Luyện thi Toán học - Chuyên đề trọng điểm bồi dưỡng học sinh giỏi Hình học không gian: Phần 2

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Nối tiếp nội dung phần 1 tài liệu Chuyên đề trọng điểm bồi dưỡng học sinh giỏi Hình học không gian, phần 2 giới thiệu tới người đọc các nội dung: Thể tích khối trụ, khối lăng trụ, mặt cầu hình trụ - Mặt nón, bài tập tổng hợp. Mời các bạn cùng tham khảo.

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Nội dung Text: Luyện thi Toán học - Chuyên đề trọng điểm bồi dưỡng học sinh giỏi Hình học không gian: Phần 2

Vay: A H •• Cho hinh chop S.ABCD c6 day ABCD la hinh binh hanh voi Cho k h o i chop S.ABCD c6 day A B C D la h i n h t h o i canh a va goc nhpn A B = a, A D = 2a, B A D = 6 O " , SC v u o n g goc v o l m a t phSng ( A B C D ) , SA h g p bang 60° va SA 1 ( A B C D ) , bie't rang khoang each t u A d e n canh SC = v o i mat p h i n g ( A B C D ) goc 45". T i n h the tich k h o i chop S.ABCD va k h o a n g T i n h the tich k h o i chop SABCD. each g i u a hai d u o n g thSng SA va BD. LOI GIAI LOI GIAI V i A B C D la h i n h t h o i c6 goc A bang 60". A p d u n g h a m cosin cho tarn giac A B D co:"^ BD^ = AB^ + A D ^ - 2 A B . A D . c o s 6 0 ° N e n A B D deu A O = -2 - 2. . .a .. 2. a .1- = .3a^ = a'^ +4a^ .2 • Xet ASAC v u o n g tai A : 1 1 1 1 .BD = aV3. AH^ A C ^ •' +AS^ — AS^ a^ Sa^ 3a^ Tir d o ta c6 A D ^ = AB^ + BD^ = 4a^ => A A B D v u o n g tai B . Goi 0 = A C n B D . 1 T r o n g A A B O c6: D i ^ n t i c h d a y A B C D V^^BCD = ^ A C . B D = ^ - 3a^ OA - V A B V B O ^ = Ja2 + The tich k h o i chop Vg ^BCD = 3 S A . S A B C D = 3 = V 4 2 = > A C = a>/7. '•" Cho k h o i chop S A B C D co day A B C D la h i n h thang v u o n g tai A va B bii Ta C O A C la h i n h chieu v u o n g goc SA tren A B = BC = a, A D = 2a, SA 1 ( A B C D ) va (SCD) h o p v o i day ( A B C D ) mot goc m p ( A B C D ) , goc g i u a SA va ( A B C D ) la goc ' 60". T i n h the tich k h o i chop S.ABCD. SAC = 45". ' '•' LOI GIAI => ASAC v u o n g can tai C nen SC = A C = 3%// . Goi E t r u n g d i e m cua A D c6 ABCE la h i n h v u o n g . D i ^ n tich A B C D : S = 2 S ^ B D = ^ A . B D = a.ayfs =a^S. Trong A A C D c 6 C E = ^ A D ( d u 6 n g trung The tich k h o i chop S.ABCD: V = - S C . S A B C D = -a^.a^^ = . • tuye'n bang n u a canh huyen) nen A A C D 3 3 3 v u o n g tai C. 'o'lr.i>-;-., K h o a n g each g i u a hai d u o n g thSng SA va BD. G D I AC C D 1 m p ( S A C ) ^ C D 1 SC. Dung A x / / B D , g o i E = A x n C D . CD I S A Vi BD//AE ^ B D / / ( S A E ) , n e n = > d ( B D , S A ) = d(BD,(SAE)) = d ( D , ( S A E ) ) . Vay goc g i i i a m p (SCD) va ( A B C D ) la goc SCA = 60*^. i / / ' ? - •'' AEICE , . , ^ , Xet A S A C c o SA = A C . t a n 6 0 " =a>/2.V3 = a V 6 . U.i
V i D la trung diem ciia CE nen d(D,(SAE)) = l d ( C , ( S A E ) ) = . j^hoang each giCra hai duong thang A B va SC. Vi A B // C D => A B // ( S C D ) , nen suy ra: ,v Cho hinh chop S.ABCD c6 day ABCD la hinh chii nhat, SA vuong goc v6i d(AB,SC) = d(AB,(SCD)) = d(A,(SCD)). day ABCD, biet AB = a,AD = a V 3 , goc giua mat phSng (SAC) va (SBC) "b^ng 60". Tinh the tich khoi chop S.ABCD va khoang each gii>a hai duong Ta c6: | t cf ^ C D 1 ( S A D ) => ( S C D ) 1 ( S A D ) , hai mat phSng nay thang AB va SC. CD -L c)A > >; LOIGIAI vuong goc voi nhau theo giao tuyen S D . Dung A E 1 SC(E 6 SC). :/ j ' r.,^., Dung A H 1 S D ( H e S D ) => A H 1 ( S C D ) . Vay d ( A , ( S C D ) ) = A H . , . : Trong mat phang (SBC) tu E ke EF 1 S C ( F e SB) . 1 1 2 1 aVlO Trong ASADco: -+• • + • • AH = ;i Suy ra goc giiia hai mat phang ( S A C ) AH^ AS^ AD^ 2a' va (SBC) la goc giua hai duong thang F aVlO AEvaFE. Ketluan d(AB,SC) = Ta c6: • Dl/BI D A I HOC K H O I A . 2006 : ; ^ BCISA ^ ' Cho hinh chop S.ABCD c6 day ABCD la hinh chir nhat voi AB = a, •BCl A F ( A F C ( S A B ) ) AD = 2a , canh SA vuong goc voi day (ABCD), canh ben SB tao voi day 1 c f SC 1 A E a\/3 Ta c6: SCl(AEF) =>SC1 A F ( A F C ( A E F ) ) (2) goc 60". Tren canh SA lay diem M sao cho A M = . Mat phang (BCM) cat SCIEF 3 T u ( l ) v a (2)suy ra: AF 1 (SBC) ^ AF 1 SB, A F I F E . SD tai N . Tinh the tich khoi chop S.BCNM. '"•^ ' • ' v ^ LOI GIAI Hai mat phcing (BCM) va (SAD) c6 M Trong AAEF co: AF = AE. sin 60" = . isi (';E3 oiaioov '.jh?J\ la diem chung thu nhat, ma BC // A D . Trong hai tarn giac SAB va SAC c6 : ; ^^^^ ' f^'iaA (fej* II'K' Suy ra giao tuyen cua 2 mat phang qua 1 1 M va .song song voi BC, AD. Giao r"K----\ (3) AF^ AS^ • + • tuyen cat SD tai 1 diem la diem N can tim. ////' '\ AB^ AS^ A F ' A B ' /a^Jr^.c'.fp* i^i.N rbiJ^r" 1 1 1 1 1 AB la hinh chie'u vuong goc cua SB (4) AE^ AS^• + • AC^ AS^ AE^ AC^ tren mp(ABCD) nen: ,0 SB,(ABCD) = SBA = 60" Tu (3) va (4) AF2 AB^ AE^ AC^ ^'^^'^S''-^ (18 f Trong A S A B vuong tai A : S A = A B . tan 60" = . 1 1 2a • AE = BCIAB , , 3AE' AE^ 4a' 3 Ta c6: =^ B C 1 m p ( S A B ) => B C 1 B M . BCISA •. .„-. „: 1, 1 1 2 r'Vay:ll- =- l •AS = ^ay tu giac B C N M ia hinh thang vuong tai B va M . AS^ AE^ AC^ 4a^ 4a^ a^ [AH I B M The tich khoi chop S.ABCD: Vg /^g^p = — SA.Sys^g(--[-, = -.——.a.a A H l B M ( l l e B M ) , vi AH I m p (BCNM). AHIBC 3 2
