^hdtt^ 5 Itdnt 2017 Nghien cdu khoa hgc
VAN DUNG PHirONG PHAP TOA DO GIAI BAI TOAN HINH HOC THUAN TUY
Applying the coordinate method in solving classical geometry o ThS. Nguyen Thong Minh-Khoa Sir phgm
TOMTAT
Noi dung chinh cua bdi bdo la de cap quan he giira hinh hgc thudn tuy vd phuang phdp tga do. Ddc biet, khai thdc uu diem cua phuang phdp tga do de gidi nhirng bdi todn hinh hgc thudn tuy phuc tgp, doi hoi phdi ve them dudng phu md hgc sinh thudng gdp phdi trong cdc ky thi tuyen sinh dgi hgc, cao ddng.
Trong bdi viet ndy, chiing toi xin giai thieu cdch van dung phirang phdp loa do de gidi bdi todn hinh hgc thudn luy.
Tu khda: Hinh hoc thudn tiiy, hinh hoc khong gian. phuang phdp loa do.
ABSTRACT
The core content of the article is to mention the relationship benveen classical geometry and coordinate method. Il special point is that it develops the strong points of coordinate method lo solve some complicated classical geometry problems demanding to draw more extra lines, with which students often face in their entrance exams. In this article, we introduce how to apply coordinate method in solving the classical geometry.
Keywords: Classical geometry, Solid geometry, coordinate method.
1. Dat van de
Hinh hpc ndi chung va hinh hpe khong gian ndi rieng la mdn hpe kha trim tuong nen da sd hpc sinh e ngai khi hpc ve phdn nay.
Thuc te eho thdy trong cdc de ihi hpe sinh gioi. dai hpe, eao dang thudng xuat hien bai loan hinh hpc ma d dd ldi gidi ddi hdi van dung kha phirc tap cac kien thiie tdng hpp nhu: chiing minh quan he song song, quan he vudng gdc, dung hinh de tinh gdc va khodng each, tinh dien tich, the lieh ... Viee tiep can cac Idi giai dd thuc te cho thay that su la mpi khd khan eho hpc sinh, tham ehi ea giao vien, chdng han bai loan tinh khoang each giiia hai dudng thang eheo nhau. Trong khi dd, nhu bd qua yeu cdu bat budc phdi dung hinh
ma chi dimg d mire dp tinh toan thi rd rang phuang phap tpa dp td ra hieu qua hon vi tdt cd mpi linh todn deu da dupe cdng thiic hda.
Bai viet nay khai thdc uu diem cua phucmg phap tpa dp. dac biet la ky nang chpn he true tpa dp de giai nhiing bai loan hinh hpc thuan tiiy phiic tap.
2. Npi dung nghien cuu
2.1. Cac budc giai bai toan hinh hpc thuan tuy bang phuong phap toa dp
Bude 1: Chpn he true tpa dp thieh hpp.
Budc 2: Tim tpa dp cac diem cd lien quan den yeu cdu bai loan theo he true tpa dp
\ ira chon.
Nghien cvtu khoa hgc '7/tiiiu/ 3 nditt 2011
Bude 3: Giai bai loan bang kien thiie Ipa dp. Sau do, chuyen kel ludn ciia bai loan hinh hpc giai tich sang tinh chat hinh hpc thudn tuy tucmg img.
Budc 4: Chuyen cac ket qua tir ngdn ngir tpa dp sang ngdn ngu hinh hpc thdng thudng.
2.2. Mpt so each chpn he true tpa do trong hinh hpc phang
Nhiing bai loan hinh hpc phang d phdn gia thilt cd nhirng dang sau thi ta giai dupe bang phuong phap tpa dp:
- Hinh da eho la hinh chii nhdt hoae hinh vudng.
- Hinh da eho la lam giae vudng, tam giac can hoac lam giac diu
- Hinh da eho la hinh thang vudng hoae hinh thoi.
Tuy nhien, vdi mdt sd bai toan ma gia thiet cho khdng thudc nhirng hinh quen thupe nhu tren, nhung chiia nhieu yh\x Id vudng gdc thi ta vdn ed thd thiSt lap he true tpa dp dh giai.
