
Vay {i-li^'P)ll(ABD').
3.2.11. Daag loan 11: C„, ha, d, vg.
Yea cda. Ball cdcl, cm ha, d, vg l,-o„g hiah hoc plldag Bie, cdcl, xi,c diah goe giirrr hai dl /Ddo
Tam vd cs. 21)10)
a) PP -. Cdch I Nil-u hai dl cit nliau thi su dung cac PP da diing irong hinh hoc phang de cm; C{',ch 2 Cm dl nay \ g vol mp chira ducmg kia: Cdch 3:
Sir dling PP V eclg Da em hai dt .-IB va CD vg ta cm Ali \ g v o i CD Tire la Ali C'D = 0.
irmh 3,2,1)
b) VD hiab 3.2.11: Cho hinh hop .ABCD.A'B C D CO tat ca c i c canh deu bang nhau. Cm ring .AC ^B D'. AB '± CD '. .ID ' 1 CB'.
G,a,.
Do ABCD la hinh binh hinh va AB - BC - a ^ABCD la hinh Ihoi (Hinh 3,2,11) 3 AC i- BD Mat kliac l i giac BDD B' co BB '//DD'. BB'
= DD' suy ra BDD B' li hinh binh lianh 3 BD//
BD V!,y.AC±B'D' Ta cd C D D ' C ' li hinh binh hinh v i CD = CC 3 CDD C' la hinh tlioi => CD' IC'D Mat khac Iii giac ADC'B' co AD/.'B C ' . AD-BC ^ ADC'B' li hinh binh hinh ^ AB'//
C'D. Vay CD 'y.AB'. Tuong tii ta co AD'l-CB' 3.2.12. Dang lodn 12: C,„ d, a vg vd, lap (/') Vi; duoc hinh. bict dinh nghia dt ss vol mp.
Nam chile cic dinh ly irong chirmig quan he vg, a)PP Ci,cl, I Cm tl vg vgi hai dt eat nhau nam trgng (P) Cdci, 2: Cm a li, true cua mp (/•) (tire la cm .MA = .MB = .MC. NA - NB = NC vgi M. Weti.
b a d i i n . - l B. CE(P)).Ci,ch3-Cn, a^(Q}.(Q}±
(P) vitayb.b- (P)r-yO). Cdcl, A: Cm ti la gl cUa hai mp cung vg vgi (P).
b) VD h„,b 3 2 12. Cho hinh chop SABCD eg da\ ABCD
Tap ehi so 19. thang 8-2016. iruimg Dai licic T i y Nguyen ISSN 1859-4611 li hinh Ihoi tam O. Biet ring SA = SC. SB =
SD. Cm ring SO_(ABCD). ,IC±SD.
G,a,:
Trong lg SBD co SB - SD ^ ASBD cin =•
5 0 ± BD Mat khie trgng tg S.AC cti SA = SC ^ iS.AC can =>SO± AC. Viy .-iO±{.4BCD].
Tuong III, do AClSO. ACLBD = .4Cy(SBC)
= > . J f „ S D (Hinh 3 2 12)
3.2.13. Dangloan 13: Ti,ahiahcbieavg.A 'caa A ,riai IPj Tinh khoaag cdch ,ic d,e,„ .1 dea a,p IP)
Yau cau. Biat pp giai Bai loan 3.2 2. bicl each su dung cac djnh li nhu Pilago. ham so sin. ham so cosin. . .
a) PP: Binrc F Dung mp (Q) qua A va vg voi (P), ((0) chira hai dt b. c cai nhau va ciing vg vai did ndm irong (P)) Buac 2 Tim gl c = (P)^^(Q).
Buac 3: Dung AA' vg voi c lai A', Khi do A" la hinh chiau v g cua A !en (P). Ivhoang each tir.*! dan iP)]aAA:AA' = d[A,{P)]
c) Chiiy: Neu ABIliP) thi d[A. {P)] = d[B. (/>)).
N a u . ' ) S n ( ? ) = / t h i / . ^ / / / S .
3.2.14. Dgng todn 14: Tim gdc giira dt a vd mp {P).
Yeu cau' Bicl dugc goe giifa dt va mp trong cac truong hop dac biel, Biat each qui ve bai loan xac dinh goe giua dt do voi hinh chieu cita no tren mp.
a) PP: Bir&c J: Tim O = a n ( P ) , Buoc 2. Chpn diem A tren a (nen chon A la diem co nhiau linh chdt lien quan), dimg AH_1_(P), ( H e ( P ) . tim H dua vao bai loan 17).
b) VD hinh 3.2.14. Cho hinh chop tg deu co canh day bang 3fl, canh ben bang 2a. Tinh goe gma canh ben va mat day,
S
Hinh 3.2.13 Hinh 3.2.14 h) r i ) / ; / „ / , i : / j ' . C h o l u d i e n . S ' . - l / ; C c 6 . . ( f l C ^ T- • , - / -—— \ la lg vu6ng can dmh B voi . I f = 2 a canh SA vg ^ " " ' ^ ' ^'"'^ -""^ (SA.{ABC)]. Gpi H la Iruc vaimp(.-IBC')v6i.S:-l ^ u Cm rdng (iS"-lfi)-!-(,V/iC) tam I g - ^ S C / l a trung di^m BC=^ . S ' / / l ( / l B C ) : ^ va lmh khoans; each lir A dan mp [SBC)
(jj^j^ '•ill la hinh chicu ciia SA tran mp (ABC)
^ {.SMirC)) . (.ST^H) . Tg vuong SAHco:
, f~r, AH V-i^l r s\nSAH - — - - ^ = V3 ^ .SAH = 60"
SA SA
=> (SAXABC)]^[SA?AH] = 60"(Hinh 3.2.M).
