Tgp chi Khoa hgc Trudng Dgi hgc Can Tho Phdn C Khoa hgc Xd hgi Nhdn van vd Gido due 33 (2014) '''' Tap chl Khoa hoc Tru''''dng Oai hoc Can Thd website sj ctu edu vn V^N D U N G PHirONG PHAP TOA D O D E GIAI[.]
Trang 1Tap chl Khoa hoc Tru'dng Oai hoc Can Thd
website: sj.ctu.edu.vn
V^N D U N G P H i r O N G P H A P T O A D O D E GIAI B A I T O A N
H I N H H O C K H O N G G I A N
Nguyen Thi Tuyen'
' HQC vien cao hgc lap Ly ludn vd phuang phdp dgy hgc bg mon Todn, khda 19, Khoa Suphgm
Thong tin chung:
Ngdy nhgn: 29/04/2014
Ngdy chdp nhgn: 29/08/2014
TUle:
Applying the coordinate
method toward the
stereometric problems
Td khda:
Phucmg phdp tga do, tga dg
hoa
Keywords:
Coordinate method,
coordinates chemical
ABSTRACT
Stereometry is an important part of the mathematics curriculum high school today.The stereometric problems are pretty complicated, requiring learners to have good and critical thinking Solving some stereometric problems is relatively difficult and takes more time, but the use of coordinate method will make them much simpler In this article, we would like to introduce how to apply coordinates method toward the stereometric problems
T6M TAT
Hinh hgc khdng gian la mgt bd phgn quan trgng cua chucmg trinh todn trung hgc phd thong hien nay Cdc bdi todn hinh hgc khong gian khd phuc tap, ddi hdi ngudi hgc phdi cd tu duy tdt Viec gidi mdt sd bdi todn hinh hgc khdng gian tuang ddi kho vd tdn nhiiu thdi gian nhung neu gidi theo phucmg phdp tga dg se dan gidn hon Trong bdi viit ndy, chiing tdi xin gidi thiiu cdch van dung phucmg phdp tga do di gidi bdi tgdn hinh hgc khdng gian
1 DAT VAN DE
,.i,::lHmhvhpp^khdh^iaff'la mdn hinh hpc kha trOru
tugng nen da sd hpc sinh e ngai khi hpc vd phdn
ndy Trong cac dd thi tuydn sinh Dai hpc - Cao
ddng gdn ddy, phdn hinh hpc khdng gian dugc ra
dudi dgng ma hgc sinh cd thd gidi bang hai phuang
phdp: Phuong p h ^ hinh hoc thudn tiiy vd phuang
pbap tpa dO Vide gidi bai todn hinh hgc klidng gian
bdng phuang phap hinh hpc thuan tfty gap nhidu
khd khdn ddi vdi hpc sinh vfta hgc xong ldp 12 vi
da phdn cde em it nhidu da quen gidi cdc bdi todn
tpa dO trong khdng gian
Vipc gidi bdi toan hinh hpc khdng gian bdng
phuong pbap tpa do cd rdt nhidu uu vidt, tuy nhidn
hpc sinh cung gap khdng it khd khdn Bdi vi,
phuong phap ndy chim dugc dd cgp nhidu ttong cac
sdeh giao khoa, hpc suih phd tiidng it dugc tidp
can, va phuang phdp ndy chi tdi uu vdi mOt ldp bdi
todn ndo dd ehu khdng phdi luc ndo nd ciing td ra hieu qua
Dd eac em hpc sinh Idp 12 cd thdm phuong phap gidi toan hinh hpc khdng gian, chuan bj cho
ki thi cudi cdp Trong khudn khd bdi bdo, chung tdi chft ydu tap trung vao cac van dd sau:
Ddu hieu nhdn bidt vd cae budc gidi bai toan hinh hpc khdng gian bang phuang phap tpa
