KHAI THAC VE DEP VE TINH THONG NHAT CUA TOAN HOC THONG QUA GlAl MOT SO BAI TOAN HINH HOC PHANG BANG PHUONG PHAP DAI Sfi HOA

Q NGHIEN Cliu U LUAN

KHAI THAC VE DEP VE TINH THONG NHAT CUA TOAN HOC THONG QUA GlAl MOT SO BAI TOAN HINH HOC PHANG

BANG PHUONG PHAP DAI Sfi HOA

NGIIYEN THANH QUANG - Tnrimg flil hoc Hnh Email: iiiauang1U@irahoo.com N G U V I N VAN THA - Tnlihig THPT Phiing Hung - TP. Htf CU M

Emair: thanialhs@gmallxoni

Tdm t&t: Phuang phdp dgi so hoa mdt sd bdi todn hlnh hoc theo hudng khai ihdc vi dep thdng nhdt cCia Todn hjt nhdm dong cdggi md, tao hUng Ihu, xdy dUng Idng lin trong hoc tdp de phdl huy tdi da ndng lUc sdng tgo cho hoesiiA.

Ciing vdl cdch gidi todn bdng hinh hoc truyen thdng. hoc sinh cdn dUde hudng ddn degiai lodn hinh hgc bdng phuatj phdp dgi so hda ciing nhung kindng chuyen doi mdl bdi todn ti/tinh vUc ndy thdnh mdt bdi lodn cua tinh vUc khde Iron^

sU thdng nhdt tron ven cua Todn hoc. TUdo, kich thich tinh to md vd sdng tao, gop phan ndng cao kindng tiip cdn glii quyit vdn de trong hoc lap mdn Todn cung nhutrong cudc song cho hoc smh

Titkhda: Todn hoc; bdi todn Hinh hoe phdng; phuang phdp Dai so hda.

(Nban bdi ngdy 19/7/2016, Nhdn kit qud phdn bien vd chinh sUa ngdy 18/8/2016; Duyit ddng ngdy 25/8/2016).

NhO'ng l<hai mem, l<et gua cung nhU phep suy luan vh phuong phap toan hoc ngay cang tham nhap sSu vho chc Ilnh vuc cua ddi song con ngudi Ngay nay, ranh gioi giOa cac nganh Toan hoc da la mot khoi thong nhat.

Che quan he hinh hoc deu co the bieu dien bSng nhQng phuong trinh dai so t h o n g qua cong cu toa do

Viec t i m toi nhieu I6i giai bSng cac ngon rgCr khac nhau cho mot bai toan Hlnh hoc giup giao vien (GV) va hpc sinh (HS) nhan thLfc dUoc g i ^ t n tham mi cua Toan hoc, CO tSc dung thiet thuc trong viec goi rod kha nang t u duy, nang luc sang tao. Cung v6i each giSi toan bSng Hinh hoc truyen thong, HS can dUoc h u o n g dan de gi^i tohn Hinh hoc bSng phuang phap Gai so hoa ciing vdi nhifng ki nang (KN) chuyen doi mot bai toan tU Ilnh vuc nay thanh mot bk\ toan cua Ilnh vuc khac trong su thong nhat tron ven cua Toan hoc, Phuong phap nay tao ra tiep can moi sinh dong, giiip ngudi hoc khai thac toi da kha nang cua ban than, tao mem vui kham pha va sang tao.

Trong bai viet nay, chung toi sir dung p h u o n g phap va cdng cu Dai so de giai mdt so bai toan Hinh hoc phSng, T u t u d n g ciia viec thUc hien cac chifng mmh nay chinh la khai thac ve dep tham m i ve tinh t h o n g nhat ciia Toan hoc trong day va hoc Hinh hoc NhiJng t i m toi nay gdp phan tich cuc hoa hoat dong hoc tap m o n Toan noi Chung va phan m o n Hinh hoc ndi neng cho HS dcac trudng pho t h d n g .

2. Oai so hda b i j toan Hinh hoc

Chung ta sil dung cong cu Oai sd de dien dat cac bai toan Hinh hoc bSng ngdn ngif p h a o n g trinh. Hon nifa, cac quan he vudng gdc, song song, lien thudc giao cat, trong Hinh hoc khi dien dat bSng ngdn n g i i 3 2 > KHOA HQC GUiO OUC

p h u o n g trlnh t r o n g B a i so rat tien Id! va hieu quei, 0^1 st cd lai the la chat che, mach lac, logic, cd nhi^u quy trififi va thuat toan chuan, p h i i h o p vdi t a m If tiep nhSn, khin pha cLia HS, Do do, p h u o n g phap nay giilp HS khi giil toan Hinh hoc tranh duoc nhCfng ki t h u a t quh d^c biH ISt leo, khong p h o bien.

