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Bach Phucme Vinh Tap chi KHOA HOC & CONG NGHE 83(07) REN LUYEN MOT SO HOAT DONG TRI TUE CHUNG CUNG VOI CAC HOAT DONG TRI TUE PHO BIEN TRONG TOAN HOC CHO HOC SINH LOfP 9 THONG QUA BAI TAP HINH HOC PHAN[.]
Bach Phucme Vinh Tap chi KHOA HOC & CONG NGHE 83(07): REN LUYEN M O T SO H O A T DONG TRI TUE CHUNG CUNG VOI CAC HOAT D O N G TRI T U E PHO BIEN TRONG T O A N H O C C H O HOC SINH LOfP T H O N G QUA BAI TAP HINH HOC PHANG Bach Phuong Vinh Trii-dng Dai hoc Su pham - DH Thdi Nguyen TOM T A T Day hgc giai bai tap hinh hgc phang d ldp nham thuc hien mdt nhQ-ng nhiem vu ciia mdn hgc la phat trien tri tue cho hgc sinh; dieu se cd y nghTa sau sac ban neu ngudi giao vien ludn tao co- bdi cho hgc sinh thuc hien cac boat ddng tri tue chung: phan tich, tdng hgp, tuong tu, khai quat hda, dac biet bda cung vdi cac boat ddng tri tue phd bien toan hgc: phan chia trudng hgp, lat ngugc van de, xet tinh giai dugc qua trinh hgc sinh di tim ldi giai va suy nghT khai thac bai tap binh hgc Tir khoa: Hogt dong tri tue, tu duy, hoc sinh, bdi tap hinh hoc, lap Phat trien tri tue cho hgc sinh (HS) la nhiem vu ciia mgi mdn hgc trudng phd thdng, nhat la ddi vdi mdn toan d trudng trung hgc ca sd (THCS) cang cd nhieu dieu kien thuan lgi de thuc hien nhiem vu Ddi vdi day hgc giai bai tap hinh hgc phang d ldp 9, de thuc hien nhiem vii tren ngudi giao vien (GV) phai ludn tao cho HS co- hdi thuc hien cac hoat ddng tri tue (HDTT) chung: phan tich, tong hgp, so sanh, tuong tir, khai quat hda, triiu tugng hda, dac biet biet hda cung vdi cac HDTT phd bien toan hgc: phan chia trudng hgp, lat ngugc van de, xet tinh giai dugc qua trinh HS di tim ldi giai cua bai toan Dieu se cd y nghTa sau sac hon neu GV ludn tao co- hdi cho HS thuc hien cac HDTT chung eung vdi cac HDTT phd bien toan hgc khdng chi d viec HS di tim ldi giai ciia bai toan hinh bgc ma d ca viec HS nghien ciru khai thac bai toan Dd cung chinh la muc dich day hgc cua mdn bgc nham phat trien tu sang tao cho HS quat hda, dae biet hda, tdng quat hda, biet lat ngugc van de va xet tinb giai dugc ciia bai toan de lira chgn nhQng PP va each thu-c phii hgp, hieu qua nham giai quyet bai toan, dua dugc ldi giai, tien tdi cd ldi giai hay, ngan ggn, ddc dao tii- dd de xuat nhung bai toan tuang tu, dac biet va cung cd the la nhQng bai toan "khai quat, tdng quat hon", nhQng bai toan mdi Trong qua trinh dd HS dugc ren luyen cac HDTT chung cimg vdi cac HDTT phd bien toan hgc, gdp phan phat trien cho HS kha nang quan sat, nang luc phat hien giai quyet van de va tu sang tao Cac dang toan hinh hgc phang ldp rat phong phii va da dang Mdi dang toan deu cd nhQng phuang phap (PP) giai ca ban va dac trung, nhien khdng phai liic nao tuan theo nhung phirang phap dd deu giai dirge bai toan; ma ddi hdi HS phai biet nhin bai toan mdt each tdng hgp de phan tich bai toan quy la ve quen biet phan chia trudng hgp so sanh khai Chimg minh rdng MA = MB + MC " Sau day la bai tap binh bgc Idp 9, xuat phat tiiviec di