dO Dua ra mgt sd each dgt hd ttuc tga dg vdi mOt sd hinh dac bidt
Trinh bay mot so bdi tap hinh hgc khdn^ gian dugc giai theo phuong phap tpa dd va mdt sd bdi tap dugc gidi theo hai phuong phap: Phucmg phap tdng hgp va phuang phdp tpa dO Dieu ndy giup hgc sinh ren luydn ki ndng gidi todn bang tpa
dO vd cd thd ttd ndn linh hoat trong viec lya chgn phuang phap giai sao cho phu hop vdi timg
Trang 22 NQI DUNG NGHIEN CUtJ
2.1 Mpt sd dau bi|u nhan biet bai toan hinh
hpc khoDg gian cd thd giai bdng phuong phap
tpa dp
Hinh da cho cd mOt dinh la tam didn vuong
Hinh chdp cd mOt canh bdn vudng gdc vdi
day va day la cdc tam gidc vudng, tam gidc deu,
hinh vudng, hinh chft nhdt,
Hinh lap phuang, hinh hOp chft nhat
- Hinh da cho cd mdt dudng thdng vudng gdc
vdi mat phdng, trong mat phdng dd cd nhtrng da
gidc ddc biet: Tam giac vudng, tam giac ddu, hinh
thoi,
MOt vdi hinh chua cd sin tam dien vudng
nhung cd the tao dugc tam didn vudng chang han:
Hai dudng thdng eheo nhau ma vudng gdc, hoac
hai mdt phdng vudng gdc
Ngodi ra, vdi mOt so bdi todn md gia thiet
khdng cho nhftng hinh quen thudc nhu dd ndu d
ttdn thi ta cd thd dya vdo tinh chdt song song,
vudng gdc efta cdc doan thdng hay dudng thdng
ttong hinh ve dd thiet lap he true tga
dO-2.2 Cac dgng todn thudng gdp
Tinh dO ddi dogn thdng, kliodng cdch tu
diem den mgt phSng, khodng cdch tft diem ddn
dudng thing, khodng cdch gifta hai dudng thdng
- Tinh gdc gifta hai dudng thdng, gdc gifta hai
dudng thdng, gdc gifta hai m|lt phdng
Tinh thd tich khdi da didn, dipn tich thidt
didn
Chiing minh quan he song song, vudng gdc
2.3 Cdc budc giai bai todn hinh hpc khdng
gian bang phirong phdp tpa dp
Budc 1: Chpn hd true tpa dO Oxyz thich
hgp vd tim tpa dO cdc didm cd lien quan ddn ydu
cau bdi toan
- Budc 2: Chuydn bdi toan dd cho vd bdi toan
hinh hgc gidi tich vd gidi
Birdc 3: Gidi bdi toan hinh hpc gidi tich
ttdn
Budc 4: Chuyen kdt lugn efta bai toan hinh
hpc gidi tieh sang tinh chdt hinh hpc tuang ftng
2.4 Thidt lap h$ tryc tpa dd
Vdn de quan ttpi^ nhdt ttong vide gidi bdi toan hinh khdng gian bdng phuong phjqi tpa do Id thidt lap he tga dO cho phu hgp Sau ddy chftr^ tdi xm
gidi thieu mdt sd phuong phdp d% thidt Igp hd
tpa
dO-(1) Thiit Igp hi tga dg doi vdi tam dien
Vdi gdc tam didn vide tpa dd hda thudng dugc
thyc hidn khd dan gidn, dac bidt vdi;
- Tam dien vudng thi bd tryc tpa dO vuong gdc dugc thidt ldp ngay ttdn tam didn dd
- Tam didn cd mOt gdc phdng vudng, khi dd
ta thidt lap mOt mat cua he true tga dO ehfta gdc phdng dd
(2) Thiet lap h^ tga dg cho hinh chdp
Vdi hinh chdp, vide tpa dO hda thudng dugc thyc bidn dya ttdn d^c tinb hinh hpc ciia chftng Ta
cd cde trudng hgp thirdng gdp sau:
Hinh chdp ddu thi he tpa dO dugc thidt ldp