-^ Thiet lap he toa do ddi vai hinh chit nhdt hoac hmh vuong
Chpn he true Oxy sao cho: Gdc tpa dp O la mdt trong bdn dinh cua hinh chir nhdt hoac hinh vudng. Hai canh xudt phat tir dinh dd nam tren hai true (hoac gdc loa dp O la tdm eiia hinh vudng).
•^ Thiet lap he toa do dSi hinh thang vuong
Chpn he true Oxy sao eho: Gdc tpa dp O la mdt trong hai dinh gdc vudng. Hai canh xuat phdt tir dinh dd ndm tren hai true.
m
•^ Thiit lap he toa do doi hinh thoi Chpn he true 0.vr sao eho: Gdc tpa do O la tdm cua hinh thoi. Hai dinh lien tiep lin lupt thudc hai true Ox, Oy.
•^ Thiet lap he toa do doi tam giac vuong
Chpn he true Oxy sao cho: Gdc tpa dp O la dinh gdc vudng. Hai canh gdc vudng \h\
lupt nam tren hai true Ox, 0 \ .
••^ Thiet lap he toa do doi tatn gidc cm Chpn he true Oxy sao cho: Gdc tpa do O la trung diem canh day. Canh day ndm tren mdt true, dinh can thupe tryc cdn lai.
2.3. Mot so each chpn he true tpa do trong hinh hpc khong gian
Nhirng bai loan hinh hpc khdng gian 6 phdn gia thiSt cd nhimg dang sau thi ta giai dupe bang phuong phap tpa dp:
- Hinh da cho cd mdt dinh la lam dien vudng.
- Hinh ehdp ed mpt canh ben vuong gdc vdi ddy va day la cac tam giae vuong, tam giac deu, hinh vudng, hinh ehCr nhat,...
- Hinh lap phuang, hinh hop chu nhat.
- Hinh da eho cd mdt dudng thdng vuong gdc vdi mat phang, trong mat phdng do cd nhiing da giac dac biet: Tam gidc vudng, lam giac deu, hinh thoi,...
Ngoai ra, vdi mdt sd bai toan ma gia thiet khdng cho nhirng hinh quen thudc nhir da neu d tren thi ta cd Ihl dya vao tinh chit song song, vudng gdc ciia cdc doan thdng hay dudng thang trong hinh ve dh thilt lap he tnjc tpa dp.
••' Thiet lap he tga do doi vai tam dien THONG TIN KHOA HOC GIAO DUC
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- Ddi voi tam dien vudng thi he true tpa dp vudng gdc duoc thiet lap ngay tren tam dien dd.
- Ddi vdi tam dien cd mpt gdc phdng vudng. khi do ta thiet lap mdt mat ciia he true tpa dp chiia gdc phang dd.
-^ Thiet lap he toa do cho hinh chop - Ddi vdi hinh chop deu thi he tpa dp dupe thiet lap dua tren gdc O trimg vdi tdm cua day va true Oz trimg vdi dudng cao cua hinh chop.
- Ddi vdi hinh chop cd mpt canh ben vudng gdc vdi day thi ta thudng chpn true pz la canh ben vudng gdc vdi day, gdc tpa 4o EHMg vdi chan dudng vudng gdc.
-^ Thiet lap he true toa do cho hinh hop chu nhdt
Vdi hinh hop chir nhat thi viec thiet lap he tpa dp kha don gian, thudng cd hai each:
- Chpn mdt dinh lam goc tpa dp va ba true triing vdi ba canh cua hinh hop chir nhdt.
- Chpn tam cua day lam gdc tpa dp va ba true song song vdi ba canh cua hinh hop chii nhat.
•^ Thiet lap he toa do cho hinh ldng tru
- Ddi vdi lang try diing thi la chpn true Oz thang diing, gdc tpa dp la mpt dinh nao dd cua day hoac tdm cua day hoac diem nam trong mat day la giao ciia hai dudng thdng vudng gdc. Cac true Oy, Ox thi dya vao tinh chat ciia da giac day ma chpn cho phii hpp.