Dai sd hda mot bai toan Hlnh hoc co the du^ vlk gih\ bai toan Hinh hoc d o ve thuc hi^n trgn may tlnh bSng chc phan m e m Tin hpc t u o n g thich. ChSng han, g\h\ cua nhieu bh\ toan Hlnh hoc phSng cd t h ^ thtfc hifn tren may t i n h bdi phan m e m Maple n h d cdng cu toil hpc la li thuyet ca sd Grobner [1].

Co sd ciia p h u a n g phap d u n g hlnh bSng cdngoi Oai so dua vao nhan xet: Moi phep d u n g hinh d^u cdchf dua ve t i m n g h i e m cua m d t he p h u o n g trinh Bai sfi dd, ChSng han, cac p h e p dUng hinh b5ng thUdc Wrf compa deu dUa den viec dUng cac d u d n g th5ng, duflnj trdn va t i m giao diem cua chiing, Khi bi^u d i l n cSchif*

phSng t r o n g toa d d Descartes (Oe - cac) vudng gdc, Mu het cac hlnh hinh hoc hoac bien cCia chiing cd th^xem la tap cac n g h i e m ciia cac da thifc va nhOng quafiW giiJa chung deu cd t h e m d t i bSng cSc phuang Irlnh*

thifc. Chiing ta xet tren mat p h 5 n g hay khdng gian thi^

nen cac da thirc c i i n g n h u tap cac nghiem phSi xft trin t r u d n g so thUc. Vi vay, cd the Dai sd hda mdt dinh li W bai toan Hinh hoc phSng d u d i dang sau;

Gid thiet:

Cho he p h u a n g trinh / , = 0 , / ; = 0 , • - • , / j = 0 . I * ludn: Khi d d moi nghi&m thuc ciia h ^ tren phdi thda r he p h u a n g trinh g , = 0 , g j = o , - • , g^ = 0, trong d6:

NGHIEN cmt Li LUAN Q

la cac ' rllAllj 3a thiJc vdi he so t h u c Cac bien d d c lap u^ la

toa d o cua cac d i e m t u o n g Lfng va cd t h e c h o n t i i y y; cac bien X,,.. ,x„ la p h u thudc, nghla la m o i bien phai xuat hien t r o n g it nhat m d t da thifc f, nao d d . Khi r = 1 ^

* ) l i m | d u o c k i h i e u l a g , ' ' hkHJH

Vi d u 1 : Churng minh rang trong tam gidc ABC ba

^'^•Uydudng cao dong quy

Chon he true tpa d d 5(0;0),C(U|;0),^(u3;u3) va

^Ihrnr ^ * ^ i ' - ^ 2 ) - Trong d o u„u2,u2 la cac bien d o c lap, .*:„r, i t e l r ' l " ^ ^ " ^ ' ^ " P ^ " ^^^"^"^ ^ ' ^ ^^ ^ " -^ ' \ ' ^ ' CH 1 AB. Ta can

^ ^ 1 2 ' ^ ^ '^'"^ ' ^ ^ -^ ^ ^ - ' ^ ^ ' ^ ° ' " ^ t ' " ^ ' ^^la-t hinh hoc se

^ M r ^ R m d ta bdi cac p h u a n g t r i n h dai so n h u s a u -

™ « k „ , D i e u k i e n B H l A C c h o t a / , - = x , ( « , - « , ) + x , i v 3 = 0 AB cho ta

D o d o / , ; = u , - - A=ur-2u,x, Tacd: ^ = (.t-,-

= («2-.r,)-

U2,-V3--T|) ,

• 5 = 0 (2),

•^ Oieu kien CH

TU d d , dieu kien c i n chung m i n h la AH 1 BC t r d t h a n h : s f e f e : g = u i ( u j - x , ) = 0 , .IOOICMK: .11. . -

', N h u vay, c h u n g ta *

^ " * f ^ ' d a Oai s d h d a bai toan

^sP[«rP^-Hinh hoc n&y t h a n h bai

ng kffei: toan: Gid thiet: Cho he hai p h u o n g t r i n h yj = 0, / ^ = 0 Bk-.r'^^^ '"^"- ^^' ^° " ^ ^ i m^^'^m thuc cua he hai p h u a n g

t n n h tren ph5i thda man p h u o n g t r l n h g = 0 That vay.