tim ldi giai va khai thac bai toan nham ren luyen cho HS mdt sd HDTT chung cimg vdi cac HDTT phd bien toan hgc Vi du "Cho tam gidc deu ABC ndi Hep dirang trdn (O) Diem M thudc cung BC • Phdn tich bdi todn tim cdch gldl Muon chirng minh MA = MB -i- MC (phan tich tach nhung thudc tinh ciia bai toan (cai toan the)) ggi cho HS lien tudng den viec tao doan thang AD nam tren MA cho AD = MC (hoac AD = MB), hinh (H 1); dd chi cdn phai chimg minh MB = MD (hoac MC = MD) Dieu cd dugc tu cac cap tam giac bang 133 Bach Phuang Vinh Tap chi KHOA HOC & CONG NGHE Neu nhin bai toan theo quan diem bien hinh (til- mdi quan he giQ-a hai each giai bai toan theo PP tdng hgp va PP bien hinh), ggi cho HS lien tudng den viec ddi MC den MA, d day siJ- dung phep quay tam B gdc quay 60° chieu quay ngugc chieu kim ddng hd; dua vao tinh chat ciia phep quay suy dieu phai chimg minh • Trinh hdy ldi gidi (HD tdng hop - hi/p Itii ctic phdn cua hdi todn ) +yi Cdch 1: Phirang phdp tong hap Lay D G A M s a o c h o M C = DA 83(07): 133- 138 i- Cdch 3: Chung minh MA = MB -i- MC, ggi cho HS lien tudng den PP chung minh dang thuc hinh hgc nhd cac ti sd cd tir hai tam giac ddng dang Xet A MBE ' ^ A MAC va A M C E ~ A M A B - ^ N B ^ _ ^ MC_ EC MA " AC ' MA " BA ^ => MB MC BE = + EC -I- MB-hMC c^ =1 MA MA AC BA MA => MA = MB + MC (dpcm), hinh (H 2) (1), cd AABD = A C B M (c.g.c) => MB = DB va BMD = 60" (gdc ndi tiep chan cung AB) => ADBM diu => MB = MD (2) Til- (1) & (2) => MA=MD + DA=MB -^ MC dieu phai chirng minh (dpcm), hinh (H I) +^ Cdch 2: Phucmg phdp bien hinh Theo gia thilt (gt): (MC, MA) = 60" => Q(B, 60°); MC -> MA C ^ A (vi AABC diu) M ^ D e MA (chieu quay ngugc chilu kim dong hd), theo tinh chat ciia phep quay => MC = DA (1) va BM = BD, MBD = 60° => A BMD diu => MB = MD (2) Til-(1)& (2) taco: MA = MD-^ DA = MB + MC (dpcm) • Khai thdc bdi todn > Khai thdc bdi todn theo hudng tim them nhieu cdch gidi khde *) HD Phdn tich bdi todn theo PP gidi 4- Chung minh MA = MB -^ MC theo each 1, dat MC tren MA bang each lay D e MA cho MC = DA va chimg minh MB = MD; ma MA, MB, MC cd vai trd nhu vi chiing deu la day cung ciia (O); *) HD tuang tw vd xet tinh gidi duac Cdch 1.1: Dat MC tren MB bang each lay D thugc tia doi ciia tia MB cho (H1.2) MD = MC Khi do, chiing minh MA = MB -I- MC ^ MA = DB C= A MAC = ADBC (c.g.c), hinh (H 1.1); Cdch L2: hoac dat MB tren MC, bang each lay D thugc tia ddi cua tia CM cho CD = MB Khi do, chirng minh MA = MB -H MC B^ ABMD la tam =^ MB = MD = BD giac diu => AABD = ACBM (c g c) =^ MC = DA ma MA < MD + DA (do xet AMAD) Dodd M A < M B - ^ M C Nhu vay, chi cac diem Me BC thda man MA = MB + MC, nen ta cd bai toan dao ciia vi du nhu sau: Bdi todn J.L Cho tam gidc deu ABC, neu MA = MB + MC thi M ndm Iren cung BC cua dirirng Irdn ngogi Hep tam gidc deu ABC *) HD tdng hap: Ket hgp vi du va bai toan 1.1 di den bai toan quy tich: Bdi todn 1.2 Cho tam gidc deu ABC Chu-ng minh rdng quy tich nhung diem M thod mdn MA = MB ^- MC Id cung BC cua dudng tron tvjoai tien taw vide deu A RC > Khai thiic bdi todn theo huang de xuat bill todn mai (bai toan tuong tu, khai quat hoa, tdng quat hda, ) *) Ren luyen HD phdn tich tdng hop, twang tir, khiii qudt hda cimg v&i HD lat ngirac van de, phan chia trudng hop vd xet tinh gidi dirge ciia hdi todn (H3) Nham tra ldi cho cau hdi: M e cung BC tbi MA = MB + MC, ngu-gc lai, nlu cd MA = MB + MC thi M cd thudc cung BC khdng? *) Ren luyen HD phan tich vdi HD phan chia trudng hop va xet tinh giai dugc: Xet cac vj tri tu-o'ng dii ciia dilm M vdi AABC va du-dng tron (0) ngoai tilp tam giac V M e BC (til- kit qua cua vi du 1) deu cd tinh chat MA = MB-^MC- (H4) 135 Bach Phuang Vinh Tap chi KHOA HOC & CONG NGHE *) Ren luyen HD tong hffp, khdi qudt hda tu ket qud cua HD phdn tich ciing v&i HD phdn chia trir&ng hap: Tu- cac ket qua cua vi du bai toan 1.1; 1.2 di den bai toan khai quat hda: Bdi todn 1.3 Trong mat phdng cho tam gidc deu ABC vd mot diem M bdt ki Chiing minh rdng MA < MB + MC Ddu bdng xay vd chi M ndm tren cung BC cua dirdvg trdn ngogi tiep tam gidc deu ABC *) Nhan xet: Tir bat dSng thu-c MA < MB + MC ggi cho HS lien tudng den bai toan cue trj hinh hgc va di den bai toan mdi sau: Bdi todn 1.4 Cho tam gidc deu ABC ndi Hep dirdng trdn (O) Hay xdc dinh vi tri ciia diem Mtren cung BC cho tdng MA + MB -I- MC cd gid tri lan nhdt Theo kit qua tren MA + MB -^ MC = 2MA MA, ta cd dai MA ludn thay ddi Neu lay mdt diem N d ngoai (O) va thudc mien gdc BAC tbi MB -^ MC + MN > AM + MN > AN 83(07): 133- 138 (H5) (H6) Do dd neu B C, N cd djnb => A cd djnb => Tdng MB -I- MC -^ MN cd gia trj nhd nhat la AN Q = AN n BC cua dudng trdn ngoai tilp A BCN, hinh (H 7) *) Nhan xet: De cd giao diem Q thi AABC phai cd cac gdc khdng Idn hon 120 Trudng hgp AABC cd gdc Idn hon 120° thi Q chinh la dinh ciia gdc Idn nhat Cd the xac dinh diem Q nhu sau: Q = B P n AC cua dudng trdn ngoai tiep tam giac deu ACP (hoac Q = C M n AB ciia dudng trdn ngoai tiep tam giac diu ABM), hinh (H 8) 136 (H7) *) HD tdng hgrp, tir kit qua ciia bai toan va cac nhan xet tren, de xuat cac bai toan chung minh sau: Bdi todn 1.6 Cho tam gidc ABC dimg cdc tam gidc deu MAB, NBC, PAC thudc mien ngodi tam gidc ABC Chimg minh rdng duang trdn ngogi Hep turn gidc deu cimg di qua mot diem Bdi todn 1.7 Cho tam gidc ABC dimg cdc tam gidc deu MAB, NBC, PAC thuoc miin ngodi tam gidc ABC Chimg minh rdng duang thdng MC, NA, PB ddng quy tai mdt Bach Phuang Vinh Tap chi KHOA HOC & CONG NGHE diem chinh Id giao diem cua ba duang trdn ngogi tiep ba tam gidc deu Bdi todn 1.8 Cho lam gidc ABC dimg cdc lam gidc deu MAB, NBC, PAC thudc mien ngodi tam gidc ABC Chimg minh rdng MC = NA = PB vd gdc tgo bai hai dogn ihdng bdng dy bdng 60" *) Nhu vay, qua trinh phan tich bai toan tim Idi giai va khai thac vi du 1, HS dugc ren luyen cac HDTT chung ciing vdi eac HDTT bien toan hgc: - Thuc hien HD lat ngugc van de, xet bai toan 1.1 la bai toan dao ciia vi du 1; - Thirc hien HD tdng hgp: ket hgp vi du va bai toan 1.1 cd bai toan 1.2 la mdt bai toan quy tich; - Thuc hien HD phan chia