- Ddi vdi lang tru xien, ta dua tren dudng cao va tinh chat cua day de chpn he tpa dp thich hop.
2.4. Mpt so bai toan minh hpa Bai toan 1: Cho hinh vudng ABCD. Gpi M la trung diem ciia BC, N la diem tren canh CD sao cho CN = 2 ND. Gpi H la giao dilm ciia AN va BD. Chirng minh AN 1 HM.
L&i gidi: Chpn he true Oxy nhu hinh ve sao cho: Gdc tpa dp O trimg diem A. Hai dinh B, D lan lupt thudc tia Ox, Oy. Dat a canh hinh vudng. (a > 0). Khi dd
A(0;0), B(a;0), C(a;a), D(0;a), Ml a;~ , A^ -;a .Xac dinh tpa dp diem H,
^AT = - ( l ; 3 ) , AN cdVecto phap tuyin (VTPT) «, = ( 3 ; - l ) ,y AN:3x-:^ = 0.
aO = a ( - l ; l ) , B D c d V T P T M^^(l;l) =>BD: x + y = a.
3 x - v = 0 Toa dd H la nghiem ciia he:. -^ <=>
'x + y = a
x = — 4 3a
D
A
I ' X
\L
/ ^ ^
/ \ c
M
Nghien ciiu khoa hoc '7hdii^ 3 nam 2011
Hay H\
3' ™ = fe-^
I 4 4 > AN.HM = 0 14 4j
Vay AN -L HM.
Bai toan 2: Cho tam giac ABC can tai A, H la trung dilm BC, D la liinh ciiieu cua H tren AB, I la trung diem HD. Ciiiing minli rang AI 1 CD.
Ldi gidi: Chon he toa dp Hxy sao cho hai diem B, C tren Hx va A tren Hy de tien cho viec tinh toan ta dat HB = HC = I va AH = b.
Khi do A(0 ; b), B(1;0) va C(-1;0) dircrng thing AB co phucmg trinh: — + — = 1 « > b x + y - b = 0 X V
1 b
do HDI AB va di qua g6c toa do H nen HD: X - by = 0 Tga dp D la nghiem he :
bx + y - b = 0 X - by = 0
1 + b' b 1 + b'
Suy ra diem I trung diem cua HD co tpa dp |
A1 = I\
[2(i+b'y 2(i+h')
Ib'.b 1.^5^2*^.1
l + b- \ + b' va do do ta co:
Vay A l l CD.
^ Nhan xet: Dira vdo de bdi co nhiiu yiu td vuong gocvd dua vdo hinh ve ta thdy bdi torn nay rat thudn lai cho viec dp dung phuang phdp toa do. Viec chgn he toa do thich hap nhu tren thuan lai cho viec bieu dien duac toa do cdc diim, phuang trinh cdc duang thing cho trong
THONG TIN KHOA HOC GIAO DUC
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bdi mpt cdch de ddng vd dan gidn nhdt. Tir do, viec kiem tra tich vo huang bdng 0 cua AI vd CD tra nen di ddng.
Bai toan 3: Cho duong tron (C) duong kinh AB, duong thang (d) vuong goc voi AB tai C c6 dinh, H la diim thay d6i treii (d), AH va BH cit ducmg tron (C) tai D va E. Chiing minh ring DE luon di qua mpt diSm c6 dinh.
L&i giai: Chpn he true toa dp Oxy sao cho C tring vol goc
toa dp; B(l - c: 0); A(-l - c; 0) va d triing voi Oy Duong tron ducmg kinh AB: (x + c)^ + y- = 1
Gia sil H(0; m), m thay ddi. Gpi Q la giao diSm cua BD va d.
Phuong trinh duong thing AH: m x - ( l + c)y+ m(l +c) = 0 Phucmg trinh duong thing BD: (1 + c)x + my + c- - 1 = 0
( l - c ' ~
=> Toa do diem Q 0;
I m
Phucmg trinh ducmg tron ducmg kinh HQ:
l - c ^ + m = X + y -
2m
1 - c ^ - m ' 2m
Khi do, DE la true dang phucmg cua hai duong tron duong kinh IH va duong tron ducmg kinhAB
> phucmg trinh DE: 2cx +
l-c"^ + m
1-c +m' -y = l - c ' C5 2cx + - v = 2 - 2 c '
l - c ' - m " 1-c" +m' 2m
\'ay. DE di qua diem co dinh K 1-c^ ;0 .
