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516.0076 C120N Th.S NGUYEN TAT THU (Nhom giao vien chuyen luyen thi Dai hoc) LUYEN TH DAI HOC HINH HOC SACH D1\ CHO HOC SINH LUitN THI DAI HOC, CAO DANG DVL.013508 HA XUAT BAN DAI m QUQJIGJA H A J O L JION 1^ Th.S NGUYEN TAT THU (Nhom giao vien chuyen luyen thi Dai hoc) Mm mum LUYEN THI DAI HOC liiNU nee (: SACHDANHCHOHOCSINHHJY$NTHIDAIHOC, CAODJNG NHA XUAT BAN DAI HQC QUOC GIA HA Npi L6I NOI D A U Cac em hpc sinh than men! Trong nhung nam gan day, de thi Dai hpc luon c6 cau thupc ve phan mon Hinh hpc so cap Cac bai toan thupc chu de hinh hpc rat da dang va sang tao loi giai da gay khong it kho khan cho cac thi sinh Hon nua, nhung chu de Hinh hpc so cap cac em hpc sinh dupe hpc ca ba khoi lop nen luong kien thuc ciia cac em hpc sinh ciing bj roi rac Voi niem dam me va nhieu nam kinh nghi^m giang day danh cho bp mon toan, thay Nguyen Tat Thu da danh thoi gian va tam huye't vie't tap sach "Cam nang luy^n thi Dai hpc Hinh hpc so cap" nham khoi day niem yeu thich toan hpc, ren luyfn ky nang tir lam bai, t u on tap Npi dung ciia bp sach dupe chia lam chuong ^ ^ ^ Chuong Phuong phdp toa mat phdng * Chuong Hinh hoc khong gian Chuong Phuong phdp toa khong gian Voi lo'i vie't khoa hpc, sinh dpng, bp sach giiip cac em tiep can mon toan mot each nh? nhang, t u nhien, khoi nguon cam t u hpc Kien thuc trpng tam ngan gpn, day dii bao gom ly thuyet, phuong phap giai cac dang bai d i t u co ban den mo rpng va chuyen sau, qua giiip cac em hieu ro ban chat, phan tich, lap luan nhuan nhuyen de chu dpng tim phuong phap giai quyet bai toan V i du minh hpa tung phan dupe phan loai, s^p xep chpn Ipc tir de den kho nham dan dat cac em den nhirng dang bai thi Dai hpc Loi giai vira chi tiet vua gpi mo de cac em tung buoc vira phan tich vua tim toi each giai chinh xac va thu vj nhat Loi binh va nhan xet cua tac gia sau moi bai giai la kinh nghi^m quy bau cho cac em Lam nhieu bai tap de nang cao nang luc t u duy, la mot each hpc toan h i f u qua nhat Cac em se thii hon dupe thu sue voi nhieu bai t^p khac va tinh huo'ng da dang c6 kem huong dan giai De su dung bp sach hi^u qua va mang lai ket qua cao nhat, cac em can kien tri tim hieu de nam chSc ly thuyet, cham chi ren luy^n ky nang lam bai thong qua cac v i du, bai tap trinh bay bp sach N^y ( ^ Hi vpng, tap sach: " Cam nang luyen thi Dai hoc Hinh hpc so cap" se tiep tyc la nguon tai li^u bo ich cho cac em hpc sinh ky thi D ^ i hpc - Cao dang toi Mac du tac gia da danh nhieu tam huye't cho cuon sach, song sy sai sot la dieu kho tranh khoi Chung toi rat mong nhan dupe su phan bi^n va gop y quy bau cua quy doc gia de nhung Ian tai ban sau cuon sach dupe hoan thi^n hon Nguyen Tat Thu Jldfi ikiiL f Theo cau tnic de thi Dai hpc - Cao dSng ciia Bp Giao d\ic thi de thi vao cac truong Dai hpc - Cao dSng c6 diem Hinh hpc dupe chia cau, moi cau diem nhu sau: , * Cau Gom cac van de ve hinh hpc khong gian tong hpp Noi dung ciia cau de thi thuong gom hai y: Tinh the ti'ch khoi da dien (khoi chop va khoi lang try) va y thii hai thuong xoay quanh cac va'n de +) Chung minh quan he vuong goc, quan h ^ song song; +) Tinh goc giiia hai duong thang cheo nhau; +) Tinh khoang each t u mot diem den mat phang; i • ' +) Tinh khoang each giiia hai duong thang cheo * Cau 7a, 7b: Gom cac van de ve phuong phap tpa dp mat phang Chii yeu ' xoay quanh cac van de sau +) Lap phuong trinh duong thang, duong tron, Elip, Hypebol; ^ +) Xac dinh tpa dp ciia mot diem Cau 8a, 8b: Gom cac van de ve phuong phap tpa dp khong gian, chii yeu xoay quanh cac chii de +) Lap phuong trinh duong thang, phuong trinh mat phang, phuong trinh mat cau; +) Chung minh v i t r i tuong doi giCra duong thang, mat phSng va mat cau; +) Xac djnh tpa dp ciia mpt diem thoa man tinh chat cho truoc * Sau day, chiing ta di phan tich de tim loi giai cac bai toan dupe trich cac de thi Dai hpc nam 2013 , ,^, K h o i A-2013 Cau Cho hinh chop S.ABC c6 day ABC la tam giac vuong tai A, ABC = 30" Tam giac SBC deu canh a va mat ben SBC vuong goc voi day Tinh theo a the tich ciia khoi chop S.ABC va khoang each t u C den (SAB) Phan tich: Yeu cau ciia bai toan c6 y: Tinh the tich khoi chop va tinh khoang each t u diem C den mat p h i n g (SAB) , j ^ +) Tinh the tich khoi chop S.ABC : De tinh the tich ciia khoi chop, truoc he't phai xac dinh dupe chan duong cao ciia hinh chop? Theo de bai, mat ph5ng (SBC) vuong goc voi day nen duong cao ciia tam giac SBC ke t u S chinh la duong cao ciia hinh chop Hon nij-a, tam giac SBC deu nen chan duong cao ciia hinh chop chinh la trung diem BC Nen the tich cua khoi chop dugc tinh nhu sau: Lai giai Cach 2: Ta co d(C,(SAB)) = 'ASAB Goi H la trung diem canh BC, suy SH BC =i> SH (ABC) va SH = Ta co: AB - BC cos B = suy S ^ B C c = -j_ ^^-^C , AC = BC sin B = - , 2 = • SM = V s r f + H M ^ = - De' tinh khoang each t u mot diem M den mat phang (a), ta c6 cac each sau = 2SM.AB, ^ S u v r a Sb^^g _ ^ aVl3 aVs buy 3a Vay d(C,(SAB)) = = V39 3N/T3 16 a^yf39 , Cau 7a Trong mat ph^ng voi he tpa Oxy, cho hinh chi> nhat ABCD co diem Cach 1: Dung hinh chie'u H ciia M len (a) va tinh M H Cach 2: Chuyen tinh khoang each t u M ve tinh khoang each t u diem khac bang each dua \o tinh chat C thuQC duong thang d:2x + y + = va A ( - ; ) Gpi M la diem doi xung cua B qua C, N la hinh chie'u vuong goc ciia B tren duong thang M D Tim toa cac diem B va C, bie't rang N(5;-4) Neu duong thSng M N ck (a) t^i I thi d(M,(a)) = ^ d ( N , ( a ) ) Diem I t a | Phan tich: Day la bai toan xac djnh thuong chon la chan duong cao ciia hinh chop toa cua mpt diem Ve mat dai so, de xac djnh tpa dp cua mpt diem ta can tim hai an, tuc la can^ thie't lap hai phuong trinh Trong hai diem can tim B va C thi diem C thupc duong thang d nen tpa dp ciia C CO dang C ( x ; - x - ) Do Cach 3: Sii dung cong thuc the tich: Xet hinh chop M.ABC, d(M,(ABC)) = ^ ^ ^ ^ M ^ Voi bai toan tren, viec dung hinh chie'u cua C len (SAB) tuang doi kho, nen ta chuyen ve tinh khoang each tu H den (SAB) Vi CH c3t (SAB) tai B va H la trung diem cua BC nen d(C,(SAB)) = 2d(H,(SAB)) Do do, ta c6 the tinh d(C,(SAB)) theo each nhu sau: Cach 1: Goi M la trung diem ciia cac doan AB va K la hinh chie'u ciia H len SM Taco: H M / / A C = ^ H M l A B , H M = -!