dya trdn gdc 0 ttftng vdi tdm efta ddy va true Oz trimg vdi dudng cao ciia hinh chdp
Hinh chdp cd mOt canh bdn vudng gdc ven
day thi ta thudng chgn ttyc Oz la canh ben vudng
gdc vdi day, gdc tga dO trftng vdi chan ducmg vudng gdc
- Trong cdc trudng hgp khdc ta dya v^o dudng eao cua hinh chdp va tinh chdt da gidc ddy
de chgn hd tpa dp phft hgp
(3) Thiit lap hi tryc tga do cho hinh hop chU nhdt
Vdi hinh hOp chft nhdt thi vide thidt ldp he tpa
dO kha dan gidn, thudng cd hai each:
Chpn mOt dinh ldm gdc tpa dO vd ba true ttung vdi ba cartii efta hinh hop chft nhat Chpn tdm cua day lam gdc tpa dd vd ba true song song vdi ba cgnh efta hinh hOp chft nhdt
(4) Thiit lap hi tga dg cho hinh lang Py Vdi lang try dftng thi ta chpn tryc Oz thing
dftng, goe tpa dO la mOt dinh ndo dd efta day hoac tam cua day ho§c didm ndm ttong m$t ddy Id giao
efta hai dudng thdng vudng gdc Cdc true Oy, Ox
thi dya vdo tinh chdt efta da gidc ddy ma chpn cho
Trang 3Vdi lang try xien, ta dya ttdn dudng cao va
tinh chdt cua ddy dd chgn he tpa dO thich hgp
Ngoai cdc trudng hgp ttdn, ttong cdc trudng
hgp khac ta dya vao quan he song song, vudng gdc
va cdc tinh chdt efta dudng cao, ddy, dk tiiidt lap
he tpa do cho thich hgp
2.5 Hf true tpa dp Oxyz
Hp true tga dO vudng gdc Oxyz ttong khdng
gian Id hd gdm ba true x'0x,y'0y,z'0z ddi mOt
vudng gdc
Diem O la gdc tga dO
Ox gpi Id tryc hoartii
Oy gpi Id tryc tung
Oz gpi Id tryc eao
Tren cdc tryc Ox, Oy, Oz Idn lugt chua ba vecta
don vi I, J, k
Cdc mat phdng (0xy),(0yz'),(0xz) ddi mOt
vudng gdc nhau
Tpa dO eua vecta: u = (x; y; z) <=^ u =
(x\y;z) ^=^ u = Xi + yJ + zk
Tpa dO cua didm: OM = ;t:? + y; + zfe <=*
M(x;y;z')
Cdch xac dinh tpa dO didm l^(XM',yM''^M)
ttong he tpa dp Oxyz
Tim hinh chidu M' efta M ttdn mdt phdng
tga dO Oxy
Tft M' ke M'l vudng gdc vdi true x'Ox tgi /
Tft M' kd M'} vudng gdc vdi tryc y'Ox tgi /
Tu M ke MK vudng gdc vdi true z'Oz tgi if
y'
I ^
K
O
\ M
>
j^x'
\^1 ,-''
N6u /,/, K Ian luat thuoc cac tia Ox, Oy, Oz C'u = 01
thi y„ = o ;
(z„ = OK = MM' Ngu l,J,K lin luat thuOc cac tia
I x„ = ~0I Ox', Oy', Oz' M\y„ = -0]
(z„ = -OK = -MM'
2.6 Mdt so bai toan
Bai toin 1: (SGK Hinh hoc NC lap 12) Cho
liinh chop 5.^BC CO duong cao 5i4 = h.i&y litam giac ABC vuong ^C,AC~b,BC = a Gpi M la tnmg diim cua AC va N \h diim sao cho 5W =
\rB
a) Tinh dp ddi doan thang MN
b) Fun sy lien he gifta a, b, h dh MN vudng gdc
Ke Ax II BC Chpn he tryc tpa dd Oxyz nhu hinh ve sao cho X = 0, C G Oy, S 6 Oz
Tacd:A(0;0;0),5(0;0;h),
SN = (x^: y„; z«), SB = (a; b; -h)
SN=ISB- ''» = 3 = ' * " 3 ' 3 ' T ^ b a b 2h
Trang 4a) Ta co: JlfW = |A*N| = J - + - + — =
i V 4 a M ^ P T l 6 F
b)Tac6: MN = ( | ; ^ ; ^ ) , S B = ( a ; 6 : - / i )
_ a' i' 2h'
MN 1SB<=>MN.SB = 0*^- ; r - = 0
3 6 3
Vay MN ISBOii 2 a ' - 4 ' - 4*^ = 0