B M = VAB2 + A M 2 = ^ +^ =^ . , y , { B M 1 SA ^ ^ mp(SAC) r-v. yr> V;:). ^ mp ( S B M ) 1 mp (SAC) ( vi B M c m p ( S B M ) ) . Co: M N / / A D ^ ^ = i ^ - U M N = f " i SA AD 3 3 J 1 2, 2 aVs a^V2 S.ARi =-•IA.IB = - . - I O . - B M = - . - 5AABi 2 2 3 3 9 2 2 6 T r o n g t a m giac v u o n g A B M : — ^ = +— = --r- + " ~ T =^ =^ ^ ^ A H ^ AB^ A M ^ a^ a^ a^ 2 Ta CO O N / / SA ( v i O N la d u o n g t r u n g b i n h cua t a m giac) nen: d(S,(?CNM)) gj^ Taco: S A n ( B C N M U M ^ - ^ 4 ^ ^ ^ V^=T^ =^ S, B C N M O N 1 ( A B C D ) , O N = I s A = I. . d(A,(BCNM)) A M (BC + M N ) . B M _ ] f _ 4a 2a lOa^Vs rQf^f!./ The tich can t i m : V ^ , ^ ^ =1^05^,^,^ =l.±.±Jl = l J l . Co SgcNM = 2 2 2a + 36 D E T H I T U Y E N S I N H D A I H Q C K H O I A N A M 2010. \ , ^ XX ^ 1 lOa^Va lOa^s/3 Cho h i n h chop S.ABCD c6 day la h i n h v u o n g canh a. M , N Ian l u g t la t r u n g Vay V3.BCNM = - d ( S , ( B C N M ) ) . S B C N M = 3 ^ - - ^ = - ^ - diem ciia A B va A D , H = C N n D M va S H v u o n g goc v o i ( A B C D ) va T U Y E N S I N H D A I H Q C K H O I B 2006. SH = a V i . T i n h the tich khoi chop S . C D N M va khoang each giua D M va SC. Cho hinh chop S.ABCD c6 day A B C D la hinh chO nhat v o i A B = a, A D = av2, : . LOIGIAI SA = a va SA v u o n g goc v o l mat phMng day A B C D . Goi M , N Ian luot la t r u n g d i e m A D va SC, I la giao d i e m ciia M B va A C . C h i i n g m i n h mat Ta CO A A D M = A D C N ( c . g . c ) => D J = q . ,, '..'T ph^ng (SAC) v u o n g goc v o i mat phang (SMB). j ^,.-^y • Ma D^ + D ^ =3 90° => q + 5 5 = 90° * " ' T i n h the tich k h o i t u d i e n A . N I B . => A C D H v u o n g tai H nen C N 1 D M . , ;^ L6IGIAI S Trong A A D M c6: Gpi O la giao d i e m cua A C va B D . ,, Xet A A B M v u o n g tai M : , ^jr.&dq ? D M = A/AD^TAM^ =. = ^ . V 4 2 2 a BM = VAB^+AM^ = a +— 'aVs' 5a' 5cDNM=|-CN.DM =i . Va: A O = - A C = i N / A B V B C ^ 2 2 1 f - 5a^ Sa^x/S V a y : VS.CDNM - - S H . S r n M M = - . a V 3 . 3—-CDNM 3-.-- g 24 ^2 2 'V'A, Goi I la h i n h chieu v u o n g goc ciia H tren canh SC V i I t r o n g t a m t a m giac A B D : • , fDMlCN fa co: CO D M 1 mp(SCN) ^ D M 1 H I DMISH "" - r v — ^ f I M = i B M . ^ . A I . ^ A O =^ . £ ^ =^ . ^ '.It.) 3 6 3 3 2 3 Tir (1) va (2) suy ra: H I la doan v u o n g goc c h u n g ciia SC va D M . «1 T r o n g A A I M c6 A M ^ = A I ^ + M I ^ = ^ Vay: d(SC,DM) = H I . • Hi! (liht mil aj"'>^'j V : lisvt
A/ - 2a ._,o 7a^ Ta c6: CD^ = C H . C N => C H = —•—;m .1. f;' - 2 . a . — .cos60 = 5 \ HI' 1 HS^ • + HC^ • 1 1 3a^ 5 + 4a^: 19 12a^ \ . CH = Ta CO- H C la h i n h chie'u v u o n g goc cua a^yT2 2aV57 SC tren mat phang (ABC) nen goc g i i i a • HI = 19 19 SC va mat phSng (ABC) la goc SCH = 60". D E T H I T U Y E N S I N H D A I H Q C K H O I D N A M 2010. T r o n g A S C H v u o n g tai H c6: Cho h i n h chop S.ABCD c6 day la h i n h v u o n g canh a, canh ben SA = a ; a ^ A SH = C H . t a n 6 0 " = ^ ' ^ 3 = . h i n h chie'u v u o n g goc cua S len ( A B C D ) la d i e m H thuoc doan Ac, 3 " 3 A H = — . C M la d u o n g cao cua tam giac SAC. C M R : M la t r u n g d i e m cua Dien tich t a m giac A B C deu: B a c 4 SA va t i n h the tich khoi t u d i ^ n SMBC. 'AABC 4 4 /^ LOI GIAI The tich k h o i chop: \(>W Goi O la giao d i e m cua A C va B D . „3 —c V,S.ABC = - . S H . S SA2-AH2=ja2-2" 3"~ ^^^^ 3 3 4 12 A H = ^ = ^ , S H T i n h k h o a n g each g i u a h a i d u a n g t h a n g S A va B C theo a. Day la Inii tinh khoang each giua canh ben va canh day, cdc ban xent Iqi phieang SH.AC = CM.SA « CM.a = ^ ^ ^ . a 7 2 .CM = phap tilth a biii khoang each gim hai duang thang cheo nhau. v. (jo , 7a2 SA Ke A x / / B C = ^ : B C / / m p ( S A x ) ( v i A x c ( S A x ) ) . a ' A M = V A C ^ - C M ^ = ^Is} - ' • '.>a,L'MA m;' 4 ~1~ I Nen: d ( B C S A ) - d ( B C m p ( S A x ) ) = d ( B , m p ( S A x ) ) . Vay M t r u n g d i e m ciia S A . ^ .., BC song song v o i mat phang (SAx) t h i k h o a n g each m o i d i e m tren d u o n g Ta c6: BO 1 A C , BO 1 S H - H T r thang BC den mat p h a n g (SAx) deu bSng nhau. V i sao Thay lai chon d i e m B ("-^ ^BOlmp(SAC). ma k h o n g chpn d i e m C la v i d i e m B nSm tren d u o n g th3ng c6 chua d i e m H u la h i n h chie'u cua d i n h , viec t i n h khoang each t u H den mat phMng ben \a42 \ a^x/M ^B.SCM - (SAx) la ra't de dang. T h o n g qua cong thue t i n h t i so k h o a n g each t h i ta tinh 3 2 •2'2 2 48 d u g c k h o a n g each t u B. D E T H I T U Y E N S I N H D A I H Q C K H O I A N A M 2012. D u n g H I 1 A x tai I . Goi K la h i n h chie'u v u o n g goc ciia H tren SI. Cho h i n h chop S.ABC c6 day la tam giac deu canh a. H i n h chie'u v u o n g •^'-^^ , , [AX 1 HI , , , , , , f' * ' ..f cua S tren m a t phSng (ABC) la d i e m H thuoc canh A B sao cho H A = 2HB Vi ] Axl(SHl)r:>mp(SAx)lmp(SHl). Goc giOa d u o n g thSng SC va mat phSng (ABC) bang 60". T i n h the tich cu^ khoi chop S.ABC va tinh khoang each giua hai d u o n g thang SA va BC t h e o ^ H a i mat phang (SAx) va (SHI) v u o n g goc v o i nhau c6 giao t u y e n SI. LOI GIAI M a H K 1 SI => H K 1 ( S A x ) . --vi:*^^',^'-- - A p d u n g d j n h ly cosin cho tam giac A H C c6: Vay d ( H , m p ( S A x ) ) - H K . ' ' C H ^ = A C ^ + A H ^ - 2AC.CH.COSCAH ,''('',)-c?r:i'.!ff; ,1, H A