trudng hgp, xet tinh giai dugc ddi vdi vj tri tuang ddi ciia cac hinh da cho ket hgp vdi HD tdng hgp va khai quat hda tu cac ket qua cua vi du 1; bai toan 1.1; bai toan 1.2 ta cd ket qua khai quat hda la bai toan 1.3; 83(07): 133- 138 - Van dung tinh chat: Trong hai tam giac cd hai gdc tuong irng bang tung ddi mdt thi gdc tuo-ng irng thu- ba cua chiing cQng bang nhau, di den dpcm *) Tri thuc PP giai bai toan 1.8 cd the sir dung de giai cac bai toan tuong tu va md rdng tubal toan 1.8 dugc de xuat bang each thay ddi dieu kien cua bai toan Bdi todn 1.9 Cho tir gidc ldi ABCD, dimg cdc tam gidc deu MAB NCD thudc mien ngodi cita tir gidc vd tam gidc deu PBC thudc mien ciia tir gidc Chirng minh rdng MP = AC, PN = BD vd gdc tcio bdi hai dogn thdng bdng bdng 60' Bdi todn 1.10 Cho diem thdng hdng A, B, C theo thir tw Tren nira mat phdng bd' AC dwng cdc tam gidc deu MAB, NBC Chirng minh rdng AN = CM vd gdc tgo bdi hai dogn thdng bdng 60" • % \ • "V - Thirc hien HD tdng hgp tir bat dang thirc cua bai toan 1.3 ggi cho HS lien tudng va di den cac bai toan cue trj: bai toan 1.4, bai toan 1.5; / "-vA -* ^'„- / ^' / - Xet cac trudng hgp cua bai toan cue tri ta phai giai quyet bai toan dung hinh va de xuat dirge bai toan chimg minh cac dudng ddng qui: chii'ng minh cac dudng trdn ddng qui bai toan 1.6; chirng minh cac dudng thang dong qui - bai toan 1.7; Tong hgp cac ket qua tren de xuat dugc bai toan 1.8 rat thii vj, ma tri thuc phuong phap hinh bai toan 1.8 dugc van dung de giai mdt chudi cac bai toan tuong tu va ind rgng cua bai toan 1.8 *) Tri thirc phuong phap ap dung de giai bai toan 1.8 (sii- dung PP tdng hgp hoac PP biln hinh): - Chung minh cac cap tam giac bang nhau; - Tir hai tam giac bang suy cac yeu td tuong irng bang nhau; '•; / I / /' N / / (H8) B^ '4' (H9) Bdi todn PIP Cho diem thdng hdng A, B, C theo thir tir dd Dimg tam gidc deu M4C', NAB thudc ve hai nira mat phdng ddi hd AC 137 Bach Phuonn Vinh Tap chi KHOA HOC & CONG NGHE Chimg minh rdng MB = NC vd gdc tgo bai dogn thdng dd bdng 60 *) Md rdng cac bai toan tren thay viec dung cac tam giac deu dung cac hinh vudng; sir dung tri thuc PP hinh d bai 1.8 chung minh dugc hai doan thang bang va gdc tao bdi giua chimg bang 90 Bdi todn 1.12 Cho tam gidc ABC Dirng cdc hinh vudng ABDE, ACFG thudc mien ngodi tam gidc ABC Chimg minh BG = CE vd gdc tgo bai hai dogn thdng dd bdng 90 Bdi todn 1.13 Cho tam gidc ABC Dimg cdc hinh vudng ABDE, ACFG thudc mien ngodi tam gidc ABC Goi M la trung diem cua BC; Ol, O2 Idn luat Id tdm ciia cdc hinh vudng tren Chimg minh MOi = MO? vd gdc tgo bai hai dogn thdng bdng 90 Sii' dung ket qua bai toan 1.12 cho bai toan 1.13 Ap dung nhanh chdng PP giai bai toan 1.8 cho bai toan 1.14 la sir dung ket qua ciia bai 1.13 Bdi todn 1.14 Cho lirgidc ldi ABCD Dimg cdc hinh vuong ABEF, BHCI, CDPQ, DARS thudc mien ngodi cua tir gidc Goi Oi, O2 O3, O4 kin lugt Id tdm ciia cdc hinh vudng tren Chirng minh O1O2 = O3O4 vd gdc tgo hdi dogn thdng bdng 90' Bdi todn 1.15 Cho diem thdng hdng A.B, C theo thir lu