-AC = Ma AB SH HK^ L^ SH^ ^ B c A M AB (SHM) => AB HK Trong tarn giac SHN ta c6: do, de tim tpa dp diem C ta chi can tim an nen can thie't lap m p t , phuong trinh Goi I la giao diem ciia AC va BC, ta co tpa dp diem J X - , - X + 3'[ W Mat khac HK S M , do H K (SAB) ' Ma VsABC=:[;^ Ta CO S M l A B i ^ >S.^AB Do do, the tich khoi chop S.ABC la: V = ^SH.S > R C = l l ^ ^ ^ ^ = ^ ^ ^^^^ 16 +) Tinh khoang each tu C den (SAB): 3V C(x;-2x - 5) Goi I la giao diem cua AC va BD, ta co I la trung diem AC nen I x - -2x + Trong tam giac vuong BND, ta c6 IN = IB = lA => IN^ = lA^ r - x + i i ^ r x + ] r2x + 13^ I ) + I J I J + I J B(3b + 17;b) ^2 Ma IB = IA = -AC=> r6b + 37^ T r b - i ^ 250 I , I J 40b^ + 440b +1120 = « b =-4,b =-7 +) Voi b - - ^ B ( ; - ) = N (loai) +) Voi b = - = > B ( - ; - ) Vay C(l;-7), B(-4;-7) Cau 7b Trong mat phang voi he toa dp Oxy, cho duong thing A : x - y = Duong tron (C) c6 ban kinh R = %/lO cat A tai hai diem A va B cho AB = 4V2 Tiep tuyen cua (C) tai A va B cat tai mpt diem thupc tia Oy Vie't phuong trinh duong tron (C) Phan tich: De vie't phuong trinh (C) ta can di xac dinh tpa dp tam I Gpi M la giao diem cua hai tiep tai A va B Khi do, gia thiet cua bai toan gom c6: lA = IB = VlO, AB = 4V2, M e tia Oy Do do, ta se di xac dinh tpa dp diem M(0;a), a > De xac dinh tpa dp diem M ta can thiet lap mpt phuong trinh, gpi H la giao diem ciia AB va IM, ta xet tam giac MAI vuong tai A va AH la duong cao, AH = — = 2V2 Tij- day, su dung cong thuc duong cao tam giac vuong ta tim dupe AM, suy MI va MH Mat khac MH chinh la khoang each tu M den AB (tuc la duong thang d) Tu day j ta tim dupe a va tpa dp diem M, do ta c6 phuong trinh MI, hon niia dp dai MI tinh dupe nen ta tim dupe tpa dp diem I Vay ta I CO loi giai nhu sau: Lai giai Gpi I la tam ciia duong tron can tim, H la trung diem ciia AB, M la giao diem cua hai tiep tuyen tai A, B Tu gia thiet ta c6 M(0;a),a > Trong tam giac vuong MAI, ta c6: 1 • AM = 2^/To —1- + 1 AM^ 10 40 AH^ Ar AM' => M I = V M A ^ + lA^ = 5V2 => M H = M I - HI = 4x/2 Hay d(M,A) = V o - a^ = 4^y2=>a-8(do a>0) hangVt^t Do do, phuong trinh IH la: x + y - = Taco I e M H = > l ( b ; - b ) , IM = 5V2 V2b^ = 5V2 ci> b = ±5 + Voi b = 5, phuong trinh ciia (C) la (x - 5)^ + (y - 3)^ = 10 + Voi b = -5, phuong trinh ciia (C) la (x + 5)^ + (y -13)^ = 10 Cau 8a Trong khong gian voi he tpa dp (Oxy), cho duong thang + l _ z +^2 va diem A (l; 7; 3) Viet phuong trinh mat phSng (P) A: x_2- _ y _2 di qua A va vuong goe voi A Tim tpa dp diem M thupc A cho AM = 2V30 Phan tich: Npi dung ciia de bai gom hai y: Viet phuong trinh mat phang va tim tpa dp diem M ; ,„ t +) De vie't phuong trinh mat phJing, ta can tim VTPT va mpt diem di qua: Trong de bai, diem di qua da eo va VTPT cua (P) chinh la VTCP cua duong thang A +) Tim tpa dp diem M: Vi M thupc A nen tpa dp cua M chi c6 an (an t) Do AM = 2V3O nen tu day ta tim dupe t, tu suy M Vay ta c6 loi giai sau: Loi giai Vi (P) A => n(-3;-2;l) la vec to phap tuyen ciia (P) do: Phuong trinh mat phang (P) la : -3(x -1) - 2(y - 7) + l(z - 3) = Hay x - y - z - = , ^ Ta CO M e A=>M(6-3t;-l-2t;-2 + t) Ta CO A M = 2N/30 A M ^ = 120 o ( - t f + ( - - t f + ( - + t f =120 H ( l + t ; t - ; l + t ) Do H € ( P ) The tich khoi chop S.ABCD la: V = I s H S o r n = ^ xac djnh dupe tpa dp diem D Vay ta eo loi giai nhu sau: Loi giai Gpi I la giao diem ciia hai duong cheo AC va BD, ta c6 A I vuong goc voi BD nen H thupc A I Phuong trinh H I : 2x - y + = Tpa dp diem I la nghiem ciia h^ < ^ ^~^o • ^~ [2x-y + = [y = => l f - - 4) ^ ' ' Ta CO Tam giac BIC vuong can tai I nen goc CBI = 45" Mat khac, ACBH vuong tai H nen BI phan giac ciia goc CBH IC = IB = I H I la trung diem CH, do Vi C € l H = * C ( c ; c + 8)=^IC = I H « ( c + 2)^+(2e + f = o c = - l , c = - Suy C ( - l ; ) Tuong t u B(6 - 2b; b) ^ IB = I H o (8 - 2b) + (4 - b) = o b = 5,b = +) Voi b ^- ^ B(0;3) Ta CO =^ = ^ ID = -316 , suy D ( - ; ) +) Voi b = 5=> B(-4;5), tuong tu ta c6 D(4;1) •-ty iiMtitiTvii Cau 7b Trong mat phMng vdi tpa dp Oxy, cho tarn giac ABC c6 chan duong cao tu dinh A la H 17 , chan duong phan giac cvia goc A la D (5; 3) va trung diem ciia canh AB la M (0; 1) Tim tpa dp dinh C Phan tich: Tir de bai ta c6 phuong trinh BC, suy phuong trinh ciia AH Do do, tpa dp ciia A chi c6 an, lay doi xung qua M ta c6 tpa dp diem B Dua vao A H vuong goc vdi BH ta tim dupe tpa dp diem A va B Lay doi xung M qua AD ta dupe dir'm N thuoc duong thang AC, tu ta viet dupe phuong trinh AC Dua vao C la giao diem ciia BC va AC ta tim dupe C Vay ta c6 loi giai nhu sau: Loi giai Ta c6, phuong trinh BC: 2x - y - = 0; phuong trinh A H : x + 2y - = V uvvH Khang Vi^t Ta cd H(3 + 2t;5 + 3t;-t)e(P)=^2(3 + 2t) + 3(5 + 3t) + t - = ^ t = - l = > H ( l ; ; l ) Vi H la trung diem A A' nen A ' ( - l ; - l ; ) Cau 8b Trong khong gian vdi h? tpa dp Oxyz, cho cac diem A(l; - ; l), B (-1; 2; 3) va duong thSng A: ^ _ y ^ _ z - -2 Viet phuong trinh duong thing di qua A, vuong gdc vdi hai duong thang qua AB va A Phan tich: De viet phuong trinh duong thang ta can tim mpt diem di qua va mot VTCP De tim VTCP, ta thudng tim hai vecto khong cimg phuong va cung vuong gdc vdi dudng thang dd Khi dd , tich cd hudng ciia hai vecto dd la VTCP ciia dudng thang Trong de bai, dudng thang d can viet phuong trinh vuong gdc vdi