Hinh 1

thKch^'''

j n | ^ c : ^ ^ ^ ' = ° ^ " - ' ' " ^ 2 " e n k h i _ / l - 0 v a / , = O t h i g = 0.

- r f ^ r - ^ * **^ ^ ' ° ' " * ' " Eu'^""): Cfiuhg mmh rdng trong tam fliD^'i^T 9'^<^'^6C r^t/c torn H, trang tdm G vd tdm O eua vong trdn

ngogi tiep tam gidc thdng hdng

Oai sd hda bai toan n h u sau: Chon he t n j c toa j - ; d 6 sao c h o S ( 0 : 0 ) ,

- ' lou

3

C ( « , . 0 ) , Aiu,.u,). H(u.:.v^):

KW^"- Ta cd G la t r o n g t a m Kr,!*.- CLia t a m giac ABC va AH e2t'''' 1 BC, dieu kien 0 la t a m giiCS" d u d n g t r d n ngoai t i e p

^•,r-" t u o n g d u o n g vdi hai dieu ga;fJ kien OB=OA, OB=OC...:

,,^Jl^Tt; B5ng each chon he true '^r^T- toa d d n h u vay, cac t i n h jjis^'- ^^^^ ^ ' " ' ^ ^9^- se d u o c m d

, t^ bdi cac phucmg trinh

dai so n h u sau: Dieu kien BH 1 AC tUong d u o n g vdi , , , " : ( ' ' 2 - " | ) + - v , ( , „ - 0 ) = 0 . H a y y , = „ f - „ , ^ , + . v , : . , . 0 , ^ f / OA = OB^ .xi + t r = (x, -^u,f+ (x, - .3 ) ^

Hinh 2

Vi vay, dieu kien ba d i e m H, G, 0 t h a n g hang la-

Hay g -= -2u.x, -x.u, + u.u, . x , u , . . . , 3 . t , - . T , » , -X,U, = 0 . N h u vay, c h i i n g ta da Dai sd hda bai toan H i n h hoc nay t h a n h bai t o a n : Gid thiet: Cho he ba p h u o n g t r i n h f\ = 0 - / 2 - 0, / 3 = 0. K-e'f iudn: Khi d d , m o i n g h i e m thuc cua he ba p h u o n g t r l n h tren phai thda man p h u o n g t r i n h g = 0.

That vay, V) u;i/^ ^ 0. nen tCr/3 = 0 suyra u,=2x,.

lis fj^Qj^^G

•^^^ou,^^u,{u,^2x,)-2x,u,-x,u,=0

^^yu,=2x,.x,.

TiS 66:

S-{-2u,x,.u,u,~.r,u,).(.r,u,.3..,x,-x,u,)-x^, iXjX^ ySx ''2{i's-2x,~x,)^x,{2xy

:,X2 2x,x,)

= 0.

^2"3

V i d u 3: Cho tam gidc ABC cdn tai A. H Id irung diem BC, D Id hlnh chieu cua H len AC. I Id trung diem HD. ChOng minhrdngAll BD.

Ta thay AH ± BC tai H nen viec Dai sd hda bai toan nay se kha t h u a n loi.Ta

chon H lam gdc toa d d . Tuy chua biet d d dai HB, HC n h u n g d o t a m giac da cho la t a m giac can nen ta c h o n ngay H B = H C = l . K h i d d , t a c h i can chon toa d d cua / roi t i n h toa d o ciia D va c h i i n g m i n h dieu kien v u d n g gdc.

Chon he true l o a

d o sao cho H(0:0), B(-1;0); C(1;0), A(0,-x), l(u,v), D(2u,2v).

Dieu kien HD 1 AC t u o n g d u o n g vdi -2u + 2xv'= 0- hay / | = - ( v + . u = 0 .

Oieu kien 0 nam tren AC t u o n g d u o n g vdi 2 i v 2 r ^ ( 2 v - . r ) ( 2 , . - l ) , h a y / , = 2 x 1 ; + 2 v - _ r = 0

Hinh 3

stfm-TiUi(Giran8-33

Q NGHIEN CUfU Li LUAN

Oieu kien de Al ± B D t u o n g d u o n g v6i

»(2K + l ) + ( v - x ) 2 v = 0 , h a y g = 2 H - + » - ^ 2 v - ' - 2 i T = 0 NhUvay, chiing ta da Dai sd hda bai toan Hlnh hoc nay thanh bai t o ^ n : Gid thiet: Cho he hai phucmg trinh

^ = 0 , / 2 = 0 . Kit ludn: Moi nghiem thUc ciia he hai p h u a n g trinh tren phai thda man p h u o n g trlnh g = 0

T U / j - 0 suyra H = . n , d o d d / ^ = 2 x ^ ' + 2 v - . t Vi vay, tif / , = 0 , ta cd g- 2x ^vf,=0.