3.2.15. Dgng todn IS: Xdc dmh gdc giira hai 'lip IP). (Q) - G6c nlii dien
Yeu cau Biet duoc khai niem, giai han duoc goe giira hai mp. xac dinh dugc goe giira hai mp irong truang hgp dac bicl (Dao Tam va cs. 2010)
a) PP Bm'jc I- Tim c - {P)rAQ) Birdc 2: Tim (/?)-Lc (uTc la lim hai dl cdl nhau eiing vg voi c).
Budc 3. Tim a = (R)r.(p). b = iR)n\Q), khi do goe giira hai mp (P) va (£J) la goe giiia hai dl a va Ta CO SA_{ABC)^BC ^.SA .Mai khac
BC ^AB ^BC ^(S-\B] ^[SBC)^(SAB) Tir .-Jdimgdi -(//vg voi.SYJiai//. Ta co BC-L(.S'..|S)
^ BC-All ^ AH - {SBC) - . diA.(.SBC))-AH
Ig ^uone SBC .o ^ = ^ ^ ^ J,
-III' .SA- All- ~ -tBC CO lif . II,
c? AB' = 2a- \'a.\
Al ' c=, 2
1 1 Au-
di I . I W i C I ) - - III - - - u j - ( l l m h J
V 2
,s:-i- All- Tu do
Tap ehi 19. thane 8-2016. truana Dai hgc Tay Nguye; ISSN 1859-4611
Hinh 3.2.15
b) Chuy: - Khi Ox = ( R ) ^ ( P ) . Oy = ( R ) ^ (Q), ta noi goe xOy la goe nhj dien. Ta thucmg lim goe nhi dien bang each chgn O Iran gl thoa man nhieu linh chat nhat. Chang han O la trung diam ctia doan gt. O la ehan duang cao ciia tg nao do ma canh day la doan gt nay, Nau co hai dl a, b thoa man a nam trong (P) vg voi ( 0 ) . b ndm trong (Q) vg vai (P) thi goe gira hai mp (P) va (Q) la goe giira hai d t a va b.
c) VD binh 3 2.15: Tif mgl diem A/ndm ngoai mp {P) nguoi la ha duong vg IvlA va hai duong xien MB, MC lai mp {P). BiSi MA = a, MB. MC dau lao voi mp {P) cac goe 30" va MB^MC. Tinh cosin goe nhj dian [M. BC.A].
Gidi-
f . ^ ^ ) ) ^(AffiX^BC)) = J . W M J ) =• .4BW = 30"
Tuangtir .4CM = 30" Taeo BC'= MB'-r MC- MB^- AM
MC^- AM
%\v\ABM -i'mACM
= 2a ^ BC- = %a' => BC ^ 2^a • Gpi / la trung diam BC (Hinh 3.2 15). Tg MBC vuong can =>MILBC M'UB = J!sMAC
=> AB = AC -^ tg ABC can => Al ± BC
=>[A/.BC..-)] = (.'1737/) Tg A/BC vuong can tai M ^.\U^-BC = asl2 ^ Xci lg vuong I.UA
™ s i n . v f f i ; i | . - ^ . J j = 3-ffl = 45"
=5[.«.BC..,|J = (.«TI/;) = 45°.
3.2.16. Dang latia 16: Cm hai lap (P). (O) vg Yeu cau- Biel xic dmh hai mp. nim duge dinh ngliia hai mp \ g
a) pp- C,'„l, 1' Cm mp na\ chua mgl dt vg voi mp kia Ca.h 2: Cm goe giira hai mp do CO so do bing 90
Hinh 3.2.16
b) VD hiah 3.2 16: Cho hinh vuong ABCD vi, tg S.4B deu eanh a nam trong hai mp vg vdi nhau.
Goi /. i^ lan lugl l i trung dicm cua.-JS \kAD. Cm ring (.SAD) ± (SAB): (SCF) ± (SID)
G,a,:
- Ta eg (.S.1B)±{ABCD). (SAB)r^(ABCD) - AB. AByAD^(ABCD) ^AD±(SAB)^(SAD)
y(.S'AB).
- Dg tg SAB ddu ^ SI 1 AB. Til (S.4B) y(ABCDy (S.AB)n(.ABCD)=AB, AB LSI c(5,-!S) ^SI±(ABCD)=^SI±CF (1). Ta cm Diy CF. Xtjl tg ADI v i tg DCF ca. AD - AC - a: DA1 = CDF = Iv. Al • DF - - z3
2
&ADI = \DCF => ADI ^ DCF .Go, K-CFr,DI.
t a c g ADiv}DC' = „ ^DCFtlDC^h- 3 DKC = Iv => CF±DI (2) T i r ( l ) v a ( 2 ) ^CF±(SID}^
(XCFl-l-(5/Z))(Hinh3 2 16).