Trong ASAM vuong tai A c6: Trong A A I H vuong tgi I c6 H I = HA.sin60° = ^ 3 • 2 Trong ASIH vuong tai H c6: AH^ A M ^ AS^ 3a^ 3a^ 3a^ 1 1 1 9 9 24 a77 .HK = Ketluan d(D,mp(SBC)) = : —:r + : HK^ HS^ H r 21a^ 3a^ 7a' 724 ' -+ Duong thSng d i qua hai diem B va H c6 giao diem voi mat phSng (SAx) la d(B,mp(SAx)) BA 3 ' ' V g ( ; - > ^ J ' ' 0 ^^-'S;'''; ^ g^^jG2: Khoi chop c6 mat ben vuong goc voi day. ^"^"^ d ( H , m p ( S A x ) ) ^ H A ^ 2 ^'"^»>>V;.y H:X^ r Cho hinh chop S.ABCD c6 day ABCD la hinh chi> nhat canh AB = a, A D = a4?>, goc giua mat phang (SAC) va (ABCD) bSng 60". Goi H la trung diem ciia AB, ^ d ( B , m ; ( S A x ) ) = |d(H,mp(SAx)) = f . ^ = ^ tY®: ' biet mat ben SAB la tam giac can tai S va thupc mat ph^ng vuong goc voi OTgt phang day ABCD. Tinh the tich khoi chop S.ABCD. DE T H I T U Y E N SINH DAI HQC KHOI D NAM 2013 LOI GIAI Cho hinh chop S.ABCD c6 day la hinh thoi canh a. Canh SA vuong goc voi Vi AB la giao tuyen cua hai mat phSng day, BAD = 120°, M la trung diem cua canh BC va SMA = 45^^ .Tinh theo a (SAB) va (ABCD) vuong goc nhau, ma the tich cua khoi chop S.ABCD va khoang each t u diem D den mat ph5ng SH 1 AB suy ra SH 1 (ABCD) . (SBC). Dung H I l A C ( l e A C ) . ] TTTT^ LOIGIAI Ta c6: AC 1 SH va AC 1 H I Vi ABCD la hinh thoi c6: BAD = 120" AC 1 SI (djnh ly ba duong vuong goc). => BAC = 60° => AABC deu. y • • »•" i"^ Vay: goc giua (SAC) va (ABCD) la goc aVs
a) . Gpi H la trung diem cua AB. L i p i I, J la hinh chieu ciia H tren AB va B C Vi A A B C deu =i. S H 1 A B . SI 1 AB, SJ ± B C . (SAB) ± (ABCD) Theo gia thiet S T H = SJH = 45". Ma : > S H I (ABCD). " (SAB) n ( A B C D ) = A B ~ Ta c6: . ASHI = ASHJ =^ H I = HJ ;r Nen B H la duong phan giac ciia AABC Vay H la chan dirang cao cua khoi chop. T u do suy ra H la trung diem ciia A C . b) . Ta CO tam giac S A B deu nen S H . Suy ra I, J Ian lugt la trung diem ciia A B va B C . / BC a HI = HJ = H S S.ABC 12 Cho tu dien A B C D co A B C la tam giac deu canh a, B C D la tam giac vuorig Cho hinh chop S . A B C D c6 day A B C D la hinh vuong canh a, tam giac SAB can tai D, ( A B C ) 1 ( B C D ) va A D hop voi (BCD) mot goc 60". Tinh the ti'ch tu deu, tam giac S C D vuong can tai S. Goi I, J, K Ian lugt la trung diem cua cac di?n A B C D . '-"rn-t -r'- • canh AB, C D , SA. Chirng minh r^ng (SIJ) vuong goc voi ( A B C D ) . Tinh the LOI GIAI tich khoi chop K . I B C D . Gpi H la trung diem ciia B C . i '- \ -J LOI GIAI ABISI^ Ta CO tam giac A B C deu nen: A H 1 B C , va A H = Ta co: •ABi(sij). ABIIJ Ma:pBC)l(BCD) ^ ' ^ ^ H K B C D ) : l(ABC)n(BCD) =BC ^ ViAHc(ABCD) ( A B C D ) 1 (SIJ). Ta c6: H D la hinh chieu vuong goc cua A D Hai mat phang nay vuong goc voi nhau tren m p ( B C D ) . theo giao tuyeri IJ. M ^ y, ; ; ( X> f>l K Dung S H 1 IJ ( H 6 IJ) ^ S H 1 ( A B C D ) Nen: AD,(BCD) = ( A D , H D ) = A D H = 60°. AH A B C la tam giac deu nen SI = — . A H A D vuong tai H : D H = ,0 2 2 2 BJL tan 60 Tam giac S C D vuong tai S co SJ = I c D = - . " ' ''^ '''' ^ ^'^^ Trong A B C D vuong can tai D c6: ^ • 2 2 .0/. .•,-r . BC = 2HD = a , DB = D C = ^ ^ ^ ABCD = -!-DB.DC = Ta co: SI^ + SJ^ = IJ^ ^ ASIJ vuong tai S, nen co: SH.IJ = SI.SJ => S H = 4 'a •
D E T H I T U Y E N S I N H D A I H Q C K H O I B N A M 2008 Ta c6: B C = N / A D ^ T D C ^ = Vil^T^ = ax/s . ( , ] • (- >/.T, . j Cho hinh chop S.ABCD c6 day ABCD la hlnh vuong canh 2a, SA = a, SB = aVa S A B C D = ^ ( A B + C D ) . A D = l ( 2 a + a).2a = 3a2. ^ ' ^ va mat phSng (SAB) vuong goc voi mat phSng day. Goi M , N Ian luot ^ trung diem cua cac canh AB, BC. Ti'nh theo a the ti'ch cua khol chop S.BMDM Ngoai ra: S^^g^D = ^,^^81 + ^ABIC + ^ACDI LOI GIAI Ha S H l A B t a i H . • => S . B . r = - 1 A B . A I - I c D . D I = Sa^ -a^ - — = — . Vi mat p h i n g (SAB) vuong goc mat 1 ... 2S..^, 3a^ 3a phSng (ABCD) c6 AB la giao tuye'n. Suy ra SH 1 mp(ABCD) . L 3a rz 3aN/l5 Xet ASAB c6: AB^ - SA^ + SB^ = Trong ASIH vuong tai H c6: SI = IH. tan 60° =^.S = •if,') Vay ASAB vuong tai S. Va : V5 5 _ 1 cTc 1 3aVl5 2 3a^N/l5 1 1 1 1 1 11 4 Vs.ABCD - 2 ABCD - ^ • ; "^a = + — + : SH^ SA^ SB^ 3a^ 3a' D E T H I T U Y E N S I N H D A I H Q C K H O I A N A M 2011 Cho hinh chop S.ABC c6 day ABC la tam giac vuong can tai B, AB = BC = 2a, hai mat phSng (SAB) va (SAC) cung vuong goc voi mat ^BMDN-^ABCD S-^^MD ^ACND phang (ABC). Goi M la trung diem cua AB; mat phang qua SM va song song voi BC cat AC tai N . Biet goc giCra hai mat phSng (SBC) va (ABC) bang = AB^ - - A D . A M - - C D . C N = 4a^ -2a^ =2a2 2 2 60". Tinh the tich khoi chop S.BCNM. LOI GIAI ^ ^S.BMDN-g-^ BMDN-^^y- ~ 3 " Taco: m p ( S A B ) n mp(SAC) = SA . D E T H I T U Y E N S I N H D A I H Q C K H O I A N A M 2009 Va hai mat phJng (SAB) va (SAC) cung Cho hinh chop S.ABCD c6 day ABCD la hinh thang vuong tai A va D; AB = vuong goc voi mat phSng (ABC). A D = 2a; CD = a; goc giiia hai mat phSng (SBC) va (ABCD) bang 60". Goi 111 Suy ra SA I m p ( A B C ) . trung diem ciia canh A D . Biet hai mat ph^ng (SBI) va (SCI) ciing vuong go^ Mat phang qua SM va song song voi vai mat ph5ng (ABCD), tinh the tich khoi chop S.ABCD theo a. BC cat AC tai N , suy ra M N // BC va N LOI GIAI trung diem ciia AC. Hai mat phang (SBI) va mat phang (SCI) CO giao tuye'n la SI cung vuong goc v o i Trong tam giac ABC co M N = ^ BC = a, BM = AB = a^" mat phang day suy ra S I 1 ( A B C D ) . Ta CO BCNM la hinh thang vuong tai B va M : j-j/ Goi H la hinh chieu vuong goc cua I tren B C . BCIIH SBCNM = ^ ( M N + BC).BM = i ( a . 2a).a = ^ 'l.:'/ I'M.; a Co: BC 1 (SIR) ^ BC 1 S I . BCISI BC I AB Ngoai1 ra : < B C l m p ( S A B ) => B C l S B . Vay goc giira hai mat phang (SBC); B C l SA Va (ABCD) la goc SHI = 60°.