Bai toan 4: Cho til dien OABC co ba canh OA, OB, OC doi mpt vuong goc nhau, OA
= a,OB = b,OC = c.
a) Tinh dp dai ducmg cao ciia tii dien ke tir dinh O;
b) Gpi 1 a.p.r lin lupt la goc giua {ABtJ) va cac mat phing (OBC), (OCA), (OAB). Chirng minh rang: cos' a + cos" ^ + cos";' = 1 •
Nghien cHu khoa hoc '^hdiu/ 5 ndin 2017
Ldi gidi: Chpn he true tpa dp Oxyz sao cho 0^ = tia Ox, OB" tia Of, C s tia Oz.
Khi do: ^(a;0;0), B(0;A;0). C(0;0;c).
a) Tinh do ddi ducmg cao cua tu dien ke tit dinh O.
De thay phucmg trinh mp(^i?0 Is
— + — + - = ] '^ bcx + cay + abz-abc = 0.
a b c
Dp dai h cua ducmg cao ciia tir dien ke tir dinh O la khoang each tir O den mp{ABC):
h = d(0,(ABC)) = \-ahc\
^(ab)-+(hc)-+(caf ^(aby+(bcf+(caf b) Chung minh cos" a + cos' /? + cos^ 7 = ^ •
V(ji a,li,r lin lugt la goc giua {ABC) va eac mat phing (OBC), (OCA), (OAB).
Dl thiy cac mat phing (ABQ, (OBQ, (OCA),(OAB) co VTPT lin lugt la, n = (bc;ca;ab) 7= (1;0;0),] = (0;l;0),t = (0;0;l). Do do:
,cos/? = ^ , c o s / = ab
^(bc)-+(caf+(abf sl(bcy + (ca)- + (ab)- ' ""'' .J(bc)- + (ca)- + (abf
Suy ra: cos^ a + cos" 13 + cos";' = 1
Bai toan 5: Cho hinh chop tir giac deu S.ABCD co AB = a, SA = aVI. Ggi M, N, P lin lugt la trung diem ciia cac canh SA, SB va CD. Chirng minh ring duong thing IViN vuong goc vol ducmg thang SP. Tinh theo a thi tich ciia kh6i tii dien AMNP.
Loi giai: Gpi O la tam ciia ABCD. Chgn he true Oxyz nhu hinh ve vol
O(0;0;0), c f ^ ; 0 ; o l i - i ^ ; 0 ; o ] , Z ) [ o ; i ^ ; 0 {sO = 4SA'-OA'=^
M, N, P lan lugt la trung diim cua cac canh SA, SB va CD ;
<°-¥4)'^(f4^«'-
0 ; - i ^ ; o l , ,sfo;0;i^)
. M | - ^ ; 0 ; ^ 1
^
THONG TIN KHOA HOC GIAO DUC
'Jhdiui 5 nani 2017 Nghien ciiu khoa hgc
Khi do .V/.V = (a^ a-Jl 4 4 f a\t2 ajl flV6 SP--
16 16 Mat Ichac, ta lai co
a-Je
= 0=>.W,ViSi'.
30^2 aV2
;o ,
AMJEA.^OAJP..
1^ 2 4 4 J L J 8
-'•'""^ 6 1 ^ - 1 I 48
Nhgnxet: Theo phuang phdp tong hap phuang an tinh the tich tic dien AhdNP phdi thong qua tinh gidn tiep the tich tu diin ABSP vd the tich khoi chop S.ABCD. Cdch tinh tren day bdng phuang phdp tga do Id hocin todn true tiep vd di dinh huang
Bai toan 6: Cho lang try diing ABCA'B'C'co day ABC vudng, AB - BC = a, canh ben AA' = a-^jl. Gpi M la trung diem ciia BC. Tinh theo a the lieh eiia khdi lang try da cho va khoang each giira hai dudng thang ^M, B'C.