Bai toan 2; Cho hinh lang uu ABC.A'B'C co
day ia tam giac deu canh bing a,AA' = /i va
vuong goe voi (ABC) Bilt r ^ g khoang each giiJa
A'B' va BC bing d Chung minh ting a =
Giai
M
•
a
C
c
Gpi M Id tning didm AB Chon he ttyc tpa do
Oaryz nhu hinh vd sao cho M = O.B G Ox.C G
Oy Ta cd:
A'{-^:0:k).B-(^:0:h)
B[-;0;0).C(0;-^:K)
WP = (o; 0; OXA^ = (a; 0-, -h)
d = d(.A'B'
>/4hZ+3a^
lf^-«^l jr^ft2+^
dy/WT3d^ = ahy/3 <=> d=^(4ft2 + 30^) =
3a^h^ ' V3(h^-d^) (dpcm)
Bdi toan 1 ndu gjai theo phuong phdp hinh hpc thudn tuy thi g ^ tid ngai d cau b Vide tim khoai^ each gifta hai dudng thdng eheo nhau cua bdi todn ndm vung phuang phdp tim khodi^ each ^ua hai dudng thang eheo nhau Ldi gjdi bdng phuong
phdp tpa dp cung ngan gpn vd kha Aaa ^dn Bdi toan 3: (DH khdi B 2007) Cho hinh chdp tft giac ddu S.ABCD cd ddy la hinh vudng cgnh a Gpi E la diem ddi xftng ciia D qua trung di&n 5^4,
M Id trung didm AE, N \k tnn^ didm BC Chiing minh MN vudng gdc vdi BD vd tinh khodi^ each giiia hm dudi^ thang MN va AC
Giai
Trang 5Phmmg phdp tdng hgp Phirong phdp tpa dy
Suy ra tft giac MICN Id hinh binh hdnh
=> MN II IC (1)
m^:[ll\%'^BDUSAC)
=i BD lie (2)
TCt (1) v4 (2) suy ra: MN ± BD (dpcm)
(MN II IC
IC cziSAC-) = • " " " (^''C).'1C c (SAC)
=> d(MN,AC) = d{MN,(SAC))
Ggi 0 Id tam cua hinh vuong ABCD Chon he true tpa dp Oxyz nhu hinh vg sao cha C G Ox,D e O y S e O z T a c o :
0(0; 0 ; 0 ) , / l ( = f ^ ; 0 ; 0 ) , 8 ( 0 : ^ : 0 )
C (2|5; 0: o),£) (0;2j?; o) ,5(0i 0; ft),ft > 0
Goi / la trung diim SA Ta co:
-ai/2 -aV2 h\
N(-^;^;0)
- d(N, (SAC))
Gpi P 14 trung diim OC Ta c6:
=> d(W, (SAC)) = WP = ^—
Vkyd(MN.AC)=~
MW = ( ^ ; 0 ; ^ ) , I C = (0;aV2;0) MW.BD = 5 ^ 0 + 0 aV2 - ^ 0 = 0
AC = (ay/2; 0; O), MA = (0; - — ; —-) , _ , -ah^f2
\MN,AC] = (0;—-—;0) _ , -^ -a^k IMN,AC].MA = (0; - ; 0 )
d(MN.AC) _ HMNMJMA^ _ I 4 I _ aVz
\\i^M\\
Ldi gidi cua bdi todn bdng phucmg phdp tdng
hgp ta thdy nd cOng ngdn ggn vd di hiiu, nhung
khi dgc de di tim ddp dn thi rdt khd phdt hien dugc
tu gidc MICN Id hinh binh hdnh, ddy Id mdu chdt
chinh di tim ra ldi gidi Viic chung minh vd tinh
khodng theo phucmg phdp tga dg rdt di ddng niu
viec tim ra tga dg cdc diim chinh xdc, nhin cd ve
ddi dong nhung phuong phdp ndy khdng ddi hdi
hgc sinh phdi tu duy cao Do dd, phuong phdp tga
dg phii hop vdi ddi tupng hgc sinh khdng co ky
phdp hinh hgc thudn tuy
Bai toan 4: (DH khdi A ndm 2012) Cho hinh chdp S ABC cd day la tam gidc ddu cgnh a Hinh chidu vudng gdc ciia S trdn (ABC) la diem H thupc cgnh AB sao cho HA ~ 2HB Gdc giua dudng tiidng SC vd mat phdng (ABC) bdng 60° Tinh thd tich cua khdi chdp S.ABC va tinh khoang each gifta hai dudng thang SA vd BC theo a
Giai
Trang 6Phinrng phdp tong h ^ Phtnmg phitp toa d$
HC la hinh chieu cua SC len (ABC)
=> G6c giiia SC va (ABC) la JCH = 60°