( S B C ) o ( A B C ) = BC Vs.ABC =^-^( Q t i ' l / i t f i mat phang (ABC) theo giao tuyen BC, h m Co: H F = | A C = ^ ( H F duong trung binh tam giac ABC). ' ,.v •1 suyra S I l l m p ( A B C ) . ^^ - / - ^ . - c i . . 1-,-L.i4.JL,ii^Hui!^^ ! . v Va SII = H,^ HF^ HS^ 3a^^3a^ " 26 ^f^^, V.:,,:' ; Trong tarn giac ABC vuong tai A c6: Vi hai diem H \ C cung nam tren duong th^ng c6 giao diem voi mat AB = BC.cos30" = — , A C = BC.sin30" = - d(C,mp(SAB)) CB 2 2 The tich khoi chop S.ABC: ' ^^^^'-''^^^^^^ phang (SAB) tai B nen: - i Ll '1 = ^ = 2 M' , d(H,mp(SAB)) HB ' - n v ciA ; v . , , „ . : \ -y. 1 1 _ 1 aVs a V 3 a a'' ^ d ( C , m p ( S A B ) ) = 2d(H,mp(SAB)) = 2. ax/39 aV39 ^2^^^^ I ' ^ - . Vc . R ^ . =-.SH.S, =-.SH.-.AB.AC = - . . .-^ — 2 '2 16 26 13 S.ABC 3 \ABC 3 2 6 2 2 2 16 DE THI TUYEN SINH DAI HQC KHOI B NAM 2013 Tinh khoang each tu C den mat phang (SAB). ./A «u'i m^ih gin;; Cho hinh chcSp S.ABCD c6 day la hinh vuong canh a, mat ben SAB la tam Cach 1: Dua vao the tich. -
M a mat phSng (SAB) v u o n g goc v o i mat 5 H N = - A M = 1BC = ^ phang day ( A B C D ) theo giao t u y e n BC, 2 4 4 suy ra SH±mp(ABCD). J. " A S H N v u o n g tai H : S H = H N . t a n 4 5 ° = The tich k h o i chop S.ABCD: ' ^,! Aur- =—-SH.SAARr" = — . S H . — A B . A C = —.——— .a = S.ABC 3 AABC 3 2 6 4 24 Cho h i n h chop S A B C D c6 A B C D la h i n h v u o n g , tarn giac SAB d e u canh a 1 aS 2 a^Vs n a m t r o n g mat p h ^ n g v u o n g goc v o i ( A B C D ) biet (SAC) hc?p v o i ( A B C D ) 3 2"' = 6 m p t goc 30° . T i n h the tich h i n h chop SABCD. I D o A B // C D va H e A B , nen: LOI GIAI d(A,mp(SCD)) = d(H,mp(SCD)). g c D v n g H I 1 A C tai I . ' Goi G t r u n g d i o m cua C D va I la h i n h chieu v u o n g goc cua H tren SG. fAC 1 H I Ta CO AClmp(SHl)^AClSI. fCDlSH A C 1 SH Co: =?CDlmp(SHG)=^CDlHI. ^ ?, , CDIHG ^ ' f .= ( | B / v 5 ] < 7 f n , f ; (SAC) n (ABCD) = A C HI±CD Vi H I 1 m p ( S C D ) =^ d ( H , m p ( S C D ) ) = H I . :jA^ Ta c6: H I I A C S I I A C t,n,n HI ISG HIC(ABCD),SIC(SAC) T r o n g ASHG v u o n g tai H c6: > ( ( S A C ) , ( A B C D ) = S I H = 30°. 1 1 1 4 1 aVH .HI = 7 • 3a'- + — 1 HI^ HS- HG'' — 3a^ V i H I va B O c u n g v u o n g goc v o i A C , nen H I = - BO = . „ V ,. is/ri" T- Ke'tlu^n d ( A , m p ( S C D ) ) = - y — . ni 'm^,t» vn>lh I A H rtv'iih x'}j :(Vi (ii< Xet A S H I v u o n g tai H : S H = H I . t a n 3 0 ° = ^,:!? Cho h i n h chop SABC c6 day A B C v u o n g can tai A v o i A B = A C = a , biot 12 tarn giac SAB can tai S va n a m t r o n g mat phang v u o n g goc v o i (ABC) • A'^ •••• O / v /f' 1 ^-.a 2 a^Ve p h a n g (SBC) h o p v o i (ABC) m g t goc 45". T i n h the tich cua SABC. ^S.ABCD - - - S H . S ABCD = 3' 12 36 LOI GIAI Cho h i n h chop S A B C D c6 A B C D la h i n h c h u nhat c6 A B = 2a, BC = 4a, (SAB) i i Goi H t r u n g d i e m BC. V i SAB can nen SH 1 A B . 1 ( A B C D ) , hai m a t ben (SBC) va (SAD) c u n g h g p v o i day A B C D m p t goc 'i, Ma ( S A B ) l ( A B C ) = ^ S H l m p ( A B C ) . ,-|ciG: _30;\h the tich h i n h chop SABCD. i'- Goi M t r u n g d i e m cua BC, t h i : c'.;;' ' i ^ LOI GIAI ,^ , i: A M 1 BC v i A A B C can tai A . ' D u n g SH 1 A B tai H . V i (SAB) 1 ( A B C D ) . Suy ra S H 1 (ABCD) T u H d u n g H N 1 BC v o i N e B C , t h i B C l AB CO BC 1 m p ( S A B ) =^ BC 1 SB. BClmp(SHN) =^BC1SN. ^ B C I S H Vay goc giira mat p h ^ n g (SBC) va (ABC) Goc gii>a hai mat phSng (SBC) va ( A B C D ) la goc SBA = 3 0 ° , v i c6 A B va SB la goc SNH = 45". Ian l u p t thupc hai m a t p h ^ n g c i i n g v u o n g goc v o i giao t u y e n . Ta c6: H N la d u o n g t r u n g b i n h cua A B A M : L y luan t u o n g t u goc giira hai mat phang (SAD) va ( A B C D ) la goc SAB = 3 0 ° . 289
Theo tren ta c6 SAB la tarn giac can t^i S nen H trung diem ciia AB. ., , ^ ^ G 3 : Khoi chop deu. Xet ASHB vuong tai H : . Cho hinh chop tam giac deu S.ABC canh day b l n g a va canh ben bang 2a. SH = B H . t a n 3 0 ° = a . ^ = ^ . Chung minh rang chan duong cao ke t u S ciia hinh chop la tam ciia tam giac 3 3 deu ABC. Tinh the tich chop deu SABC . •.I " L d l GIAI Di?n tich day S^BCD = AB.AD = 2a.4a = 8a^ . D i ^ g SO 1 (ABC). Ta c6 SA = SB = SC. 1 BN/S g^2 _8a^%/3 V.S.ABCD = -.SH.S ABCD Suy ra OA = OB = OC (duong xien bang 1 3' 3 • ^ • 9 i nhau thi hinh chie'u vuong goc ciia chiing bang nhau). H M n,' r>*»A Vs-n W'itv Cho hinh chop S.ABC c6 SA = 3a, duong thang SA hgp voi mat pharig Vay O la tam ciia tam giac deu ABC. (ABC) goc 60". Tarn giac ABC vuong tai B, ACB = 30°, G la trpng tarn ciia tam giac ABC. Hai mat phang (SGB) va (SGC) cung vuong goc voi mat Vi A ABC deu c6 A O = - A H - — . 3 3 phSng (ABC). Tinh the tich khoi chop S.ABC theo a. / • ! LOIGIAI 5 V'-"'" TrongASAOc6 SO = V S A ^ - O A ^ = .La^ -—=t Hai mat phiing (SGB) va (SGC) cimg .i:?