V i d y 4 : Cho tam giac ABC vdi d u d n g cao AD, d la d u d n g t h i n g di qua D, (ay E, F e d, khdng triing vdi D sao cho AE±EB, A F I F C Goi M, N lan luot la t r u n g diem cua cac doan thSng BC vh EF. Chtfng minh rSng A N I M N .

Hinh 4

Nhin vao Hinh 4, GV va HS cd the thay viec chii'ng minh true tiep bSng Hinh hoc gap nhieu khd khan, phiJc tap, Vdi chil y rSng, trong chc gia thiet da cho cd den yeu to vudng gdc va yeu td ba diem t h i n g hang, chinh dieu nhy nay smh y tUdng Oai sd hda bai toan. Ta se chpn A la goc toa do vh dat toa d d eho ba diem D, E, F. Tir dd, si^

dung cae dieu kien vudng gdc de viet p h u o n g trinh cae dudng t h i n g t h i n g AD. AE, Af, BC, BE, CF de t i m ra toa d d ciia cae diem B, C. M. Tiep dd, GV yeu cau HS kiem t i n h vudng gde bSng cdng cu n'ch vd hUdng. Neu dat toa d d bat ki cho D, f, F thi d i i n g 6 an sd se qua phiJe tap. Do do, de dan gi^n hon, ta gan tung do ciia D.E.F b i n g -1 hay ndi cheh khac, ba diem nay n5m tren d u d n g t h i n g y = - 1 , Luc nay b^i toan chi edn 3 an sd.

Chon he true toa dd sao cho A(0;0), Ax//d, Di,l.-l).E{<'.-[).F(f.-l) vdt d , e , f doi mdt khac nhau!

Ta cd V " W

I P h u o n g t r i n h c a c d u o n g t h i n g lan luot nhusau- .^D .»+,/i = 0 . .AE A + t'r = 0

--!/•" . w / . i = 0 , BC dx~^-cJ- -\=o.

BE c.-,-^-e'-\^Q: CF /V^ , • _ / - _ i . Q

34 • NHOA HOC GIAO DUC

Toa d p eac d i ^ m lan l u o t la: B[d + e;de~l]'

Do d d AN 1 MN vl

(-0-

^{de + df}

Qua vide Oai so hda cac bai toan Hinh hoc, ta nhjn thay KN chon he true toa d o va gan tpa d p cho moi di&i da cho hoae diem can t i m sao c h o phii h o p v^ t i f n fch H m d t KN quan t r o n g GV can trang bi va boi dUdng cho HS, Viec chon tpa do nao la bien p h u thudc hoac bi^n dtft lap la KN can cd khi t h u e hien Oai so hda mdt bai toin Hinh hoc Ngoai ra, KN phien djch moi tfnh chat Hlnh hoc thanh n h i i n g t i n h chat Dai so, So hoc thdng qua c3c cong cu nhtf p h u o n g t r i n h , he p h u o n g trlnh, h^m thCfc, cdng thu'c dai sd !^ can t h i ^ t t r o n g luydn thp giJI toan cho HS.

Trong qua t r l n h day g i l l c^c bai t3p da neu, GV ed t h e h u d n g dan HS t i m ldi gi^i bSng nhCfng cau hfii dan d^t, cd tfnh chat g o i m d p h i l hop. C h l n g han, dfii vdl b^i toan nay, ta ndn ehon gdc va cac true toa fld nhu the nao? Hay xac d i n h toa d d ciia cac d i ^ m vdi toa &) da chon? Hay dien dat t i n h chat Hinh hoe nhy b i n g cik p h u a n g trinh Oai sd? NhUng toa d d n^o \h bi^n phy thudc va bien dpc lap?

Vdi t h d l gian hoc tap ngoai gid, GV cd t h ^ t?o ra n h i f n g san chOi b d fch c h o HS luyen tap dien dat m$tsfi t i n h chat Hinh hoc b i n g ngdn ngCfDai sd difdi dangtil d o n gi^n den phUc tap. C h l n g han, HS ph^t bi^u Djnhli Pitago trong Hinh hoe bang n g d n ngC dien tfch, sd hpc hoac luang giac; phat bieu v e d i e u kidn vudng, nhon,tii ciia m d t gdc t r o n g t a m giac bSng b^t d i n g thu'c trfin sfi do ciia eac canh,,.