Gpi D la trung diem canh ^ S Ta co: CD =
2 3 ~ 6
Xet AH DC vuong tai H, ta co:
-•,HD HC = -JHD^ + CD' = J(f)' + [if]' = i f
Xet ASHC vuong tai H, ta co:
SH = HC tan 60° = 2 ;,V3=221
Gpi D la trung diem AB;kiHE H DC, E e BC
Chpn h^ tryc tpa dO Oxyz nhu hinh ve sao cho
H = 0,B eOx,EeOy,Se Oz TIL CO:
H ( 0 ; 0 ; 0 ) , y 4 ( ^ ; 0 ; o ) , B ( | ; 0 ; 0 ) ,
SC = (-f ; ^ ; - 2 ) * = CO;0:1) GOC giiiaiC
va (ABC) bang 60° nen ta co:
|sc.S| Ml sin 60° =
ThS tich ciia khoi chop S.ABC la Vs,tBc = -SH.S^^
1 a ^ aV3 _ a^/7 _
3 ' 3 • 4 12 ^ ^ •*
K6 Ax II BC, Ice HW II Ax, N £Ax
Ta co: BC II (iXN) nen ii(5>4, BC) = d(BC, (SAN)) =
d(B, (SAN))
mm'
^ ) , M = ( f ; 0 ; ^ )
Vi HB n (i/lN) = ,4 nen ta co: d^aXSAN)) _ AB
d(H.{SAN)) ~ AH~
= -m
AW 1 (SHN)
a = ( ^ ; 0 ; [S2,S5] = ( 0 ; ^ ; 0 )
, ^ 6 2 ' 3 -'
The tieh ciia khoi chop S.ABC la
''^•^??#-ilMFli^l#-^ <^™'
BC = ( Y ; 2 f ; 0 ) , I s = (a; 0;0)
=» (SAN) 1 (SHN)
MSt khac: (SAN) n (SHN) = SN
Tir H UHK 1 SN (K E SN) Khi do: HK 1 (SAN) h a y r a g f l = f-a'^.-a'^m.-a'j3\
d(H,(SAN))=HK <- • ' ( 2 ' e • 3 J
AH = ^AHN = AH.sin60<>=i^ d(SA,BC) = ^^^Si^ J ^ 'J^
Xet ASHN vudng tgi H, ta cd: HK.SN = NH.SH =^
(*)=> d(B, (S>17V)) = |d(W, (^SAN)) = ^
V d y d ( S > l , S C ) = ^
Z.OT gidi bdi todn trin cOng chung minh dugc
uu diim cOa phuang phdp tga dg
Bdi todn 5: (Gidi todn Hinh hgc 11, NXB Gido
Due) Cho hmh ldp phuang ABCD.A'B'C'D' canh
a Gpi M,N Id hai didm ndm ttdn hai canh B'C vd
CD sao B'M =^B'C'.CN = 'CD Chftng minh AM_ J, BN vd tinh khodng cdch giu:a AM vd BN
Trang 7Pbmmg phap tong hyp Phirong phap tpa dp
- — i J ^
4i \
c
E
Ke ME II CC,(EeBC);I = BNnAE
MBE = ABCN (c c c) => BSC = 3 I B
=»Tll giic INCE nOi tilp => NIE = 90°
Hay BN 1 AE (1) Mat khac:
{M;;rc"'^^ «c.Bco)
=>ME IBN (2)
TCr(l)va(2)suyra:BW 1 (AME) => BN LAM
MJM c (AME)
°-\(AME) IBN,(ISBN)
Trong (AME), tir lUIH 1 AM, H E AM
S\iyr^:d(AM,BN) = IH
Xet AABE vuong tai E, ta co:
AE = 4AmTBE^ = ^!^;AB'=AI.AE
3
" ' Vl3o m
Xft AAME vuong tai £, ta co:
AM = 4APTJm= | H £ ! + a 2 = J!H
\ 9 3
A>i;// - tiAME(g o) =s — = —
^ " ^-^ ME AM
_ MEAM _ 9 a
'z
\
D
o-V~-^^
r
y,
Chpn he true tpa dO Oxyz nhu hinh ve sao cho
A = 0,B' E0X,D' EOy,A 6 Oz Taco:
yl(0; 0; 0), Jl« fa; Y : o) ,B(a; 0; o),
AM -{a:f,-a) — -2a
.BN = (—-,a;a)
— , — -2a' -2a' AM.BN =—:r- + —— = 0
=>AM IBN
AB = (a;0;0), ,— — ^ ,2a' 13a^
\AM,m\ = (a';-^:-^)
_ \[AM,B!'i].'AB\ _ \a^ _ 9 a d(AM,BN)
•-\[AM.BN\\ |2B6^^ V286'
ilH =
Ddi vdi hinh lap phuang thi viic gidi bdi todn
bdng phuang phdp tga do cd nhieu thudn lai nhdt
Viic chgn hi true tga do vd tim tga do cdc diem
cung dan gidn Do dd, ldi gidi hang phucmg phdp
ndy ngdn ggn han Di gidi duoc bdi todn ndy bang
kien thiic cua hinh hoc phdng vd hinh hpc Idiong gian
U'u diem cua phtnmg phap tpa dd