Lc^igidi: Tir gid thiet ta cd tam gidc ddy ABC vudng cdn lai B, ket hop vdi tinh chdt cua lang tru dirng, la chpn he true G.viz nhu hinh \ e, vdi
BsO(0;0;0), C(^;0;0), A(0;a:0), B'(0;0;aV2).
De thay V cf42
Ba} gid la linh khoang each giiia AM va B'C.
M la trung diem eua BC
•A/ -:0;0
-Mat khac, B ^ = (a;0;-aV2) = * [ Z w , i r c ] = a"V2;^^-^;fl-
Nghien ciiu khoa hgc 'Jhdiia 3 nain 2017
_ IAM.B'C].AC\ - ~ ^^
Lai CO /^C = ( a ; - a ; 0 ) = ^ i / ( . - t , t / . B ' C ) = ' ir i - , , ' = — ^ = ^ — . r . 4 A / , 5 ' c ] flV7 7
%/2
Aftow K7.' r/jeo phuang phdp tdng hap, viec tinh Ichodng cdch giiia hai duang thang AMvd B 'C trong bdi todn ndy hodn todn khong de, doi hoi dung dugc mat phdng chira AAfva song song vdi B V. Ldi gidi bdng tga do ro rdng Id rdt ngdn ggn vd true tiep.
Bai toan 7: Cho hinh lap phucmg A B C D . A'B'C'D' canh a. Gpi M, N lan lupt la trung diim ciia AB va C D ' . Tinh khoang each tir B' den (A'MCN).
Phucmg phap tong hgp Phuong phap toa do
Bon tam giac vuong A A ' M , B C M , C C ' N , A D N bang nhau (c g c)
=> A'M = MC = CN = N A '
=i A ' M C N lac hinli thoi
Hai hinh chop B A MCN va B A NC co chung ducmg cao ve tir dinh B va .S , = 2 S ,
,\'MCN • A'NC
nen V , = 2 V
li .\'MCN • B' A ' N C
ma V = V = - CC' S
B ANC C A'B'N -, "-^.y
1 1 a'
= —.a. —.a.a = — 3 2 6
Chon he true toa do Oxyz, vai Dx, Dy, Dz doi mot vuong goc Khi do
A(a, 0, 0). B(a; a, 0), C(0, a; 0), D(0, 0, 0), A (a; 0, a), B (a, a, a), C (0, a. a), D (0, 0, a). M | a : - ; 0 , N 0 ; - ; a
T a c o A ' C = ( - a ; a ; - a ) , M N = ( - a ; 0 ; a ) [A'C; M N ] = ( a ' ; 2a-; a") = a ' ( l ; 2; I)
= a ^ n vdi n = ( l ; 2; I).
Phuong trinh mp (.\ MCX) qua C(0, a, 0) voi V T P T n : l ( x - 0 ) + 2 ( y - a ) + l ( / - 0 ) = 0
m
THONG TIN KHOA HOC GIAO DUC'Jhdiuf 3 iiuiii 2017 Nghien cdu khoa hgc
Taco: S^,^^ =-.A'CMN.vai A'C = a^^
^
MN = B C = a N ^ = > S , „ ™ = — . ' A'MCN 2 Gpi H la hinh chieu ciia B ' tren ( A ' M C N ) , ta
<=*: \.'ucs=l-^'^\'«cs
S.'MCK 3 2 3
• » (A'MCN) : x + 2 y + z - 2 a = 0.
KhoSng each d tir B'{a; a; a) den mp(A'MCN):
|a + 2a + a - 2 a | _ 2a _ a.-j6 Vl + 4 + 1 v'e 3
Nhdn xet: Vai phuang phdp long hap de tinh duac khodng cdch tir B' din mat phang (A'MCN) phdi thong qua cdc khoi chop co ciing duang cao, yiu to nay khong phai liic nao ciing nhan ra. Phuang phdp toa do thudn lai hem khi dua bdi todn vi tinh khodng cdch tu diem B'uen mat phdng (A MCN).