3. K^t l u a n

Vide dinh h u d n g cho ldi g i l i bai t o l n Hlnh hgc b5ng p h u o n g p h ^ p Dai sd n h i m khai t h i c nhCng v^ d?p t h d n g nhat eiia Toan hoe la dieu khd khan ddi vdi HS t r u n g hoe p h o t h d n g . Da so HS giSi toan trong linh vilc nao thl t h u d n g sU d u n g cdng cu va phuong p h i p ci}a linh VUc do, Vdl kien thu'c ea b i n ve Dai sd vh Hinh hoc eung su h u d n g dan ciia GV, viec n h i n dang, phSn N de van d u n g , sang tao dUa ra each g i l i dep vh hi^u qui la didu h o l n toan cd the thUe hidn duoc trong g i l i loin Hinh hoc b i n g edng eu Oai sd va nguac lai Trong thi/ctl g i l n g day, chung tdi da lua chon duac mdt he thong bJi tap p h i l hop va k h i o sat, danh gia kha nang vSn dgng vao thue te hoc tap va tfnh k h i thi eda c i c bidn ph^pd*

xuat theo h u d n g da ndu d tren ddi vdi t i m g HS, Tao h i l n g t h i i va niem dam me cho HS trong dff hoc Toan n h i m nang cao hieu q u i ciia q u ^ trinh dff va hoc Qua kmh n g h i e m g i l n g day thUe te. c i c t l c g B .

NGHIEN COU LJ LUAN Q

™tBNNiri...j

PflUttlOKIm, iBaisofiH-t,

Jidi 110111-- IMiSilftii!';,

plutajjrf;-

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• nhan thay viee van d u n g kien thiJc Dai sd vao gi^i cac bai toan Hinh hpc g i u p HS cd each nhin l o n g q u a n ve sU lien he chat che gida Hinh hpc va Dai s6.TCf d d , kieh thieh t i n h t o m d va sang tao, g d p phan nang cao KN t i e p can giai quyet van d e t r o n g hoe t a p m d n Toan c u n g n h u t r o n g cupc sdng cho HS. N h d su c l i tien t r o n g day va hoc, vide tranh luan edi m d vdi HS va d e HS trao ddi' p h i n bien ldi g i l i e l c bai toan Hinh hoc g i u p GV vCfng vang hOn t r o n g nghe nghiep cua m i n h .

Khai thac ve d e p t h o n g nhat cua Toan hoc k h d n g chi h u d n g d e n t l n h ydu Toan hoc ma cdn truyen e l m hilng cho HS v u o n t d i su sang t a o de t i m t d i n h i ^ g cai m d i , cai dep t r o n g eupc song. C I c vf d u trdn day n h u m m h c h u t i g cu t h e ve khai thac vd d e p ciia sU t h d n g

nhat giCfa hai linh vUc Oai so va Hinh hoe t r o n g c h u o n g t r i n h Toan hpc p h d t h d n g .

T A I u l u T H A M K H A O

[1], Le Tuan Hoa, (2003), Dal so mdy tinh ca sd Groebner. NXB Dai hpe Qudc gia Ha Ndi,

[2]. Ha Huy Khoai, (2007), Cdc nhd Todn hoc duac Gidi thudng Fields (1936 - 2006), NXB Giao duc, Ha Ndi.

[3]. Theoni Pappas, (1994), Suki diiu cua Todn hoc (Ban djch tieng Viet), NXB Dan Trf, Ha Ndi.

[4]. Rene Vidal - Yi Ma - S. Sahankar Sastry, (2016), Gerenatized Principal Component Analysis, Spnnger Verlag New York.

i&Miii

EXPLOITING THE BEAUTY OF THE MATHEMATICS UNITY T H R O U G H D O I N G SOME PLANE GEOMETRY EXERCISES BY USING ALGEBRA M E T H O D

N g u y e n T h a n h Q u a n g - Vinh Unhrersity Emaih ntquang 144@yahoo.eom N g u y e n V a n Tha - Phung Hung high school -Hochiminh city Email: thamaths@gmail,eom

0 i m s t a : a T e Z 2 2 : ' l 7 e T s ° 2 t T d : n r f ' ' ' ° " ^ ^ ^ ^ ^

Keywords: Maths; plane geometry exercise; Algebra method

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