^, vuong goc voi mat phSng (ABC). (AS) : 12 1 a^V3 aVTT a W Vay: V = 1 S A B C - S O = 4. Suy ra giao tuyen SG vuong goc voi mat 3' 4 • V3 12 ; phSng (ABC). '" ' Cho khoi chop t u giac SABCD c6 tat ca cac canh c6 dp dai bang a . Ta CO AG la hinh chie'u vuong goc ciia SA ^ a) . Chung minh rang S.ABCD la chop t u giac deu. - ^,, { * « tren mp(ABC). b) . Tinh the rich khoi chop SABCD = LCJIGIAI Suy ra: (SA, (ABC)) = S'AG = 60°. Dyng SO 1 mp(ABCD) n Ta CO SA = SB = SC = SD nen OA = OB = OC = OD. V^y A G = SA.COS60° = — , SG = SA.sin60° ^ 2 2 , Co ABCD la hinh thoi va c6 duong tron ngoai riep tam O nen ABCD la ''*«*0 Xet trong AABC c6 AE - - AG = - . — = — . ,,7 ^ 2 2 2 4 hinh vuong. D Dat AB = X > 0, BC = A B : tan 30° = XN/3 . ' '"^ ' ' ^ Ta c6: SA^ + SC^ = AC^. •- - 3x2 9a 9aV7 — = X=' Nen ASAC vuong tai S nen SO = ^ AE = V A B ^ + BE^ AE = 2 4 2 14 = i.SO.S ABCD 1 aV2 2_a^72 ^S.ABCD HA LJ 3" 2 ~ .Si ( 9aV7^ 243a^ Cho khoi t u di^n deu ABCD canh bang a, M la trung diem DC. .-AB.BC= ^ Vs.ABC - T - ^ ^ ^ - ^ A A B C " 14 ' 112 3 2 2 4 a). Tinh the tich khoi t u di^n deu ABCD. • i ^ Tinh khoang each tii M deh mp(ABC). Suy ra the rich hinh chop MABC. LOT GIAI 290
a). Gpi O la tarn ciia tarn giac A B C suy ra D O 1 ( A B C ) OIISH Taco • 01 ± ( S A B ) . Trong A A B C c6 O C = |ci =^ Va -- Oil AB ;>B, . _ . „ , •>£ i l l ;< Suy ra SI la hinh chieu vuong goe cua r>:;; « Trong A D O C vuong tai O c6 : •2 iM SO tren m|it phSng (SAB). Nen goe giua S O va mat phang (SAB) DO = De-oc^ = la goe OSI = 3 0 ^ . 1 1 aTe a^y/S a^yjl Trong A S O H vuong tai O : O H = SO. tan 30° = b). K e M H // D O , khoang each tu M den m p ( A B C ) la M H .{-Jfidc : i AB.>/3 r Ma H C = 3 H O = hVs . Ngoai ra H C = . A B = 2 : ^ = 2h. 1 „ 1., 1 a^V2 a^yjl 2 ^/3 M H = - D O = ^ V ^ , . A B C = 2VD.ABC = 2- 12 24 The tich khoi chop: Cho hinh chop tarn giac deu S . A B C c6 cgnh day bang a, mat ben hpp voi day goe 60° • Tinh the tich khol chop S.ABC. Vs.ABC - 3 •SO.S^ABC = 3 S O - - ^ = 3 h . - j - = — . , LOIGIAI Cho hinh chop tam giac deu c6 duong cao h va mat ben c6 goe 6 dinh bang Gpi O la tarn eua day thi S O 1 ( A B C ) . ., I 60°. Tinh the tich hinh chop. Gpi H trung diem ciia B C thi C H 1 A B vi A A B C deu. I LOIGIAI Ta c6: < fABlHC •AB1(SHC)=>AB1SH. Gpi O la tam cua day thi S O 1 ( A B C ) . ^ ^= Me A B I S H Gpi H trung diem cua B C thi C H 1 A B vi A A B C deu. Vay ( ( S A B ) , ( A B C ) j = S H O -60°.-^j^^ j Gia s u goe A S B = 60° ASAB la tam giac deu. Dat dp dai doan A B = x > 0. T ' ur^ 1 ur- 1 ^N/S aVs Ta co: H O = - H C = . Suy ra: O B = O C = ^ H C =^ ^^"^ = ^ 3 3 2 6 3 3 2 3 • Trong A S H O c 6 S O = H O . t a n 6 0 ° ^^.S^^. Pitago cho ASOB vuong tai O: 1 l a a V3 a N/3 X = The tich khoi chop: Vg A B C = 3 S O - S ^ A B C = 3 • 2 " ~jr' ' ' SB2 = SO^ + OB^ « . h2 + « x2 = Cho chop tam giac deu c6 duong cao la h hpp voi mpt mat ben mot goe 30". The tich khoi chop: ,. , ^ ^ - . ^ 4-^ m . i . ;^nu,M Tinh the tich hinh chop. \2 LOIGIAI U ..Hi.!-, I', • ^s.ABC - 3 :50.b^ABC = 3 - S O . — - — = - h . Gpi O la tam ciia day thi S O 1 ( A B C ) . D l / BJ D A I H Q C K H O I D 2006 Gpi H trung diem cua B C thi C H 1 A B vi A A B C deu. ^ . „, Cho hinh chop tu giac deu S . A B C D co canh day bang a. Gpi S H la duong ABIHC Ta c6: • AB 1 (SHC) ^ A B 1 SH . cao ciia hinh chop. Khoang each tix trung diem I o i a S H den mat phJing ABISH ben (SBC) bang b. Tinh the tich khoi chop S . A B C D . D v n g 0 1 1 S H tai H . L6I GIAI
Vi S.ABCD hinh chop t u giac deu va SH J. ( A B C D ) . SO „ „ a72 ^ tan (p = — => SO = — . tancp . Nen H la tarn ciia day ABCD. Gpi E trung diem ciia C D . n-is'i > n 1 4 a^-ieb^ 2ab Gpi I trung diem ciia AB. » • SH = HN^ HS^ HE^ HS^ 4b^ a^ 4a2b2 16b^ O la tam ciia day thi S O 1 ( A B C ) . j? 6=^ ) ^ i.c 1 ,2 2ab 2a\ ^, ABICI , , ^S.ABCD - 2 ' ^ A B C D S H - -.a . AB 1 SO ^ mp(SCl) AB 1 SC. Va^-16b^ • 3Va^-16b^ A C O = - C I = va (ABCD) theo cp va tinh the tich khoi chop S.ABCD theo a va cp."'' ' 2 3 3 LOI GIAI 2 -V33 Trong A S O C vuong tai O c6: S O = V S C ^ - O C ^ = Ma^ - ^ - ^ Gpi O la tam ciia day thi SO 1 ( A B C D ) . 3 3 Gpi I trung diem ciia CD, thi O i l C D . 2~_aVTT Ta CO CD 1 mp(SOl) => CD 1 SO. Trong ASCI c6: IH.SC = SO.IC => I H - 2a Vay goc giiia mat p h i n g (SCD) va m^t 1 a>/TT a^yfn .. . .... i'.nii !r' " p h i n g (ABCD) la goc SIO. ' ' = ^' ' ' ' Suy ra S'AABH = i H I . A B = - . ^ ^ . a = 2 2 4 8 Ta CO OD la hinh chieu vuong goc cua 3a^ lla^ a SD tren mat phang (ABCD). Trong ACHI vuong tai H c6 C H - Vci^-IH^ = ^ 16 Vay goc giCia SD va mat phang (ABCD) la goc SDO = (p. , , Taco: SH = S C - H C = 2 a - - = — . 4 4 Trong ASDO vuong tai O: * Ketluan: Vc ARH =-.SH.S S.ABH 'AABH 3' 4• 8 ~ 96