Bai toan 8: Cho hinh ehdp S.ABC cd day ABC la lam gidc vudng lai B, AB = a, BC ^ 2a. canh SA vudng gde day va SA = 2a. Gpi M la trung diem eiia SC. Chiing minh AMAB can va tinh dien tich AMAB theo a.
Phuong phap tong hgp
A
\ i v l
B
Ta c6: S.A J. (ABC) => SA ± AC.
Dc do ASAC \-u6ng tai -•>. co ,AM la trung tuyen nen W.\= — SC.
f S A ± ( A B C ) Ta lai CO \
[.AB L BC (AABC vuong tai B)
Phucmg phap toa do
2a
A s
\ %."'
/ \ / H \ C >•
5 - / a B
AABC vuong tai B co:
AC- = A B - + B C ' = a N / 5 Dung BH ± AC (H e AC), ta co
AH = ^ = ^ = ^
AC aV5 S
Nghien citu khoa hgc IhuiKf 3 ndnt 2017
=> SB ± BC (dinh ly 3 duong vuong goc) Do do ASBC vuong tai B co BM la trung tuyen nen MB = - S C .
2
Suy ra: MA ^ MB => AMAB can tai M Dung M H / / S A va H K / / BC
(H e AC; K e AB) SA J. (ABC) B C J . A B
1VIHJ_(ABC) H K ± A B
5 2a
^ = > B H = ^
V5 _1_^_J_ _ 1 _ ^
BH^ AB'' BC^ 4a' Dung he true tpa dp vuong goc Axyz:
A(0; a, 0), C(0; a^/5; 0), S(0; 0; 2 a ) , f 2 a a ^ X B —j=; —7=; 0 Tpa do trung diem M c u a S C l a M 0; ^ ^ ; a . T a c o :
M H = - S A = a H K = i B C = a I 2 AMHK vuong can tai H co:
MK^ = MH^ + HK^ = 2a^ => M K = a%^
Dien tich AMAB:
S M A B = | M K . A B = i . a > ^ . a = ^ ^
•(«¥'•]=
2a 3a
nen: MA = MB => AMAB can tai M
=> | [ M A ; M B ] | = a ' , ^
S M A B = | | [ M A ; M B ] | = i . a ^ , ^ ^
^
Bai toan 9: Cho hinh chop SABC co day ABC la tam giac deu co canh bjng 2a^/2 , SA vuong goc vol (ABC) va SA = a. Gpi E, F IJn lugt la trung diSm cua canh AB, BC. Tinh goc va khoang each gifl-a hai ducmg thjng SE va AF.
Phuang phap tong hgp
U~ -i
AV\ . \ c
^^~~~~~~^ \ ^ y i
B
Goi M la trung diem ciia BF ^:> EM // AF
Phucmg phap tpa do
Dung he true toa do Axyz, vdi Ax, Ay, Az THONG TIN KHOA HOC GIAO DUC
Q/i, 'anij.5 naiii 2017 Nghien cHu khoa hgc
=> (SA; AF) = (EM; AF) = SEM ASAE vuong tai A co SE^ = S A^ + A E = a= + 2a^ = 3a^
^ S E = a,/3,AF = ?5^^I:^ = aV^
2
=> E M = B M = M F = . ^ ; B F = R-Jl 2
SB^ = SA- + AB^ = a^ + Sa' = 9a^ => SB = 3a SF- = S A ' + AF- = a- + 6a- = 7a- => SF = a V ^ Ap dyng dinh ly duong trung tuyen SM trong ASBFco. S B - + S F - = 2 . S M - ^ - - B F -
2 0 9 a - + 7 a = = 2 S M = + - . 2 a - « . S M = = ^ ^
2 2 Goi A la goc nhon tao boi SE va AF
Ap dung dinh ly Cosin vao ASEM co
cos a = cos SEM = E S ' + E M ' - S M '
2.^.aV^
2
•J^ • a = 45".
Dung AK X ME; AH 1SK. Ta co
• J2
AK = MF = ^ ^ vao AH ± (SME) Vi AF//ME => d(SE; AF) = d(AF; (SME))= AH.