Phuong phdp tpa dO giup gidi mpt sd bai todn
hinli hpc khdng gian dan gian hon khi giai bdng
phuang phap hinh hpc tiiudn tfty-'
Lupng kien thftc vd kT nang dd giup hpc sinh cd
the gidi cac bai todn hinh hpc thdng qua phuang
p h ^ ndy khdng nhidu ehu ydu la cac kidn thuc vd
tpa dO vecto ttong khdng gian, phuang ttinh dudng
tiidng, m$t phdng, mdi quan he gifta chftng
Phuang phap ndy khdng qud khd ndn ddi vdi
cac em hpc sinh trung binh yeu vide sft dung
phuong phdp ndy dan gian han nhidu so vdi
phuang phdp tdng hpp, chft ydu Id dgy cdc em cdch dat he true tpa dO sao cho phu hop
Nhiryc didm ciia phuang phdp tpa dp Khdng phai tdt cd cdc bdi todn ve hinh hpc khdng gian ddu co thd sft dung phuong phap tpa dO
dk gidi, chi vdi nhung hinh dac biet cd nhung cgnh
cd quan he vudng gdc vdi nhau thi ta mdi nen su dung phuang phdp ndy vi neu khdng vide tinh tpa
dp cdc didm rdt khd khSn
Sft dyng phuang phdp ndy ddi hdi phdi cd ki nang tinh toan khd tdt vd phdi nhd duprc cac cdng thftc vd phuang trinh ciia dudng thang, mgt phang, cdc cdng thftc vd tinh gdc vd khoang cdch MOt sd cdng thftc khd gidng nhau ndn ddi khi dd gdy ndn
sy nhdm lan
3 T H V C N G H I ^ M
3.1 Muc dich thyc nghifm Thye nghidm nhdm kidm tta tinh khd tiu vd
Trang 8^ a i bdi tap hinh hpc khdng gian bdng phuang p h ^
tpa dO, kdt hpp dieu tra va phdng van
3 J Npi dang va phinmg phap thyc nghidm
Thye n ^ e m dupe tien hanh tai Idp 12A,
trudng Trung hpc Phd thdng Hda Binh Lap 12A
gdm 40 hpc sinh cd ket qua hpc tap tucmg ddi ddng
ddu do thdy Tran Ngi^dn Khai Hung ^ang day
mdn toan Thdy Himg da cd ttdn 7 nam kinh
nghidm gidng day mdn todn ldp 12 Trudc kia, khi
day hpc gidi cdc bdi tgp vd tpa dp troi^ khdng gian
thdy Hung it gidi thidu cdc bdi tap hinh hpc khdng
gian cd thd gidi bdng phuang phdp tpa do Do dd,
hpc smh khdi^ dupe ren luyen {diuong phdp tpa dO
hda vd edm thdy e ng^ trong ky thi tdt nghidp vd
dgi hpc vi cd bai ky thi ddu cd cdu bdi tap hinh hpc
cdc em khdng mdy ty tin bdi cd mOt so bdi tap
ttong thdi gian ngdn khdng thd tim dupe len gidi
Qud trinh thyc nghidm duoc tidn hdnh ttong cde
tidt gjdi bdi tap vd daiy hpc mdt sd bdi tap hinh hpc
khdng gian theo phuong phdp tpa dp vd cdc tidt day
ndy duoc phdn bd sau Idii hpc cac bdi phuang phdp
tpa dO ttong khdng gian Trudc khi thyc nghiem,
chiing tdi cung vdi thdy Hung da cftng nhau ttao
tpa dp khdng ^an sao cho phft hpp va dd dang tim
dupe tpa dO cdc didm ttong bdi todn
Ben cgnh dd, chung tdi trao ddi xin y kidn efta
rapt sd ^ao vidn, phdng 20 hpc sinh dang hpc ldp
12 vd 9 hpc sinh viia hpc xong ldp 12
Cdc cdu hdi phong -van
- Sau khi hpc xong chuang 4 ciia chuang
trinh hinh hpc ldp 12 thi em gidi cdc bdi t|p hinh
hpc khdng gian bdng phuang p h ^ thudn tuy hay
phuong p h ^ tpa dp?