Cho hinh chop tam giac S.ABC c6 AB = 5a, BC = 6a, CA = 7a. Cac mat ben Suy ra: Mat phJing (AMN) song song SAB, SBC, SCA tgo voi day mpt goc 60° .Tinh the tich khoi chop. voi mat phiing (SHC). LOI GIAI Trong hinh vuong A B C D c6: Gpi H la hinh chieu vuong goc ciia S tren mat phSng ( A B C ) , SH 1 ( A B C ) . ABCP = A C D H (c.g.c) nen BJ = q . n DungHE±AB,HFlBC,HJlAC. s Ma B i + ^ = 9 0 " ^ C i + ^ = 9 0 ° z ^ C H l P B . Suy ra S E 1 AB, S F 1 BC, S J 1 AC . BPICH I Ta c6: • BP 1 (SHC) Theode SEH = SFH = SJH = 60°. ' BPISH Vay: ASHF = ASHE = ASHJ => HE = HF = HJ . I =>BP±(AMN):^BP1AM. Suy ra H la tam duong tron npi tie'p AABC. Co M trung diem ciia SB nen: Mat khac ta lai c6: AB + BC + CA 5a + 6a + 7a d ( M , ( A B C D ) ) = l d ( S , ( A B C D ) ) = isH = ^ . P= • = 9a H p i ? n tich tam giac PCN: Va: S ^ B c = Vp(p-a)(p-b)(p-c) » A P C N = T C P . C N = 1^.^ =^ O /! = J9a{9a -5a)(9a -6a)(9a -7a) = 6a2>/6 . 2 2'2'2 8 Ta lai c6: S^gc = P' (^^^ ^ '^'"'^ duong tron npi tie'p AABC). VMPCN = id(M,(ABCD)).S,pcN = = "'"^ 3' 4 • 8 96 2 r Suyra: pr = 6a v6 r = HF = • 9a 3 3 r T u do suy ra chieu cao SH, tam giac vuong SHF: LQAIJI: THE T I C H L A N G TRV ^ SH = H F t a n 6 0 0 = ^ V 3 = 2a>/2. - B A N G J ; K h o i lang try dung c6 chieu cao hay canh day. " ' ^ ^ ^'^ ' Day cua lang try dung tam giac ABC.A'B'C, day la tam giac ABC vuong Vay: Vg ABC = - S A B C S H = -ea^Slayll = ia^.lS = 8a^S can tai A c6 canh BC = a ^2 va biet A'B = 3a. Tinh the rich khoi lang tru. 3 3 c DE THITUYEN SINH D A I HQC K H O I A N A M 2007 Ta CO AABC vuong can tai A nen AB = AC = a. Cho hinh chop S.ABCD c6 day la hinh vuong canh a, mat ben SAD Ki ABC.A'B'C la lang tru dung=> A A ' 1 AB tam giac deu va nam trong mat phang vuong goc voi day. Gpi M, N, P Ian lupt la trung diem ciia cac canh SB, BC, CD. Chung minh A M vuong goc v o i •\B c 6 : A A ' 2 = A ' B 2 - A B 2 = 8 a 2 y. BP va tinh the rich ciia khoi t u d i f n CMNP. =>AA' = 2 a N / 2 . ' " • • • • LOI GIAI Vay V = B.h = SABC .AA' =d?-j2 ^ ,73 Ha SH 1 A D tai H. Vi SAD la tam giac deu nen SH = Vi mat p h i n g (SAD) vuong goc mat phang (ABCD) c6 A D la giao tuyen Suy ra S H l m p ( A B C D ) . ^ho ISng tru t u giac deu ABCD.A'B'C'D' c6 canh ben bSng 4a va duong cheo C g i j m h the rich khoi lang try nay. Ta CO A N // HC va M N // SC. \ If-R,;^ " ,:,V' : n ? i i t l • LOI GIAI Ma A M , M N cz ( A M N ) ; Va HC,SC c (SHC). ' ^
A B C D . A ' B ' C ' D ' la lang tru dung nen: Vi A B C D la hinh thoi c6: S^BCD = T A C . B D = ^.aV3.a = £ . ^ ^ Trong A B D D ' v u o n g tai D c6: 1 ' 2 2 B D = V B D ' 2 - D D ' ^ = 3a . A' Ket luan: V^BCD. A • B • c • D • = A A ' .S ABCD 4a \5a A B C D la hinh vuong nen c6: 0 BD 3a ^ 9a^ "2 2 ' « ' AB=-7=^--p= 'ABCD = AB' = "pflHI TUYEN SINH D A I HQC K H O I D N A M 2008 Cho lang tru dung ABC.A'B'C c6 day ABC la tarn giac vuong, AB = BC = a, Vay: V A B C D . A ' B ' C D ' = A A ' . S A B C D = 4 a . — = 18a . - cgnh ben A A ' = a\/2 . Gpi M la trung diem cua canh BC. Tinh theo a the tich cua khoi lang tru ABC.A'B'C va khoang each giira hai duong thang A M , B'C Cho lang tru dung tarn giac ABC.A'B'C la tarn giac deu canh a = 4 va biet LOI GIAI A dien tich tarn giac A'BC bang 8. Tinh the tich khoi lang tru. The tich ciia khoi lang try: , . 5^ , ^, , ,^.-^ LOIGIAI GQillatrungdiemBC. *^ " V A B C . A B C = AA'.S^A3C = A A ' . i B A . B C By Ta c6: AABC deu nen A I 1 BC va A I = -= 2S. = \aV^.a^ = ^ ' ^ . . . , * ' ! l . ^ 2 2 ' BC 1 A I Goi H trung diem cua BB' thi H M // B'C "? Ta c6: • z^BClA'I;:^ , , • B C I A A (djnh ly ba duong vuong goc ). M Nen d ( A M , C B ' ) = d ( C B ' , m p ( A M H ) ) = d (B', m p ( A M H ) SAABC = - - A ' T - B C ^ A'l = AA BC 2 BC ~ 4 =4 . Qpi h la khoang each t u diem B deh mat phSng ( A M H ) . Tu dien B . A M H c6 B A , B M , B H doi mot vuong goc nen c6 Trong AAIA' vuong tai A: 1 1 1 4 2 AA' = V l A ' 2 - A l 2 = V I ^ = 2 . ^^^.^ •h= BA^ - + BH^ BM^ a^ a^ a^ " 7 Vay: V ^ B C A ' B ' C = A A ' . S A B C = 2 . ^ = 8^3 . Duong thang di qua B va B' c6 giao diem voi mat phang (AMH) la diem H n e n c6: . ^ ( B ' - " ^ P ( A M H ) ) ^ B2i^^ ; : m^ una Cho hinh hop dung c6 day la hinh thoi canh a va c6 goc nhon bang 60". " d(B,mp(AMH)) BH 4*0 1 ; ' ; : Duong cheo Ion cua day bang duong cheo nho cua lang tru. Tinh the ticli hinh hop. d (B', mp ( A M H ) ) = d (B, mp ( A M H ) ) = ^ . D' LOI GIAI Gia sir B A D = 60° thi A C la duong T B I TUYEN SINH D A I HQC K H O I D N A M 2012 cheo Ion cua day, D B ' la duong cheo Cho hinh hop dung ABCD.A'B'C'D' c6 day la hinh vuong, tarn giac A ' A C A' B* ^^ong can, A'C = a. Tinh the tich khoi t u dien ABB'C va khoang each t u nho ciia hinh hop. .,„ ^'.^nA den mat phang (BCD') theo a. Ta CO tarn giac A B D deu nen: B D = a, A C = aVs . ^ -'' Ldl GIAI Theode: A C = D B ' = a V 3 . Xet A B D B ' v u o n g tai B: . ^> tarn giac A'AC vuong can c6: AC = A A ' = ^ = 4 - = — • %/2 N/2 2 * B B ' = V D B ' 2 - B D ^ = V 3 a ^ - a ^ =a>/2. '"i* < '*