AB va A nen AB A U^^ la VTCP ciia d Vay ta cd loi giai nhu sau: Loi giai Gpi d la dudng thang can lap phuong trinh Ta cd AB = (-2; 3; 2) va u = (-2;1;3) la VTCP ciia dudng thing A Vi A € A H ^ A ( - a ; a ) ^ B ( a - ; - a ) Vi d vuong gdc vdi A va AB nen a = AB A u = (6; 2; 4) la VTCP ciia d Vi AH.HB = nentaco =^a = ^ A ( - ; ) , B ( ; - ) Phuong trinh AD : y = => N (0; 5) la diem doi xung ciia M qua AD eAC => Phuong trinh AC : 2x - 3y + 15 = va phuong trinh BC : 2x - y - = Vay C (9; 11) Cau 8a Trong khong gian vdi h^ tpa dp Oxyz, cho diem A (3; 5; 0) va mat phSng (P) : 2x + 3y - z - = Viet phuong trinh duong thing di qua A vuong goc vdi (P) Tim tpa dp diem doi xung ciia A qua (P) Phan tich: Duong thang d can viet phuong trinh vuong goc vdi (P) nen nhan VTPT ciia (P) lam VTCP, tu ta viet dupe phuong trinh d Gpi A' doi xung vdi A qua (P) va H la giao diem ciia d va (P), ta cd H la trung diem AA' Do do, xac dinh dupe H ta se cd diem A' Vay ta cd loi giai nhu sau: Vay phuong trinh d : -• j - l _ y + _ z - * KhoiD-2013 Cau Cho hinh chdp S.ABCD cd day ABCD la hinh thoi canh a, canh ben SA vuong gdc vdi day, BAD = 120" M la trung diem canh BC va SMA = 45*^ Tinh theo a the tich khdi chdp S.ABCD va khoang each tu Dden (SBC) Phan tich: De bai cho SA la dudng cao ciia hinh chdp nen ta di tinh SA dua vao tarn giac SAM vuong tai A Vi AD//(SCB) nen d(D,(SBC)) = d(A,(SBC)) Vaytacdldi giai nhu sau: Loi giai Loi giai Vi B'AD = 120" => ABC = 60" Loi giai Ta cd n = (2;3;-l) la VTPT ciia (P) Suy duong thSng d di qua A => AABC deu, suy AM BC va AM = va vuong gdc vdi (P) nhan n lam VTCP x = + 2t Phuong trinh d : y = + 3t, t e R z = -t Tarn giac SAM vuong tai M va SMA = 45° Gpi H la giao diem ciia d vdi (P) va A' la diem doi xung vdi A qua (P) • SA = AM = aVi • , D Ma S A B C D = S ^ B C = - Cant nang luyen thi DH Hinh Hoc - Nguyen latThti 1 a V s a^S The tich k h o i chop S.ABCD la: V = - S A S A B C D " • ~ ~ ~ V i AD//BC ^ AD//(SBC) Ta c6: IC^ = l A ^ _3a^ ~8~' ri Do BC S A • BC1(SAM)=>BC1 AH T u d o suy A H (SBC) => d(A,(SBC)) = A H Ma 1 AH^ AM^ _8_ - +• SA^ AM^ +5=0 c = l,c = Voi a = -5=> A ( - ; - ) , B ( - ; ) ^ B H = ( ; - l ) Phuongtrinh A C : x - y + = 0=>C(c;2c + 8) ' " T a c o : I C ' = I B =>(c + l ) c =-l,c +(2c + 7) = 25 « c^ + 6c + = ' ' =-5 Suy C ( - l ; ) >AH = aV6 Vay C ( ; l ) hoac C ( - l ; ) 3a' j,,, Cau 7b T r o n g m a t phang v o i he tpa dp Oxy, cho d u o n g t r o n (C): (x -1)^ + (y - 1)'^ = : r < va d u o n g thang A : y - = T a m giac M N P c6 true tam t r u n g v o i tam cua (C), cac d i n h N va P thupc A , d i n h M va t r u n g d i e m Vay d(D,(SBC)) = ciia canh M N thupc (C) T i m tpa dp d i e m P C a u 7a T r o n g m a t p h n g O x y cho tam giac A B C c6 M '2'2 la t r u n g d i e m Phan t i c h : Ta thay (C) «- va A tiep xuc v o i tai T, ma tam I la true tam nen M la giao cua T I v o i (C) A B , d i e m H ( - ; ) , I ( - ; ) Ian l u o t la chan d u o n g cao ve t u B va t a m ducmg Goi J la t r u n g d i e m M N , suy IJ la d u o n g t r u n g b i n h nen IJ song song v o i t r o n ngoai tiep t a m giac ABC T i m toa dp d i n h C A va J thupc (C) nen ta t i m d u p e tpa dp d i e m J, lay d o i x u n g ta c6 d i e m N Phan tich: V i M va I la t r u n g d i e m canh A B va t a m d u o n g t r o n ngoai tiep V i P thupc A nen tpa d p cau P chi c6 an, dua vao N I v u o n g goc v o i M P ta t a m giac A B C nen I M v u o n g goc v o i AB, t u day ta c6 p h u o n g t r i n h cua A B t i m dupe P Vay ta c6 l o i giai n h u sau: Ta goi toa d o cua A (chi c6 an) lay d o i x u n g qua M ta c6 toa d o ciia B L a i giai D u o n g tron (C) c6 tam 1(1; 1), R = D u a vao A H va B H v u o n g goc ta t i m d u p e tpa d p ciia A va B Ta C O d ( l , A ) = K Co A , H nen ta c6 p h u o n g t r i n h A C D u a vao l A = IC ta t i m d u p e tpa dp d i e m C 2'2 nen suy A tiep xuc (C) tai T Do L o i giai Ta c6 I M = la true tam tam giac P M N nen M I v u o n g goc V i I M A B nen p h u o n g t r i n h A B : A A, suy x^^ - x, =^1 7x - y + 33 = M a M thupc (C) nen M ( l ; -1) Suy A ( a ; 7a + 33) H A = (a + 2; 7a + 29) Gpi J la t r u n g d i e m M N suy B ( - a - 9;-7a - 30) ^ B H = (a + 7;7a + 34) IJ la d u o n g t r u n g b i n h cua tam Do BH A H o giac M T N , suy y j = y , = BH.HA = M a J thupc (C) nen J(3; 1) hay J(-l; 1) (a + 2)(a + 7) + (7a + 29)(7a + 34) = a^ + 9a + 20 = P(3; 3) C a u 8a T r o n g k h o n g gian v o i h^ tpa dp Oxyz, cho cac d i e m A ( - l ; -1; -2), B(0; 1; 1) va mat ph^ng ( P ) : x + y + z - l = T i m tpa d p h i n h chieu Ldm ridHg lU^fH THI UH HMH Hl?e - Nguyen i nur vuong goc cvia A tren (P) Viet phuong trinh mat phang di qua A, B va vuong goc voi (P) Phan tich: De lap phuong trinh mat phang ta can tim mot diem di qua va VTPT De tim VTPT ta thuang tim hai vecto khong cimg phuong c6 gia song song hoac nam mat phing Voi bai toan tren, mat phang can viet di qua A,B va vuong goc voi (P) nen AB A np la VTPT Vay ta c6 loi giai sau: Loi giai Gpi (a) la mat phang can lap Ta c6 n = (l;l;l) la VTPT cua (P) Vi (a) di qua A , B va vuong goc voi (P) nen n ' = AB A n = (-l;2;-l) la VTPT ciia (a) Phuong trinh (a) la: x - 2y + z - = Cau 8b Trong khong gian voi he tQa Oxyz, cho diem A(-l; 3; -2) va mat phang (P): X - 2y - 2z + = Tinh khoang each tu A den (P) Viet phuong trinh mat ph^ng di qua A va song song voi (P) , - l - + + 5| Khoang each tu A den mat phang (P): d( A,(P)) = — , =— \/l + + Goi (Q) la mat phang can tim (Q) di qua A va c6 mpt vecto phap tuyen la n = (l;-2;-2) = > ( Q ) : X - 2y - 2z +3 = De thi thu truang THPT Chuyen Luong The Vinh nam 2014 * KhoiA Cau 5, Cho hinh chop S A B C D c6 day A B C D la hinh thoi canh a va = 60^' Hinh chieu ciia S len mat phing ( A B C D ) la trpng tam tam giac A B C Goc giiia mat phSng ( A B C D ) va ( S A B ) bSng 60° Tinh the tich khoi chop S A B C D va khoang each giua hai duong thang SC va A B BAD Loi giai la tam tam giac A B C , suy SH ( A B C D ) Ke MH vuong goc voi AB, M thuoc AB Ta CO SMH la goc giua hai mat phSng ( S A B ) va ( A B C D ) , do SMH = 60" CQ'I H HB AT.