3.2.17. Dangloan 17: Tiah khnaag cdch giira ha, d, cheo nhaa - D,n,g doaa vg chuag caa ba, d, cheo nhaa a. b.
Ytiu ciu. Ilii;u duge khai niem, ,\iie dinh duge hai dl C i c dinh li hen quan den do dai canh ciia Ig Irong Ilinh hge phang.
a) PP Cdch I Bade I: Tim mp (P) vg vtii c, tim 0-= o^(P}. Bade 2- Tim hinh chiiu ii'cua dt b Ircn mp (P). (Tim / = b n (/>) Liy dijm M e 4.
qua .1/ ke dt .MK^(P). ta eo IK li, hinh chicu b' cua b iren (P)). Bade 3: Trong mp (P) dimg OH 1 b ' khi dd ta co O / / - d\a. b] ~Badc 4 Diing lIBIIz.
B cb. Bade 5: Dung BAIIOH. Aea ta eo AB la doan vg ehung cua „ va b Cdch 2: Tiln mp (P) chira dl tl \ a ss vtin dl b. Khi do d[ab] = il\b.(P)\
= t^.l/. (/')). (.\/li diem tiiy ; ttin b).
Tap chl sti 19. thing 8-2016. trirbng Dai hoe Tiiy N"guyt:n ISSN l859-4{ri
J c Hinh 3.2.17
b) VD hinh 3.2.17' Hinh chop S.iBCD e6 ABCD lii hinh vudng eanh a. SA = SB = SC ^ SD
~ 0-12 . Ggi /, y lan luot la trung diem eua AD va BC. Tinh khging cich giira hai dt AD va SB.
G,a,:
Tir / kc dt vg vtri SJ tai H. Tir / / dung dt ss vol AD cil SB tai .M. Til ,Udung dt ss vgi IH ck AD lai P. Ta ct> IH LSI. IIILBC => IHy(SBC) ma NPi/IH =>NPy(SBC)^NPySB. AD±(SI.I)
=>AD1 dutmg \
NPLAD. Vay W U
; chung cua AD v i SB. Trong t vuong
J _ _1_
IH' ~ SI- ,' la'- .
.Vai sr-'Srf-liy ,
IH'-SI'*U'
-"-4i-
Vjy4 (—
d{AD.SB}^IH=aJ2- (Hinh 2.17).
4. KET LUAiV
Bii bag trinh bay PP giii mpt s6 dang loan HHKG, dua ra cic ehu y v i VD cii the cho time dang, Ngi dung chtnh l i neu len 17 dang loan co bin nhai cimg vtri 17 VD tirong ijrng d i qua do giup HS cd cii nhin day du hon, siu sic hon khi hgc phan HHHG - mgt trong nhung mon hgc kho g chugng Irinh pho thong noi chung, chuong trinh DBDH noi rieng. Chiing toi hy bii bio se giup giim bdt phin nio kho khin ehg GV v i HS trong viee dtiy va Iioc mon HHKG. Tir do gtip phin nang eag chat Iirong day v i hoc mon Toan trgng chuong Irinh DBDH.
METHODOLOGIES OF SOLVING SOME MATH PROBLEMS GEO.METRIC SPACE IN P R E - U M VERSITY PROGRAM
Received Date. 20.'7/2016. Re\i
KJcu Manh Hung' .ed Dale: 01/8/2016; Accepted for Publication: 15/8/2016
SUMMARY
hi this paper, ivt, present methodologies to solve some basic math problems in geometric space, oec He si' """u " . ' ' " " " ' " ™ ' ° '"'"""^ ""= S="='-"l ^"l-'ions, we also give sgme notes, o i s f d le m U l l " Z'"""' °' '""'"'""•'• ''•'"™'S'' " " = - - - " P l - - •h'y - I - help student consolidale methodologies and training skills lo solve each corresponding math problem
;• .yiace. pre - ,m,vers„y s„„len,K. skills. solv„,g „,alhp,-oblem Keyu-ards Ge<
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Nguven Mong H, (Chu bicn) (2008), H„,h hac 10 NXB Giao due, Nguyen .M.ing Hy (Chu bicn) (2007) Had, bac II NXB Giao due.
Nguven Ba Kim (2004). P ; T O / „ „ ) „ - ; ; „ ) „ NXB DH Su Pham,
Dio ram.Chu Irong T h a n h v a \ . . i i M ; n C h i i n T h - i n n l 7 n i n w i ; , ,
n NXBDHSPllN ^ " - ' ^ " U l i " ! lhang(20IO) Day hoc iheo chada hia Ihire kiadagTow,
•"Y',«IC.,.„„„gS,„„,ev -Pa, Nguven U.„- mh Ihiivi, Phi/iie 1)949195';^
Lnuiil kmhiiiifidbinu gmuil a