Cho lang tru dung tam giac A B C A ' B ' C c6 day A B C la tam giac vuong tai Vi ABCD la hinh vuong c6: A voi A C = a, A C B = 6 0 ° , biet B C h
L6I GIAI A lai mat phang ( A B C ) va ( A B C ) c6 giac Trong (ABC), ke CH 1 A B ( H e A B ) . jyen la B C , hai duang thang A ' B va A B Ian thupc hai mat phang cung vuong goc B' voi giao tuyeh. Nen goc giira hai mat phJing - {cHiAl-"'^^^^^^^'^')' la goc giu'a A B va A ' B chinh la goc nen A ' H la hinh chieu vuong goc ciia A i A ' = 60°. ! A'C tren (ABB'A'). _» \ \ f Xrong A B A A ' vuong tai A c6: A /2a AA' = AB.tan60° =aV3 . H " Do C C // A A ' ^ C C // mp(ABB'A') . « VABC.ABC- = AA'.SAABC =a^^4-^^ • Suy ra: d ( A ' B, CC') = d ( C C ( A B B ' A•)) = d (C, (ABB• A ' ) ) = C H Day ciia lang tru dung tam giac A B C . A ' B ' C la tam giac deu . Mat ( A ' B C ) tao Trong tam giac ABC c6 : S^gc = " AC.BC. sin 120° = voi day mpt goc 30° va di?n tich tam giac A ' B C bang 8. Tinh the tich khoi ling try. \f d r v f , -.M- ... AB^ = AC^ + BC^ - 2AC.BC.cosl20° = 7a^ AB = 3%/? 2.S.^Br- aV2] . Gpi I trung diem ciia BC. • ,\i>rinfii & i Dat dp dai canh day bang a. B' Ketlu^n: d ( A ' B X C - ) = ^ . • i Trong ACHA' c6: A ' C = Co B C l A A ' , B C 1 A I ' ^ a^^^ \ sin 30,0 => BC 1 A ' l (Djnh ly ba duang vuong goc). B •'9 Trong A C H A ' c6: A A ' = N / C A - A C^2 ^ =a Vay goc gitfa hai mat phSng (A'BC) va (ABC) la goc A I A ' = 30°. aV3 aV35 a2^/3 a^VlOS The tich khoi lang tru: V = A A ' S ^ g c = Trong A A I A ' vuong tai A, c6: A ' I = = =a; 14 •rli- .;;tV' cos30° V3 AA' = A l . t a n 3 0 ° = ^ . : ^ = i . ; D A N G 3: Lang tru dung c6 goc giua hai mat phang. 2 3 2 ^rr'- S AABC A A B C= = ^ A ' I i.D(^c;>o .BC8 = = —.a.aa l.a. =16a = 4 . Cho lang tru dung tam giac ABC.A'B'C c6 day ABC la tam giac vuong can tai B voi BA = BC = a, biet (A'BC) h(?p vai day (ABC) mpt goc 60o.Tinh the tich lang try. '"'' , ^ A B C . A B C = A A -SAABC = T - ^ = — T — = 8V3 2 4 8 LOI GIAI B C l AB Cho lang try t u giac deu ABCD A'B'C'D' c6 canh day a va mat phSng (BDC) Ta c6: B C 1 ( A B B ' A " ) => B C 1 B A ' ( B A ' c ( A B B ' A ' ) ) . BCIAA' vai day (ABCD) mpt goc 60°. Tinh the tich khoi hpp tren. LOI GIAI 302
G p i O la t a i n cua h i n h v u o n g A B C D . C" j g ^ N G 4: Khoi lang try xien. T a c o BD_L A C B D I C C C h o l a n g t r y x i e n t a r n g i a c A B C . A ' B ' C c 6 d a y A B C l a t a r n g i a c d e u c a n h a, => B D J . C O ( d i n h l y b a d u o n g v u o n g goc). ^, A' bie't c g n h b e n l a aV3 v a h p p v o i d a y A B C m p t g o c 60°.Tinh t h e t i c h l a n g t r y . V g y goc g i u a m a t p h a n g (C'BD) va m a t L6I G I A I C ( A B C D ) la goc C O C ' = . G p i H la h i n h c h i e u v u o n g goc ciia C T r o n g A C ' C O v u o n g t a i C c6: tren m a t p h a n g ( A B C ) , C ' H 1 ( A B C ) aV2 fz a^/6 V ^ y goc g i i i a C C v a m a t p h a n g ( A B C ) CC' = CO.tan60" = l a goc C C H = 6 0 ° . 'ABCD = = => V ^ B C D - A B C D ' = CC'.SABCD T r o n g A C H C ' v u o n g t a i H c6: ^ ^ V i d u 4: C h o h i n h h o p c h u n h a t A B C D . A ' B ' C ' D ' c 6 A A ' = 2a; m a t ph3ng C'H = CC'.sin60°=aV3.^ = — . , " ( A ' B C ) hop v o l d a y ( A B C D ) m p t goc 60° v a A ' C hop v o l day (ABCD) mot AB^Vs ^a^>/3 g o c 30°. T i n h t h e t i c h k h o i h o p c h i i ' n h a t . T a m giac A B C d e u n e n S ^ g c = 4 4 , , .V LOIGIAI The tich can t i m : V = CH.S ^ 3a a V s _3aV3 T a c6: A A ' l ( A B C D ) n e n A C la h i n h ^AABC 2 • 4 ~ 8 c h i e u cua A ' C tren ( A B C D ) . B- Cho l a n g t r y A B C D A'B'C'D'co d a y A B C D la h i n h v u o n g canh a va biet c a n h b e n b a n g 2a h p p v o i d a y A B C m p t g o c 30°.Tinh t h e t i c h l a n g t r y . V a y goc A ' C , ( A B C D ) = A ' C A = 30' i 2a LOT G I A I A ; , D ' " BC 1 A B Vi: BClmp(ABB'A') G p i H la h i n h c h i e u v u o n g goc cua A ' \ B C l AA' tren m p ( A B C D ) . /6q'!'' BCIA'B. tA^^ogBii'.. Theodebai A ^ A H = 30°. r . 2a TTA- -•*ii'A-:i,a/A V a y g o c g i u a m a t p h ^ n g ( A ' B C ) v a m a t ( A B C D ) l a goc A B A ' - 6 0 ° . sin A ' A H = - ^ H A ' = AA'.sin30''=a; AA' A T r o n g A C A A ' v u o n g t a i A c o : A C = A A ' : tan30° = 2 a : ^ = 2 a V 3 . S^BCD = ^ (ABCD hinh vuong). A ^ABCD.A'B'C'D' - ^ • ^ • S A B C D = a" ^ 0 2a\/3r r T r o n g ABA A ' v u o n g tai A c o A B = A A ' : t a n 60 =2a:V3=— C h o h i n h h o p A B C D . A ' B ' C ' D ' c o A B = a, A D = b , A A ' = c v a B A D = 3 0 ° va T r o n g A A B C v u o n g t a i B c6: t f - .„j b i e t c a n h b e n A A ' h p p v o i d a y A B C m p t g o c 60°. T i n h t h e t i c h l a n g t r y . LOI GIAI 2a V3' ^ ^ 4aN/6 'ii>', ; G p i H la h i n h c h i e u v u o n g goc cua A ' BC = V A C 2 - A B ^ = (2aV3) - tren m p ( A B C D ) . • ^ 2aVs 4aN/6 \bd^4l Theodebai: A ^ ^ = 30°. The tich h i n h hop: V = A A ' . A B . A D = 2 a . — = ^ — smA'AH =l ^ ^ H A ' = AA'.sin60°=£:^ AA' 2 304