ASAK vuong CO ; - = -H = AH- SA- AK-
=> AH = — Vay. d(SE; AF) = —
3 3
•• • ..Anc HOC Khi do A(0,0,0),
B(a%/2; ax/6; 0),C(-....-. '~n^
S(0;0;i.),E['^,^;ol;F(0;a76.0) : / l ; i ^ ; - J ; X F = (a;ax/^;0)
u[i
-; -d-JE; 0 Suy ra.iSi = fi^;aV6;-a|
Goi A la gdc nhon tao boi SE va AF 0 . i ^ + a 7 6 . ^ 0 ( - a ) 2 2 L ^ ^ r, /a- 3a^
V0 + 6a- + 0 . , \ ha- V 2 2
- • ' » ' ^ ^
ax/e.aV? 2
[5i;sS] = [ ^ ; 0 ; ^ )
= i ! ^ ( V ^ ; 0 ; , ) = ^ h ,
2 2 vol a = (%/2; 0; I)
Phuong trinh mat phiing (SEM) qua S voi VTPT n: V2x + z - a = 0.
Khoang each til' A den (SEM) 0 + 0 - a | d-Jl d ( A ; S E M ) = i 3
•Jl+l Vi A F / / E M = > A F / / ( S E M )
=> d(SE; AF) = d(A; SEM) Vay, d(SE; AF) = — ^ .
Nghien cHu khoa hgc CThdjui 3 ndin 2017
Nhdn xet: Bdi todn ndy niu gidi phuang phdp tdng hgp. ta phdi thong qua nhieu cong thiec vd nhiiu buac tinh todn. Trong khi do, vai phuang phdp tga do chi cdn quy ve tinh khodng cdch tit diin A den mat phdng (SEM).
3. Thirc nghiem 3.1. Muc dich thirc nghiem
Thuc nghiem dupe tiSn hanh nham muc dich kiSm nghiem tinh hieu qua, kha thi cua phuong phap day hpc giai bai tap hinh hpc khong gian icrp 12 bang phucmg phap tpa dp hoa so vai phucmg phap hinh hpc tong hpp.
3.2. Noi dung va kit qua thu-c nghiem
Tac gia tiln hanh day thuc nghiem dupe tai Icrp 12A, (lop thuc nghiem) va I2A3 (Icip doi chimg) tai Trucmg Trung hpc ph6 thong TrSn Ngpc HoSng, nam hpc 2013 - 2014. Lop 12A, CO sTs6 39, lop 12A3 CO si s6 37. Ca hai lop co trinh dp hpc luc tucmg dirctng nhau va lop 12A, chua dupe hpc phuomg phap nay.
Sau cac day tilt day va giai bai tap bSng phuang phap tpa do 6 icip thuc nghiem 12A,, tac gia cho hai Icrp lam bai kiem tra va th6ng ke dupe ket qua nhu sau:
Bdng 3. J. Biing thong ke diem bai kiem tra
^ \ Diem
L(.Tp ^ \ Thuc nghiem
12 A, Ddi ehiing
12 A3
STsd
39 37
Kem [0;3]
SL 0
2 TL%
0
5,4 Ylu [3.5;4.5]
SL 4 7
TL%
10,3
18,9 TB [5;6]
SL 14
19 TL%
3S.9
51.4 Kha [6.5;7.5]
SL 16
8 TL%
41
21,6 Gioi [8;10]
SL 5 1
TL%
12,8
2,7
^~"~'"'"~----^ Ldp
Loai ^"""""---....^^^
Diem trung binh Tl le dal yeu cdu Tl le diem yeu kem Ti le diem Irung binh
Thue nghiem 6,29 89,7%
10,3%
35,9%
Ddi chiing 5,43 75,7%
24,3%
51,4%
THONG TIN KHOA HOC GIAO DUC
'Jhaiui 5 iiatn 2017 Nghien ciiu khoa hgc
Ti le diem kha Ti le diem gioi
41,0%
12,8%
21,6%
2.7%
Bang 3.1 cho thdy dihm trung binh cdng; li le dat yeu cdu; ti le dal diem khd, gidi d ldp thuc nghiem cao hem so vdi ldp ddi chiing. Cdu hdi dal ra la: Cd phai phucmg phap day d ldp thuc nghiem tdt hon phuong phap day d ldp ddi chiing khdng, hay chi do ngau nhien ma cd?