\s ax/s f suy SH = MH tan 60° = | Mat khac tam giac ABD deu canh a nen S^BCD = ^ S ^ B D = - j — = - y - The tich khoi chop S.ABCD la V = -3S H S^^^^ ^ e D3 = 2- - - 2- ^ = Ta CO AB//(SCD) r:>d(AB,SC) = d(AB,(SCD)) = d(B,(SCD)) ^ = |d(H,(SCD)) ^ Gpi N, K theo thu hx la hinh chieu cua H len CD va SN, d(H,(SCD)) = HK ViHN=2d(B,CD)=2£V3^aV3 ' ' ^ Vay d(AB,SC) = ^ ^ HK= ^"'^^ VSH^+HN^ • 11 Cau 7a Trong mat ph^ng Oxy cho hinh vuong ABCD c6 A(1;1), AB = Goi M la trung diem canh BC, K - ; — la hinh chieu vuong goc ciia D len AM Tim toa cac dinh lai ciia hinh vuong, biet Xg < Loi giai Gpi N la giao diem ciia DK va AB Khi ADAN = AABM AN = BM => N la trung diem canh AB -— r 8^ Taco AK= , phuong trinh V^ ^J AM:2x + y - = 0, D K : x - y - = Vi N G DK ^ N(2n + 3;n) =:> AN = (2n + 2;n - ) Cty TNHH MTV DWHKhang Phuong trinh mat cau (S): (x -1)^ + (y -1)^ + (z -1)^ = « n = -l,n = - l +) Voi n = - i = ^ X B = X N =^>2 I-) Voi t = 5=^l(-5;7;13), R = 12 (loai) Phuong trinh mat cau (S): (x + 5)^ + (y - jf +) Voi n = - l = > X B = l < , y B = - ^ B ( l ; - ) d : ^ =^ Phuongtrinh C D : X = = > D ( ; ) Cau 7b Trong mat phSng Oxycho tarn giac ABCco true tarn H(-6;7), tarn = | , ^ • ^ = Z^ = ^^ duong thSng BC Tim tpa dinh A Lai giii Duong thing d di qua B(-1;-1;0) va c6 u = (-2;l;2) la VTCP BI = (t + 3; -t +1; 2t + 3) => BIA u = (-4t -1; -6t -12; -t + 5) Goi M la trung diem canh B C , ta c6 I M = d(l,BC) = N/S Kc duong kinh BB', dcS AHB'C la hinh binh hanh Dodo d(l,d) = non A H = B ' C - M - N / A ( ~ 2a;a) ^ A H = {2a -14;7 - a) BIAU ^(4t + l)2+(6t + f + ( t - f V53t2+142t + 170 Theo de bai, ta c6 d{l,d) = lA 53t^ + 142t +170 = 54t^ + 54t + 81 Suy (2a -14^ + ( a - ^ = ^ ( a - ^ = => a = 9,a = o t ^ - 8 t - = o t = - l , t = 89 Vay A (2; 5) hoac A (-10; 9) +) Voi t = - l : ^ l ( l ; l ; l ) , R = IA = ^ y-2 Cau 8a Trong khong gian Oxyz cho duong thang ^ ' — ^ z- = mat phang (u): x + 2y + 2z + = 0, (p): 2x - y - 2z + = Viet phuong trinh mat cau (S) ccS tarn nam tren duong thing d va (S) tiep xuc voi hai mat phang (a) va ((i) Lai giai Goi I la tam cua mat cau (S), l e d nen l(-t;2 + t;3 + 2t) Vi (S) tiep xiic voi hai mat phing (a) va (p) non d(I,(a)) = d(I,(P)) Phuong trinh mat cau (S): (x -1)^ + (y -1)^ + (z -1)^ = +) Voi t = 89=>l(91;-89;18l), R = IA = 748069 Phuong trinh mat cau (5): (x - 9l)^ + (y + 89f + (z - 18l)^ = 48069 * Kh6iD Cau Cho hinh lang try dung A B C A ' B ' C c6 tam giac ABC vuong t^i A, AB ^- a, BC = 2a va A A ' = 2a G(?i M la trung diem ciia canh BB' Tinh the tich khoi chop BMCA' va c6 sin cua goc giua hai duong thing A'M va BC Lai 5t + n = 7t + l o t = 5,t = - l +) Voi t = - l ^ I ( l ; l ; l ) , R = 2, ' Suy AI = (t;-t-3;2t)=> lA = V6t^+6t+ suy phuong trinh B C : x - y + = Phuong trinh D H : x + 2y - = 7t + l va diem A(2;3;3) Viet phuong Gpi I la tam cua mat cau (S), ta c6 l(2 + t;-t;3 + 2t) Loi giai 5t + l l thing trinh mat cau (S) di qua A , c6 tam nam tren duong thing A va tiep xiic voi duong thing d duong tron ngoai tiep l ( l ; l ) va D(0;4) la hinh chieu vuong goc ciia A len Vi A G D H + (z -13)^ = 144 Cau 8b Trong khong gian Oxyz cho hai duong thing duong Phuang trinh BC: y = -3 => C (5; -3) Ta CO HD = (b;-3), Vi^t giii Ve duong cao A ' H ciia tam giac A ' B ' C Ta CO A'H (BMC) va A'H = ^ ' B ' A ' C B'C :—» Cam nang luifftt thi DH Hinh hpc - Nguyen Tat Thu ua tryc duong tron GO' voi AB la khoang each giua GO' va mat phSng (AA'BB') Cho M = O thi G A ^ + G B ^ + G C ^ + G D ^ = 4(R^ - O G ^ ) 4(R2 Vithe'4 = - -OG^) _ GA^+GB^+GC^+GD^ R^-OG^ R^-OG^ GB^ GA" G A G A ' ^ GB.GB' Hay Gpi H la trung diem ciia AB, thi O H la khoang each giua OO' va m|it phSng (AA'BB').Tac6 OH = G C^ GC^ GD^ GC.GC' GD.GD' •>44 GA GB GC GA'GB'GC'GD' pai 2.7.2 Cho hinh non dinh S, duong cao SO Gpi A, B la hai diem thupc G A ' GB' GC' GD' duong tron day cua hinh non cho khoang each t u O den AB bang a va GA GD SAO = 30°, SAB = 60° Tinh di^n tich xung quarih va the tich a i a hinh non GB GC Mat khac, G la trQng tarn cua tw dif n nen V,G C ' D ' A ' V G D ' A ' B ^ , ^ G A - B ' C ' ^ A ' B ' C ' D ' _ "^GB'C'D' ^ABCD _ ^ GD VA B C D VA B C D VA B C D ^GB'CD' V,G C ' D ' A ' ^GD'A-B' ^ VGA'B'C YGBCD 4V,G C D A YGDAB 4V,G A B C VA B C D fGB'.GC'.GD' GC'.GD'.GA' GD'.GA'.GB' ~4 GB.GC.GD ^ GC.GD.GA ^ GD.GA.GB GA' GB' G C GD' ^1 GA GB GC GD Huong d i n giai Ggi I la trung diem doan thSng AB thi O i l A B , S I l A B , O I = a Tae6AO = SA.eos30° GA'.GB'.GC' GA.GB.GC , (2) Tir (1) va (2) suy dieu phai chung minh Dau d3ng thuc xay G = O, hay ABCD la t i i di^n gan deu Bai 2.7.1 Mpt khoi tru c6 ban kinh R va chieu cao 1) Ti'nh di^n tich ciia thiet di qua AB va song song voi true cua khoi try 2) Tinh goc giua hai ban kinh day di qua A, B 3) Cho hai diem A, B Ian luQt nkm tren hai day tron cho goc giija AB va tryc hinh try bSng 30° Tinh khoang each giua AB va tryc cua hinh try Huong dan giai 1) T u A, B ta ke AA',BB' song song voi tryc hinh try Khi thiet di$n la hinh chu nhat A A ' B B ' , ta CO ABB' = 30° la goc giiia AB va tryc hinh try Di^n tich thiet d i f n 'AA'BB = BB' B" A = V S R V S R tan 30° = N/SR^ 2) Goc giiia A O va BO' la AOB' = 60°, tarn giac AOB' deu 3) Ta CO mat phang ( A A ' B B ' ) chua AB va song song v o i true duong tron Do khoang each =^.SA; A I = SAcos60°=isA Tu suy AI AO AI Ta lai c6 —— = cosIAO AO •sinIAO = * Bai tap van dyng / OA AO = Xet tarn giac S A O ,ta c6 S A = AO cos 30° = ^a D i f n tich xung quanh ciia hinh non la: S^^ = Trrl = yjsna^ The tich khoi