Pren nua mat phdng bd AC dimg cdc hinh vudng ABDE, BCFG Chimg minh rdng AG = CD vd gdc tgo bai hai dogn thdng dd bdng 90 Bdi todn 1.16 Cho diem thdng hdng A,B, C theo thir dd Dimg cdc hinh vudng ACDE, ABFG thudc ve hai nira mat phdng ddi bd AC Chimg minh rdng BE = CG vd gdc tgo bdi hai dogn thdng dd bdng 90 83(07): 133- 138 Xuk phat til- viec di tim ldi giai va khai thac phat trien bai toan hinh hgc mot each cd he thdng vdi nhQng HD toan hgc, GV tao dieu kien eho HS tu nhin nhan phan tich bai toan tim each giai va de xuat nhung bai toan mdi bai toan tuong tu, bai toan md rdng cua bai toan ban dau, tir bai toan 1.1 den bai toan 1.16 Dd la ca qua trinh HS dugc ren luyen cac HDTT chung: phan tich, tdng hgp, tuong tir, khai quat hda ciing vdi cac HDTT phd bien toan hgc: lat ngugc van de, phan chia trudng hgp va xet tinh giai duge thdng qua giai cac dang toan hinh hgc d ldp 9, da gdp phan khdng nhd vao viec phat trien tri tue va tu sang tao cho hgc sinh TAI LIEU THAM KHAO [I] Hoang Chung (1997), PPDH Todn hoc a trudng ihdng THCS, Nxb Giao d'uc [2] Phan Diic Chinh, Ton Than, Nguyen Huy Doan Pham Gia Due, Truang Cong Thanh Nguyen Duy Thuan (2008), Todn - Tap vd 2, Nxb Giao due [3] Vu Huu Binh (2008>, Ndng cao vd phdt trien todn - tiip 2, Nxb GD [4] Nguyen Ba Kim, Vuang Duang Minh, Ton Than (1999), Khuyen khich mot sd hogt ddng tri tue cita HS qua mdn Todn d Trudng THCS, Nxb Giao due [5] Ton Than, Pham Gia Dire, Truong Cong Thanh, Nguyen Duy Thuan (2008), Bdi tap todn - Tap 2, Nxb Giao due [6] Vu Duang Thuy, Pham Gia Dire, Hoang Nggc Hung, Dang Dinh Lang (1998), Thuc hdnh giai Todn, Nxb Giao due SUMMARY PRACTISING SOME OPERATIONS WITH GENERAL INTELLECTUAL ACTIVITIES IN COMMON INTELLECTUAL MATHEMATICAL GRADE STUDENTS THROUGH EXERCISE PLANE GEOMETRY Bach Phuong Vinh College of Education - TNU Teaching award exercises the plane geometry in grade to fulfill one's duties is subject to the intellectual development of students will have a deeper meaning if the teacher is always an opportunity for students to perform general intellectual activities: analysis, synthesis, similar generalizations, especially of with intellectual activity common in mathematics: division cases, reverse the problem, considering the resolution is in the process of students finding the solution and thinking exercises exploit geometry Key words: intellectual activity, thinking, students, homework geometry, grade 138 ... giai duge thdng qua giai cac dang toan hinh hgc d ldp 9, da gdp phan khdng nhd vao viec phat trien tri tue va tu sang tao cho hgc sinh TAI LIEU THAM KHAO [I] Hoang Chung ( 199 7), PPDH Todn hoc... dogn thdng bdng 90 Sii'' dung ket qua bai toan 1.12 cho bai toan 1.13 Ap dung nhanh chdng PP giai bai toan 1.8 cho bai toan 1.14 la sir dung ket qua ciia bai 1.13 Bdi todn 1.14 Cho lirgidc ldi... da cho ket hgp vdi HD tdng hgp va khai quat hda tu cac ket qua cua vi du 1; bai toan 1.1; bai toan 1.2 ta cd ket qua khai quat hda la bai toan 1.3; 83(07): 133- 138 - Van dung tinh chat: Trong