Chiing ldi de ra Gid thuyet thdng ke H,,: "Khong co su khdc nhau giica hai phuang phdp " va sir dung kiem dinh z nham bac bd H..
^~~~~~~~~~...._^^^ T h a m s d L d p ^~~~~~~~~~~....,^
Thue nghiem Ddi chiing
Bdng 3.2. E
X
6,29 5.43
dng tham sd dac trung s^
1,96 1,73
s 1,40 1,32
s' 2,01 1,78
s' 1,42 1,33 Trong dd cae dai lupng dupe tinh theo cdng thiic:
Vdi mdu X = ixi,x,,....,x„] va H, la tan sd xudt hien ciia x^,i = \.n .
~ - \ -^
+ Gia tri trung binh eua mau: x - — > x,/7,
1 " - ^ + Phucmg sai ciia mdu: s^ ^ — /'^nJx^-x) + Dp lech chudn ciia mau s = J—^'^, (^,
+ Phuong sai hieu chinh cua mau: s'' 1 V
X"a^'-^
+ Dd lech chuan hieu chinh cua mdu: s' - j / //, (.v, - x ]
\ n - \ ^ '
Ap dung cdng thiie z - • , trong dd x,,Xj,s\ ,s\ ,n.,n-2 Idn lupl la diem
Nghien cHu khoa hgc <^hdiuj^ 5 ndiii 2017
trung binh, phuong sai hieu chinh, so lupng hpc sinh cua lijp thuc nghiem va lap doi chimg.
. , .; . -1 -2 6,29-5,43 . ., Vay ta co ket qua: z = , = , ^ >* 2, /
•' ^ r r , TT /2,01 1,78 39 "^ 37
Vol mtic y nghia o = 0,05 thi gia tri tai han z, = 1,96 .Vi z = 2,7 > 1,96 = z, nen Gia thuylt H^ bi bac bo. Vay phucmg phap day a Icjp thuc nghiem t6t ban so vcji phuang phap day tr lop doi chung.
4. Ket luan
Phuong phap tpa dp hoa giup giai mpt so bai toan hinh hpc thuan ttiy don gian han khi giai bang phuong phap tong hap. Lutmg kien thiic va ky nang de giiip hpc sinh co the giai cac bai toan hinh hpc thong qua phucmg phap nay khong nhieu va la phuang phap khong qua kho doi vol cac em hpc sinh trung binh. Trong bai viet nay chiing toi tap trung vao viec chpn he true tpa dp de giai bai toan hinh hpc thuan tuy. Day la phan quan trpng nhat de giai thanh cong mpt bai toan hinh hpc bjng phuang phap tpa dp. Cac bai toan tren day da phin nao khing dinh cac im diem, tinh kha thi va hieu qua cua phuang phap tpa dp hoa trong day hpc toan,
TAI LIEU THAIVI KHAO
Van Nhu Cucmg (chu bien), Pham Kh5c Ban, Ta Man, 2007, Bai lap hinh hoc 11 ndng cao, Nxb Qiao due.
Van Nhu Cuong (chu bien), Hoang Ngoc Himg, Do Manh Hiing, Hoang Trpng Thai, 2005. Kmh hpc sir cip vd thuc hdnh gidi todn, Nxb DHSP.
Doan Quynh (tong chii bien). Van Nhu Cuong, Pham KJik Ban, Le Huy Hiing, Ta Man, 2007. Wnhhx 11 ndng cao, Nxb Giao due.
Doan Quynh (tdng chii bien). Van Nhu Cuong, Pham Khac Ban, Le Huy Hiing, Ta Man, 2010. Hinh hoc 12 ndng cao, Nxb Giao due.
Vo Thanh Van (chO bien), Le Hien Duong, Nguyen Ngoc Giang, 2010. Chuyen di img dung toa di troiif gidi loan hinh hoc khdng gian. Nxb Dai hoc Su pham.
Tuyin lap cac dS thi Dai hoc, Cao dSng mon loan cac nam (2008 - 2011).
THONG TIN KHOA HOC GIAO DUC