non la: V = ^TiR^h = I T I O A ^ S O - — na^ 3 ^ai 2.7.3 Cho hai mat phSng (P) va (Q) vuong goc voi c6 giao tuyen la duong thMng A Tren A lay hai diem A, B voi AB = a Trong mat phSng (P) lay diem C, mat phJing (Q) lay diem D cho AC, BD cung vuong goc voi A Va AC = BD = AB Tinh ban kinh mat eau ngoai tiep t i i di|n ABCD va tinh •dioang each t u A den mat phing (BCD) theo a (De thi tuyen sinh dai hpc ^^^61D nam 2003) Huong dan giiii ^ > (P) (Q) va C A A nen C A (Q) => C A A D Tuong t v B D _L BC, nen cac diem B, A cung nhin doan C D duoi mpt goc vuong, do m|it cau ngoai tiep t u di$n A B C D c6 tam la trung diem C D va c6 ban kinh a2 The tich khoi lang tru ABC.A'B'C laV = AA'.S^gc _= ^3 a^ 7— ' = 373 MG M H Goi H la trpng tam cua tam giac ABC, ta c6 nen G H / / A A ' ^ ^ MA' MA =:> G H (ABC) Do H cung la tam duong tron ngoai tiep tam giac ABC nen GH la true duong tron ngooai tiep tam giac ABC G Q I I la giao die'm CD R=— cua GH voi trung true cua GA (qua trung diem N cua GA) thi la tam Ap dung dinh l i Pitago mat cau ngoai tiep t u di?n GABC cho cac tam giac A B D , A C D ta c6 R= VAC^ + AD2 =^UC' + AB2 + BD2 Ta CO hai tam giac IGN va A G H dong dang, nen ^'^ AG = :^a Ke A H B C thi H la trung diem cua BC (do tam giac ABC vuong can A) GA.GN ^ GA^ GH 2GH AA' Vi GH = ^ = - nen GA^ ^ GH^ + HA^ = -^a^ Vay R = — a 12 '-^ 12 Ta CO A H = I B C = I N / A C ^ T A B ^ = :|^a Vi BD (ABC) =^ BD A H nen A H ± (CBD) Vay d(A,(BCD)) = A H = Suy ban kinh mat cau R = GH Bai 2.7.5 Cho hinh chop S ABCD c6 day ABCD la hinh thoi canh a, BAD = 60° ^a Bai 2.7.4 Cho hinh lang try tam giac deu ABC.A'B'C c6 AB = a, goc giua hai va cac canh ben SA = SB = SD Xac dinh tam va tinh ban kinh m^t cau ngoai tiep hinh tu di?n SBCD biet BSD = 90° mat phSng (A'BC) va (ABC) bang 60° Gpi G la trpng tam tam giac A'BC Huong dan giai Tinh the tich khoi lang tru da cho va tinh ban kinh mat cau ngoai tiep tu Gpi O la giao diem hai duong cheo cua hinh thoi ABCD di?n GABC theo a (De thi tuyen sinh dai hpc khoi B nam 2010) Theo bai ta c6 tam giac ABD la tam giac deu c^nh a ^ BD = a Huang dan giai Goi M la trung diem cua BC v nen SB = SD = : ^ a , S O = ^ 2 Do tam giac ABC deu nen BC A M A ' M BC (dinh l i ba duong vuong goc) Vay A ' M A = 60° Taco A M = — a A' Ma tam giac SBD vuong tai S nen A ' A = A M t a n A ' M A = - a Gpi H la hinh chieu cua S tren mat phang day thi H la tam duong tron ngoai tiep tam giac ABD (do cac canh ben SA = SB - SC ) Ta CO SH = V S O ^ - O H ^ = -a SC = s H + H C = : ^ a Gpi K la tam cua tam giac deu BCD thi K la h-ung diem cua HC, tryc duong tron ngoai tiep tam giac BCD di qua K va song song voi SH nen la trung trirc cua HC cat SC tai diem I Ta c6 I la trung diem cua SC nen IS = IC, do I chinh tam m3t cau ngo^i tiep hinh t u di§n SBCD Ban kinh GQ\ la trung diem ciia BC, ta c6 A I ± BC Mat khac (SBC) (ABC) nen A I ± (SBC) => A I S I mat cau la R = ^SC = Bai 2.7.6 Cho t u di^n deu ABCD c6 tam mat cau ngo^i tiep t u di^n la O H la hinh chieu cua A len mat phSng (BCD) to«w;jU.ii, O A 1) Tinh t y 1? k = OH 2) Gia su m|t cau ngoai tiep t u di?n c6 ban kfnh bling Ti'nh dp dai cac c^nh' i SI la duong cao ciia tam giac SBC Ta CO AABI = AASI ^ IS = IB = IC I la tam duong tron ngoai tiep ASBC Gpi O la tam hinh cau ngoai tiep hinh chop S.ABC ta c6 OS = OB = OC nen O thupc thSng d qua I vuong goc voi (SBC) Ta CO (SBC) (ABC) H u a n g dan giai => d c (ABC) => O G (ABC) => O la tam duong tron ngoai tiep AABC 1) D o t u di?n ABCD deu nen cac khoi ^ Gpi K la giao ciia A I voi duong tron ngoai tiep tam giac ABC chop O.ABQ O.ACD, O.ABD, O.BCD CO the tich bSng Ta CO : AB^ = AI.AK = AI.2R => R = ^A.BCD = "^Vo.BCD /IO; AB = 6;Oi02 = >/2T Huang dan giai Gpi dj,d2 Ian lupt la hai duong thang di qua O ^ O j va vuong goc voi (P) TaCO R = O A = l = > O H = i va (F) Gpi M la trung diem AB ta c6 ( M O j O j ) AB => d ^ d j c= (MO1O2) • Tam giac B O H vuong tai H nen B H = V o B ^ - O H ^ = V R ^ - O H ^ = ^2N^ Gpi O la giao diem cua d j va d j Ta c6 O la tam mat cau chua (Oj) va (02).Bankinh R = OA Ta CO B H la ban kinh duong tron ngoai tiep tam giac deu BCD nen Taco M O j = BH = : / ^ = > B C = ViBH = Ta CO t u giac MO1OO2 la ti^giac npi tiep ^ Vgiy canh ciia t u di?n deu la BC - i4e Bai 2.7.7 Cho hinh chop S.ABC c6 (SBC) (ABC) va cac canh AB = AC ^ S SB = a Xac dinh tam va ban kinh hinh cau ngoai tiep hinh chop SC - cosM = - M A ^ = 4;M02 = ^jrl -MA^ = 2M01.M02 => j M = 120*^ O1OO2 = 60° 5at X - O O , y = C)Oi,z = O M A p dyng djnh ly cosin cho A j 0 Ta c6: , ^ = O O j + OOl ^ - ZOOj.CX^j + y^ - xy = 21 , Ch^ng minh rS„g (1).- Do t u giac M O j O O j npi tiep nen MOj-OOj + M O j O O , = MO.O1O2 o 4x + y = V2T.Z (2) OM^ = MO? + OjO^ = MO^ + O2O2 (3) = 16 + y2 = + x^ Giai h? gom ba phuong trinh (1),(2) va (3) ta tim du-gfc: x = 3V3;y = 2N/3;z = 2V7 =^ OA^ = AO? + OjO^ = 37 => R = OA = N/37 Bai 2.7.9 Cho hinh chop S.ABCD c6 day ABCD la hinh thang can va AB//CD Duong tron tarn O npi tiep hinh thang c6 ban kinh r Bie't SO (ABCD) va SO = 2r Xac djnh tam va tinh ban kinh mat cau npi tiep hinh chop S.ABCD Huong dan giai Gpi M , N , P , Q la cac tiep diem cua Do S O I (ABCD) nen cac tam giac la gia t r j 3V Idnong doi, ma r = - — nen r Ion nha't S^^ dat gia trj nho nha't D l t AB = 2a, A M = x, B N = y Tu A M + B N = M N suy xy = 2a^ Di^n t'ch toan phan ciia t u di^n la S, = a(x + y) + i yVx^ +4a^ + x-^y^ +4a^ ^ 2V J Six dyng bat dang thuc Cauchy, ta tim dupe gia trj nho rdiat cua S,p dat ^A^ + 4a^ nen minR = l-Jla x = y a\/2, hay dat dupe tuong ung voi ban kinh mat cau npi tiep dat gia trj Ion nha't mpi diem tren SO each deu cac mgt ben ciia hinh chop Tam mat cau npi tiep la giao ciia phan giac goc SNO voi SO ^ TS Theo tmh chat phan giac: — = ^ ^ IS NS Suy ban kinh mat cau npi tiep hinh