- Phuang phap tpa dp hda de tidp thu hay
khdng?
- S ^ khi dupe trang bi thdm phuong phdp
tpa dp hda thi em cd an tdm hon khi ldm cdu hinh
bpc khdng gian ttong ki thi dgi hpc khdng?
3 J Phan tfch ket qua thurc nghidm
Sau khi tidn hdnh thyc nghiem, qua kdt qud bdi
kidm tra chung minh dupe rang cac em hpc sinh
trung binh khd ed the tiep thu dupe phuong phdp
nay de ddng vd cung mOt b ^ tap hinh hpc khdng
gian thi sd luong hpc sinh giai duoc bdng phuang
phdp toa dp nhidu hon sd hpc sinh ^di bdng
Qua trao ddi vdi mdt sd gido vien cd nhidu ndm kinh nghidm day Idp 12 thi cac gjao vidn eung nhan dinh rang van dm^ phiror^ p h ^ tpa dp de gidi cdc bai tap hinh hpc khdng gian cd nhidu thudn lgi Y kidn ciia cde em dang hpc ldp 12 thi nhdn xet rdng phuong phdp n ^ dl hieu han phuang phdp tdng hop va cd the tiep thu dd dang, cdn cdc em viia hpc xong Idp 12 thi cho rdng phuong phdp tpa
dO hda rdt hay, cdc em edm thdy ty tin hon khi budc vdo ky thi Dai hpc
Muon qud trinh nay dat hidu qud, can phoi hpp day hpc bdng phuang phdp tpa dd dd hpc sinh cd nhidu ea hOi gidi dupe cdc bdi tap binh hpc khdng
^an Phuong phdp nay khdng nhftng giftp hpc sinh
dn lai kidn thuc tpa dp tiong khdng gian ciia chuong 4 ttong chuang trinh Hinh hpc ldp 12 ma
nd cdn la cdng cu dac lyc dd hd tra cdc em ttong cdc ky thi, dac bidt Id ky thi ti^dn sinh Dai hpc Dieu dd da chftng td uu diem ndi bat cua phucmg phap tpa dp
» KETLU.4N Trong bai bao nay chftng tdi tap trung vdo vide dua hd tryc toa dp dd gidi cdc bdi todn hinh hpc
khdng gian Ddy Id phdn quan ttpng nhdt iik gidi
thdnh cdng mOt bdi todn hinh hpc khdng gian bdng phuang phap tpa dp Cdc vi dy ttdn ddy ciing vdi kdt qud thyc nghidm su pham d trudng phd thdng da khdng dinh cdc uu didm, tinh khd thi vd hpc todn
TAI LIEU THAM KHAO
1 Dodn Quynh tdng chu bidn Van Nhu Cuang chu bidn, Pham Khdc Ban, Ld Huy
Hung, Ta Mdn (2010), Hinh hoc 12 ndng cao, Nxb Gido dye
2 Van Nhu Cuong (chu bien), Pham Khdc
Ban, Ta Man (2007), Bdi tap hinh hgc 11 ndng cao, NXB Gido dye
3 Tnrang Ngpc Dung (2008), Gidi todn hinh hgc lap 11, NXB Gido Due
4 Vd Thanh Van (chu bidn), TS Ld Hidn Duong, N^iydn Ngpc Giang (2010),
Chuyen de ung dung tga do trong gidi torn hinh hoc khong gian, NXB Dai hpc su
pham, Hd Chl Minh
5 Le Hdng Ddc, Nguyen Dftc Tri (2007),
Phuang ph^ gidi todn binh hgc gidi tich