chop la R = IO = Ta tinh dupe V = R = ^x^ + y2 + 4a^ > ^Ixy + Aa^ = Vsa^ = Isfla SOM,SON, SOP SOQ bang va SN = VSO^ + ON^ = AB"' 2) Ban kinh mat cau ngoai tiep t u d i f n la R = •y/x^ + cac canh ciia hinh thang CO Hirong dan giai 1) Gpi V la the tich t u di^n A B M N dupe X = y = a-Jl, hay A M = B N = duong tron npi tiep hinh thang voi Ta ban k,„h „ , „ ca„ „go,i Hep ^ d i , „ ABMNir^Tho OS.ON 2r S-\ NO + NS i + Vs -r Bai 2.7.10 Cho hai nua duong th5ng Ax, By cheo va vuong goc vO' nhau, nh^n AB lam duong vuong goc chung Tren tia Ax lay diem M , tren tia By lay diem N cho A M + BN = M N 1) Tim vj tri ciia M , N cho ban kinh mat cau npi tiep t u di?n A B M N la lo''^ nha't I Bii 2.7.11 Cho SABC la hinh chop tam giac deu voi c^nh day AB = a, duong cao SH = h 1- Tinh theo a va h cac ban kinh r, R ciia hinh cau npi tiep va ngoai tiep hinh chop 2- Gia su a eo djnh va h thay doi xac dinh h de — Ion nha't R Huong dan giAi TinhR: Tam hinh cau ngoai tiep t u d i f n S A B C nam tren SH (trye ciia duong ngoai tiep tam giac ABC) va tren duong trung trye eua S A ve mp(SAH) Tu gi^c AHOJ npi tiep la trung diem eiia SA), Cavf iian/l2h^ Ta 6ah' 6a R X (a + x) \ -1a , t g a -1 b SABCD = -.a cos^a V a tga + b 1cos^a f A I SB AI1SC ^"^^ ^ ^ ^ ^ ^ ^ ^ : AI l i e ' ' max = - neu va chi ne'u x = 3a o h Zcosa ^' L H ciing nhin AS duoi goc vuong nen hinh cau ngoai tiep khoi da d i f n '^SIJH la hinh cau duong kinh AS Tam la diem giiia ciia AS, ban kinh bSng ^^a _ 2 ' ' ' (a + 3af +3a Bang bien thien ta tim dua x>a 12 r ^- b ^ Nhu v^y : I , J va B cung nhin AC duoi goc vuong nen hinh cau ngoai tiep da di?n ABCIJ la hinh cau duong kinh AC Tam la diem giiia ca AC, ban kinh (a + Vl2h^TI^).(3h2 ^ a2 ^ Dat: x = N / l h + a ^ o - -b^ Tam va ban kinh hinh cau ngoai tiep cac da di^n ABCIJ va ASIJH 2V3 ^ Dinhhder/Rlonnhat R ^ CD^ - • -b2r^CD = , I cos^a cos a Dodo: SACD-TT-N + h AD = b x/l2h^+a^ Sl2=h2 Ta CO- AB = a =i BC = a.tga Do do:SABC = ^ a ^ t a n a Tinh chat cua phan giac cho: OH HI , h.HI r.SI = HI.(h -i)=>i = SI + H I O'S SI HI = Vift pai 2.7.^2 Trong mat phJing (P) cho AABC vuong tai B, c6 canh AB, ABC = a va ^ D C vuong tai D CO A D = b (B va D khong ciing phia doi voi AB) Tren duong thing vuong goc voi mat phang (P) tai A ta lay diem S cho SA = x ^ Tinh the tich hinh chop SABCD Al, AJ, A H la duong cao xuat phat h i A cua tam giac SAB, SAC, SAD Hay tim tam va ban kinh cua hinh cau ngoai tiep cac da d i f n ABCIJ, ASIJH Xir xac djnh giao cua hinh cau noi tren H u o n g dan giai '* ' - d v a d ' d n g p h S n g K h i d o x a y rabatrudnghpp C2: Lap phuong trinh mp(P) di qua M vuong goe voi A Tim giao diem H i) d va d ' c i t o [u, u'] ^ va tpa dp gia diem la nghi^m cua h | : cua (P) voi A Kiii dp dai M H la khoang each can tim \f) Khoang each g i u hai duong thang cheo nhau: a b b' c Cho hai duong thang cheo A di qua Mg eo VTCP u va A' d i qua MQ ' c' CO VTCP u ' Khi khoang each giiia hai duong thang A va A' dupe tinh Iheo cae each sau: u,u' [u,u''l = d i i ) d//d'c:> C l : Sir dyng eong thuc: d(A, A') = [u,MM>6 u/U' Iu,u'] = iii) d s d ' o * C2: Tim doan vuong goe chung M N Khi dp dai M N la khoang each can tim [U,MM'']=6 C3: Lap phuong trinh mp{P) di qua A va song song voi A ' Khi khoang Neu [u,u']MM'?i => d va d ' cheo V i tri tuong doi giua duong thing va mat phing Cho mp(a): Ax + By + Cz + D = c6 n = (A;B;C) la VTPT va duang each can tim la khoang each t u mpt diem bat ki tren A ' den (P) th.ing IV GOC Goe giua hai duong thang: ^ ^ ^ ^ ZzytL^iZlo u=(a;b;c) la VTCP va d i qua M(,(xo;yo;z„) a b c • A c3t (a) n va u khong cimg phuong Aa + Bb + Cc 5t Khi tpi Ax + By + Cz + D = a duong t h i n g A ' : ^ ^ a' (a) giao diem la nghi^m cua h^ : j x - X(, _ y - y(, _ z - z^ « = (A,A'), Tu (b) X - X , at, y yo ^ bt, z = ct the vao (a) => t fAa + Bb + Cc = n 1u A//(a) o M „ ^ ( a ) ^ l A x o + Byo+Czo + D ^ O ^ :lZ]!^ ^IZJo ^^Z3 c6 VTCP u=(a;b;e) va Cho hai duong t h i n g giao diem =^ bb ' b e ^ =^ ^ do: eosa = cos|u,u'j Ac(a)o n 1u M„^.(a)'^lAxo+Byo+Czo +D = A (a) cv n va u cung phuong o n = k.u aa'+bb'+cc' 2- G 6c giiia duong t h i n g va mat p h i n g ! Cho mp ( a ) : Ax + By + Cz + D = c6 n - ( A ; B ; C ) la VTPT va duong t h i n g I : i Z o = i Z f o C O u-(a;b;c) a b c '^P(a) va duong thang A , ta c6: A:^^lil = Aa + Bb + Cc = ec CO VTCP u'' = (a';b';e') Dat e' sin(p = cos {n,u) la VTCP Gpi (p la goe giua Aa + Bb + Ce VA^ + B^+C^Va^ + b^+e^ ' 371 Goc giua hai mat phing Cho hai mat p h i n g (a): Ax + By+ Cz + D = c6 VTPT n^ = ( A ; B ; C ) va (P):A'x + B ' y + C'z + D ' = c6 VTPT = ' I A A ' + B B ' + C C ' § T f C H C H I / O N G C U A H A I V^C T O V A ONG l3 - - 1-2 2, DUNG • Di^n tich tain giac: Dien tich tarn giac ABC duoc tinh boi cong thiic Vi SABCD = T ^ H C D 3) T a c : V A B C D =7 nen suy B H = ^^ABCD ^ 2>/62 CD "ViJ 27651 35 BA B C B D Gpi h = d ( A , ( B C D ) ) , t h e t h i h = ^^^^^BCD AB,AC 21 ^_35 2762 • ='ABCD x = + 5t • The tich: +) H i n h h p p : The tich hinh hpp ABCD.A'B'C'D' dirpc tinh boi cong thuc 'ABCD.A'B'C'D' AB,AD AA +) T u di?n: The tich t u di#n ABCD dirac tinh boi cong thuc 4) Ta CO phuong trinh A B : y = + t , s u y r a M ( + 5t;2 + t ; l - 2t) z-l-2t Do do: C M = (2 + 5t;4 + 2t; - t ) II AB,AC AD 6lL • Dieu ki^n vecto dong phang: V ABCD +) Ba vec to a, b, c dong phang va chi a,b c = +) Bon diem A , B , C , D dong phang va chi AB, AC AD = 3.1.1 Trong khong gian Oxyz cho bon diem A(4;2;l), B(-l;0;3), C(2;-2;0), D ( - ; ; l ) 1) Chung minh rang A, B, C, D khong dong ph4ng; 2) Tinh dien tich tam giac BCD va duang cao BH ciia tarn giac BCD; 3) Tinh the tich t u di?n ABCD va duong cao ciia t u di^n t u A; 4) Tim tpa dp diem M nam tren duong thSng AB cho tam giac MCD di^n tich nho nha't CHCD] 1) Taco BA = (-5;-2;2), BC - (3;-2;-3), BD = (-2;2;-2),CD = (-5;4;1) [ C M , C D ] = ( l t ; t - ; t + 28) =i7l025t2+1610t + 833 \2 Do 1025t^ + 1610t + 833 = 1025 t + 161 ^8232^8232 41 ~ 41 I 205) Dang thuc xay t = - 161 205 Vay S^j^(~|-, nho nha't va chi t = 205 hay M 88 527' U l ' 2— ' 2— ,/ V» du 3.1.2 Trong khong gian Oxyz cho A(0; 1; 0), B(2; 2; 2), C(-2; 3; 1) thoa man AB = 2>/l0 va AOB = 45° 1) Tim toa dp diem C nam tren tia Oz cho the tich tu di^n OABC bang 2) Tim toa dp diem M thupc Ox, diem N thupc Oz cho tarn giac AMB can tai M va t u di^n A B M N c6 the tich b^ng 20 Loi gidi Taco: OA = (4;0;0), O B - ( x o ; y o ; ) suy OA.OB = 4xo Theo gia thiet bai toan ta c6 he phuang trinh sau: (xo-4)'+yg=40 4x, n = ±2 vuong voi A(1;0;1), B(2;0;0),C(0;1;0) The tich ciia khoi lang tru bkng NB]| = ^^321^ + 128t +146 V i d y 3.1.3, Trong khong gian x^+yo-8^o=24 V2xo=^f4+yl Xo=6 yo = A M , A B = (0;0;60) Do do: A M , A B •AN = 60n => V ABMN = 10|n| , 17 hay t = —— 4 t= Taco: N € ( A ) = > N ( l + t ; - - t ; + 2t) Suy S^3N = OA, OB = (0;0;24), OC = (0;0;m) B(6;6;0) The tich cua khoi lang tru la nen A A ' = ^ABC.A'B'C ^ ^ ^ABC Tir A A ' = t.'' A B , A C = (t;2t;t) suy = 1J • Neu t = l thi A'(2;2;2),B'(3;2;1),C'(1;3;1) ' Neu t = - thi A'(0; - 2; 0), B'(l; -2; -1), C ' ( - l ; - ; -1) ) Diem I each deu tat ca cac dinh cua lang tru la trung diem cua duong noi fam hai day Gpi M , M ' Ian lupt la trung diem cua BC,B'C' B'(3;2;1),C'(1;3;1) thi M ,M' Voi B ' ( l ; - ; - l ) , C ' ( - l ; - l ; - l ) thi M ' f ^ (3 1] 2'2'2 =>I _1 _1^ 2' 2' cam nang luy?n thi BH Hinh HQC - Nguyht Tat Thu Cty V i dv 3.1.5 Trong khong gian voi h? tpa dp Oxyz cho ba diem A(2;l;0), B(0;4;0), C(0;2;-l) va duong t h i n g d : x-1 y + _ z-2 Suy S^g^D - ^AABC + S^B^D = ^ AB A AC + Lap phuongy — TNHH MTV DCADI; = DWH Khans Vi$t ^ Nen Vg ABCD = SM.SABCD = |sM.766 = 66 => SM = 2^66 trinh duong t h i n g A vuong goc voi mat phSng (ABC) va cSt duang thSng d tai diem D cho bon diem A,B,C,D tao mpt t u di^n c6 the ot^+ (8tf +1^ = 4.66 t^ = t = Hh2 Tu- ta tim dugc hai diem S(7;15;3)va S ( ; - ; - ) tich bang — J x+2 Lai gidi Vi dv 3.1.7 Trong mat phang Oxyz cho duong thang A : Ta c6: AB = (-2; 3; 0), AC = (-2; 1; 1) => AB A AC = (3; 2; 4) Vi D € d nen D ( l + t ; - l + t;2 + t ) ^ A D = ( t - l ; t - ; t + 2) |20t + l 19 Lai gidi Taco M ( - + t ; l + t ; - - t ) A B = (-1;-2;1), A M = ( t ; t ; - t - 6) • Voi t = - , ta CO D ( - l ; - ; -1) va phuong trinh A : x+1 y +2 _ z+1 ^ | | u y A B A A M = (t + ; - t - ; - t ) nen S^^MB = | V3t^ + 36t +180 7_ 14 va phuong trinh A la: • Voi t = — , ta CO D " l OJ " l O ' ' 10 14 ^ ^ S ^ M B =3V5 nen taco: t ^ + t + 60 - 60 t = 0,t =-12 Tu ta tim dugc hai diem M ( - ; l ; - ) va M(-14;-35;19) GQi C ( - + t ; l + t ; - - t ) , D ( X ; ; ) y^To Ma CD = ( x - t + ; - t - l ; t + 5) nen A B CD o A B C D = V i du 3.1.6 Cho hinh chop S.ABCD voi A( ; - l; ) , B ( - l; ; - l) , C ( ; ; - ) va -(x - + 2) + 2(3t + l ) + 2t + = o x = 9t + D(10;-2;4) Gpi M la trung diem CD Biet SM vuong goc voi m?t phang Matkhac: A C = ( t ; t ; - t - ) , A D = (x + ; - l ; - l ) (ABCD) va the tich khoi chop S.ABCD hkng 66 (dvdt) Tim toa dQ die'mS Suy A B A A C = (t + ; - t - ; - t ) Lai gidi Taco AB = ( - ; l ; - ) , AC = (-4;l;-4), A D = ( ; - l; ) ( ( A B A A C ) A D - (t + 7)(x + 2) + 2t + = 9t2 + 117t + 90 Suy AB A AC = ( - l ; - ; - l ) , (AB A AC).AD = Do A , B , C , D dong ^°^°VA3C^=1|(ABAAC)AD p h i n g va n = (1; 8; 1) la VTPT ciia mat p h i n g (ABCD) Vi M la trung diem CD nen M(5; - ; 1), suy phuong trinh SM: X - y +1 _ z-1 ^~~~8~~ 2 Tim toa d o diem C thuoc A va diem D thuQC true Ox cho t u dien ABCD CO the tich bang va AB v u o n g goc voi CD Swy t = - l , t = — ^-T z+5 = va hai diem A ( - ; l ; l ) , B ( - ; - l ; ) Tim toa diem M thupc A cho tarn giac A B M c6 di?n tich bang 3V5 Do A l ( A B C ) nen u ^ =(3; 2;4) Suy V^BCD = ( ^ B A A C ) A D y—1 = Do S(5 + t ; - l + t ; l + t) Taco: DB = ( - l l ; ; - ) , DC = (-10;2;-6) =:> DB A DC = (-2;-16;-2) i t + n t + 72 = 9t2 + n t +108 = t = 9t'' + n t + 90j = -13 + >yi37 t = - l , t = -12 C j ( - ; - ; - ) , D j ( - ; ; ) ; C^ (-14;-35;19),D2 (-103;0;0) Cam nang luy^n thi DH Hinh hgc - Nguyen Tat Thu V i dy 3.1.8 Trong khong gian Oxyz cho hinh chop S.OABC c6 day OABC la hinh thang vuong t^ii O va A(3;0;0), AB = OA = ^ O C , O S A O M = (-12; ] 2; -9), ( O S A M ) O N = S(0;3;4) va Vay VsoMN = ~ yc>o1) Tim toa dp cac dinh lai va tinh the tich ciia hinh chop S.OABC trung diem C C , biet A M B ' M Chpn h^ true Oxyz cho A = , C Tinh the tich khoi chop SOMN thuQctia Ox, A ' thupc tia Oz va B thuQc mien goc x O y Lai gidi 1) Do ABCD la thang vuong tai A va O, 1) Xac djnh tpa dp cac dinh cua lang try, ^ 2) Tren cac canh A ' B ' , A ' C , BB' Ian lugt lay cac diem N , P, Q thoa dong thoi A e Ox, YQ > 0,OC = A ' N = N B ' , A ' P = 2C'P, B'Q = 3BQ.Tinh the tich khoi da dign AMPNQ Nen ta suy duoc C(0;6;0) Lai gidi Tuong t u ta c6: B(3;3;0) Dat A A ' = 2x, x > Vi S(0;3;4), nen ta gpi H(0;3;0) 1) Taco A(0;0;0), C(0;a;0), A'(0;0;2x), C'(0;a;2x) la trung diem OC thi ta c6 S H I (OABC) Gpi K la hinh chieu cua B len Oy, ta c6: OA(AB + O C ) ^ (dvtt) V i dy 3.1.9 Cho lang try deu A B C A ' B ' C c6 canh day bang a Gpi M la 2) Mpt mat phang (a) d i qua O va vuong goc voi SA cat SB,SC t ^ i M va N Ta c6: SQABC = ' = AB.sineo" = ~ Nen B SH = nen the tich khoi AK = I , B' chop S.OABC la: Suyra M ( ; a ; x ) = ^ A M = (0;a;x), V = l s H S o A B C - f ^ = 18 (dvtt) B'M = f aV3 a 2) Ta c6: SB = (3;-3;-4), suy phuong trinh mat phang (a) 3x - 3y - 4z = Vi SB = (3;0;-4), SC = (0;3;-4) nen ta c6 phuong trinh x = 3t SB: y =3 z = 4-4t x=0 j e n s u y x = i ^ va SC: y = + 3t' z = -4t' '^a A M B ' M ^ O I A'{0;0;av/^) va T u ta tim dup'c cac giao diem cua (a) voi SB,SC la: L.96._72^ M ( ; ; ) va N ' ' Suy OS = (0;3;4), O M - ( ; ; ) , O N = ^)Ta ^ 96 72' wi^.l,^^^ '2' a r CO A ' N = ^ A ' B ' = > N aS ,A'P = | A - C : ';-7;aV2 aV3 a 3a72^ va M 0;a; '2'~T~ 0;f;aV^ Cam nang luyen thi DH Hinh faVi 4 Suy A N = AQ = ^aVs hgc - Nguyen a 3aV2' '2'4 Thu Huong dan giai ,AP = AM = 0;|;aV^ [AP,AQ' = AP,AQ AP,AQ.AN ' ^A.MPQ BL - r AP,AQ A M 13a\/6 ^AMPNQ * - ^A.MPQ ^A.MI'Q M e O x = > M ( m ; ; ) , N € Oy => N(0;n;0) • L y r a A M = ( m - ; ; - ) , A N = (-3;n + 2;-4) AM = 24 Ian lupt la hinh chie'u ciia A len cac tryc Ox,Oy,Oz m, Bj,B2,B3 la hinh chie'u cua A len cac mat ph^ng tpa dp (Oxy),(Oyz),(Ozx) KPO „,,, - ^ | j ) Gpi A , , A , A • f c - a c o : A,(3;0;0),A2(0;-2;0),A3(0;0;4) va Bi(3;-2;0),B2(3;0;4),B3(0;-2;4) a ^ ^ a^V^ r AP,AQ! D o d o VA M P Q la't 24 •' AX^TNT fAM.AN = ^ • l a m giac A M N vuong can tai A nen ta co -j AM^ = AN^ ^-3(m-3) + 2(n + 2) + ]6 = ,(m - 3)^ + 2^ + (-4)2 = (-3)2 + (n + if + (-4)2 „ _ , ( ^ ^ ,1,; Bai tap van dung Bai 3.1.1 Trong khong gian Oxyz cho cac vec to: a = (2;3;l), b = (-3;-2;0), c = (x;2;-3) 1) Tim X de a A b vuong goc voi c 2) Tim X de goc giiia hai vec to a A b va bang 120° 2(n + 2) + 16 -i2 (2) {n + 2f+5 Ta c6: ( ) « 4(n + if + 64(n + 2) + 256 = 9(n + if + 45 Huong dan giai Ta CO a A b = (2;-3;5) n +2 = o ( n + 2)2-64(n + ) - n = 0o n + = 32 - 37231 1) a A b l c < : : ^ ( a A b ) c = 0c:>2x-3.2 + 5.(-3) = 0x = y (aAb).c 2) Yeu cau bai toan 2x-21 Vx^ +13.V38 \z—TTTTV a Ab J = cosl20" = — 21 x< — l l x ^ + 168x-635 = m = m = 21 X < — — o 2 2(2x-21)^ =19(x2 +13) 189 + 67231 189 - 67231' V a y CO h a i b p thoa y e u c a u bai toan: M, 189 + 67231 15 r ;0;0 11 Bai 3.1.2 Trong khong gian voi h^ tpa dp Oxyz cho diem A (3;-2; 4) 1) Tim tpa dp cac hinh chieu ciia A len cac true tpa dp va cac mat phang tpa dp2) Tim M € Ox,N € Oy cho tam giac A M N vuong can tai A 3) Tim tpa dp diem E thupc mat phang (Oyz) cho tam giac AEB can tai E va CO di^n tich bang 3^29 voi B ( - ; ; - ) hoac M- 189-67231 15 0; J -84±Vl4041 X = • 32 + 37231 \ ;0;0 0; 22 + 37231 ^ '° J 22 - 37231 ^)Vi Ee(Oyz) nen E(0;x;y) Suyra AE = (-3;y + ; z - ) , BE = ( l ; y - ; z + 4) ^ [AE, BE] = (8y + 6z - 8; 4z + 8; 10 - 4y) n = n = 22 + 37231 22-37231 A E ^ = BE^ AE^ = BE^ Nen t u gia thie't bai toan ta c6: A E = B E « •A E , B E I AE,BE + (y + ) + { z - ) = l + ( y - ) + ( z + 4)2 « y ' = 1044 o (8y + 6z - 50Z-16 = 3^29 AE,BE -(4z + 8)' = Ta = 1044 • Vai z = =^y = ^ 25 25 , nen 25' U'2' 2 ,BN = '2'T 141 CM.BN 2x/6555 ' =l+t CEACD ,3 = (9t + ; - t - ; t + 5) ^ 14 ^ 27 1) Chu ng minh rSng A, B, C, D idiong dong phang Tinh the tich cua tu dien ABCD t u A 141 > CM.BN = - S ^ c D E = ^ CE A CD = -j\/243t^ +360t + 138 A ( l ; 2; 1), B(2; - ; -3), C(-2; 0; 3), D(0; 3; 4) 2) Tinh chieu cao ve t u B ciia tarn giac BCD va chieu cao cua t u di^n ABCD ve _„ 13 CM.BN = (t + 3;-3t + ; - t - ) , Bai 3.1.3 Trong khong gian Oxyz cho bon diem 742 ^ ~ Vay S^^-f),: nho nha't va chi E 38 107 27'9'27^, , , Bai 3.1.4 Trong khong gian Oxyz, cho ba diem 3) Goi M, N Ian luot la trung diem cua AB va CD Tinh c6 sin cua goc giu-a hai duong thang CM va BN 4) Tim E tren duang thang AB cho tarn giac ECD c6 di^n tich nho nha't A BD = (-17;16;-14), ( B C A BDJBA = ^) Tinh dp dai duong cao t u dinh A, ban kinh duong tron ngoai tiep, ban kinh duong tron noi tiep cua tam giac ABC ^0 Vay A , B, C, D khong dong phang va the tich khoi t u di^n ABCD la: v = l( B C A B D ) B A =1 (dvtt) CD = (2;3;1) ^ CD = Vi4,S,ABCD 1) Chung minh rang cac diem A, B, C khong thang hang 3) Tinh cos A,sin B,tanC cua tam giac ABC 1) Taco BC = (-4;1;6), BD = (-2;4;7), BA = {-1;3;4) Suyra BC A ( l ; ; l ) , B(5;l;-2), C(7;9;l) 2) Tim tpa diem D cho ABCD la hinh binh hanh Huong dan giai 2) Taco: N 4) Ta CO phuong trinh A B : y = - t , s u y r a E(l + t;2 - t ; l - 4t) z = l-4t 3uy 25 , X CE 37 2'2' ^ Men cos(CM, BN) = sf + (4z + S)^ + (lO - y ) ' = 1044 26-16zi •_1044 = 0c:>z = 2,z = - 34 — M guy C M = 4z + l • Voi z = 2=>y = nen E(0;3;2) 34 CO 7741 BCABD ^) Tim toa giao diem ciia phan giac trong, phan giac ngoai goc A voi duong thang I5C Huong dan giai ' j T a CO AB(4;0;- 3), AC(6;8;0) nen cac diem A, B, C thSng hang va chi = 6k Suy duong cao BB = ^ ^» • (v6 li) Vay A,B,C -3 = O.k 3VABCD 'ABCD N/24T' Chieu cao ciia hinh chop ve t u A la: h = — 382 tgi so thi^c k cho AB = kAC, tuc la < = 8k •^hong thSnghang 383 ; V A, B, C h „ g h i n g S „ g nen ABCD 7-XD=4 Mnh ^i^^^^^^ Cty TNHH MTV DWH Khang Vift chi tt, XD=3 17 7-XE=-2(5-XE) DC = AB, hay 9-yE =-2(l-yE) 1_ZE=-2(-2-ZE) 3)Tac6 A § = x / o M ^ = 5, AC! = 10, BCl = N/77 AB.AC _ ^ —/-Mj./^^^ _ - - _ 25 Bai 3'l-5- Trong khong gian Oxyz, cho tam giac ABC c6 A(2;3;l),B(-l;2;0),C(l;l;-2) R ^ Tuong t u cosB =— , c ocosC s « ^ = ^ ^^ - V i B,C€(0;7i) nen 1) Tim tpa dp chan duong vuong goc ke tir A xuong BC 2) Tim tpa dp H la tryc tam ciia tam giac ABC 3) Tim tpa dp I la tam duong tron ngoai tiep ciia tam giac ABC +) sin B > => sin B = V l - c o s ^ B = | ^ +) tan C, cosC ciing da'u tanC= _^^V48T V cos C 3o 4) Dien tich tam giac ABC la S = -BA.BC.sinB = 4) Gpi G la trpng tam cua tam giac ABC Chung minh rSng cac diem G , H , I nSm tren mot duong thang '481 = ylm 77 2S BC ^^^"ZsinB p , '481 z = -2t V AC 77 2sinB 481 2S AB + BC + CA Suy K(2t - ; - ; - 2t), AK(2t - 3; - - ; - - 2t) Vi AK BC o AK.BC = A (1; 2 ^ • [6x-y-4 =0 ^ ' Vay B ( -1 5 ; -1 6),C (-3 ;-7 )... a-2 = - ^ , b ^ b-2' b-2 i N2 b -1 + l-(b-2)2+(b-l)2 (a-2)2+l = (b-2)2+(b-l)2 b-2 b -1 MA = -3 MB: (x -1 )2 + (y + if = 16 tam I va diem A(l + Do b = khong thoa man, v$y a-2 = MA = 3MB YB =-3 ^^ 6-3 ... M nen: (a-2)(b-2 )-( b-l) = MA.MB = ^y(a-2)2+l=V(b-2)^+(b-l)^' ,b^2 a=2 a-2 = ^ , b ^ b-2 (b-2)2+(b-l)^ (b-2)^ A , B e ( C ) nen ^B +^^6 ''A=4xM-3xB= 4-3 xg yA=4yM-3yB =-4 -3 yB (xB-4)^+yB'=25