ab 'ABCD - 2S^BCD = 2.-CB.CD.sinBCD = BC 1 A H BC ± m p ( A H I A ' ) => BC ± H I abc>/3 B C l A'O ^ A B C D . A ' B C ' D ' - HA .S^gCD ' "^//^^'=.BciBB'. • ;:• A"'^>^-:.. Cho lang tru tam giac ABC A'B'C c6 day ABC la tam giac deu, va diem A' HIIBC '• ' .73 each deu ba diem A, B, C bie't A A ' = , AA' hop voi mat phang day Vay: BCC'B' la hinh (Mx nhat. > , BC = ( A B C n B C C ' B ' ) ; (ABC) mQt goc 60°. Tinh the tich lang tru. BCl AH; BCl HI. , v T i i A .,. '„ LOIGIAI Goi O la tam ciia day (ABC). s ; V^y: [(ABC),(BCC'B') = IHx = 60° = A ' A O Ta C O OA = OB = OC (tinh chat tam giac deu ). A'O sin A ' A O = A ' O = AA'.sin60° = ^^ Theo de bai c6 A'A = A'B = A'C. AA' ' Vay O la hinh chie'u vuong goc ciia A' tren mp (ABC). AO la hinh chie'u 'AABC vuong goc ciia AA' tren mat phang ^ 3a 3a^sf3 (ABC) Theo de bai yVAO = 60°. ^ A B C A B C -O-^'-^ABC " 2 • 4 8 Trong tam giac A'AO vuong tai O c6 : Cho lang tru ABC A'B'C c6 day ABC la tam giac deu voi tam O. Canh ben sin^VAO = — ^ O A ' = AA'.sin60'' = a ; CC = a hop voi day ABC mpt goc 60° va C c6 hinh chie'u tren ABC trimg AA' voi tam O. ' " ' ' ' OA cos A ' A O = •OA = AA'.cos60°= — Chiing minh rang AA'B'B la hinh chu nhat. Tinh di^n tich AA'B'B. -KJ'"' AA' Tinh the tich lang tru ABC.A'B'C. Trong tam giac ABC deu c6: AO = - A H = - . " ^ ^ ' ' ^ = => AB = a . V , LOI GIAI Goi I trung diem ciia A'B'.'H trung diem ciia AB. AB^Vs a^Vs IH//B'B JIH//CC' 'AABC ^ABCABC -OA'.SAg(- - a. Suy ra: I H = B'B ^ [ I H = CC' • ^ Cho lang tru ABC A'B'C c6 day ABC la tam giac deu canh a, dinh A' c6 hinh Vay: t u giac C H I C la hinh binh hanh. chieu tren (ABC) nam tren duong cao A H cua tam giac ABC bie't mat ben BB'C'C h^p voi day ABC mot goc 60°. ABICH Ta c6: • A B I O C AB I m p ( C H I C ) a). Chung minh rang BB'C'C la hinh chii nhat. b). Tinh the tich lang tru ABC.A'B'C. A B I H I LOI GIAI Ma JHW/BB'^ ABIBB'. Gpi O la hinh chie'u vuong goc ciia A' tren mp(ABC). Goi I trung diem ciia B'C HI 1 AB Ta c6: H trung diem ciia BC. Vay: ABB'A' la hinh chCr nhat. I H / / B'B IH//A'A C O la hinh hie'u vuong goc ciia C C tren m p ( A B C ) . Suy ra: 'A A I H = B'B IH = A'A Vay: [ C C ' , ( A B C ) 1 = C ' ^ = 60° . „, Vay: A H I A ' la hinh binh hanh. 306
Theodebai C ' 0 ± m p ( A B C ) . oc Trong ACOC' vuong tai O c6: sin OCC' = =:> OC' = CC'. sin 60 = Hai hinh binh hanh BCCB' va A C C A ' bang nhau, c6 chung canh CC. cos OCC' = — ^ O C = CC•.cos60''=- Ha B L l C C ' t h i A L I C C . CC• 2 ' ( B C C B ' ) , ( A C C A ' ) ] = ALB = 90° . 1CH CO = ^ CH = 1 = f ^ ^^ = = ^^ ^^ A C O . V=3 = ^ A BB == CO.V^ 3 3 2 V3 2 ^ , . „ aVs ay/s a Taco: A H = . Di?n tich hinh chii nhat ABB'A': S^gBA' = AB.BB' = -.a = 2 2 ' 2 Trong tam giac vuong A H A ' c6" AB^N/3 33^73 VSa 3a^S 9a^ A H ^73 ' VABC.ABC =OC'.SABC = cosA'AH = A ' A H = 30 'AABC 16 16 32 AA' 2 Cho lang tru ABC A'B'C c6 day ABC la tarn giac deu canh a biet chan A'H , sin A ' A H = duong vuong goc ha tu A' tren ABC triing voi trung diem ciia BC va AA' = a, AA' V. Tim goc hpp boi canh ben voi day lang tru. Tinh the tich lang tru. A'H = AA'.sin30°=- 2 ~. 7. LOIGIAI ~ Gpi H la trung diem cua BC. 'AABC 4 U''^ \ir,,i; r'i'..} r w o l i' ,i: -/ic :'>'^'' Theo de bai A ' H 1 m p ( A B C ) . A H la hinh chieu vuong goc ciia A A ' tren a a^S a^S • 111) CL'. r / v h •,,n:yh 1 > 'ABC . A B C -HA'.SABC = mp(ABC), nen goc AA',(ABC) = A ' A H Cho hinh hop ABCD.A'B'C'D' c6 6 mat la hinh thoi canh a,hinh chieu a vuong goc cua A' tren(ABCD) nSm trong hinh thoi, cac c?inh xuat phat tu Tac6:AH = ^ A ciia hpp doi mpt tao voi nhau mptgoc 60° . r u , / ,iaf. r . (duong cao ciia tam giac deu). a) . Chung minh rang H nam tren duong cheo AC aia ABCD. 1 b) . Tinh di^n tich cac mat cheo A C C A ' va BDD'B'. , "' Trong lam giac vuong AHA' c6: c) . Tinh the tich ciia hpp. ' ?• A'AH AH S . A ' A H = 30". COS d) . Tinh goc giua hai mat phang (AA'D'D) va mat phing (ABCD). AA' .fS.,l(.i/» . " 1 * sin A ' A H = — A ' H = A A ' . sin 30" - J LOI GIAI AA' 2 a) Theo de c6: D A A ' = BAD = BAA' = 60° a a ^.\ABC ^ = — ^ ' VABC.ABC - " ^ •^ABC-2-~^ 8 nen ba tam giac deu ADAA' = ABAA' = ABAD (c.g.c). - - -:r-'" c - • Cho lang try xien ABC A'B'C c6 day ABC la tam giac deu voi tam O. H" Suy ra A ' A = A ' B ' = A ' D chieu cua C tren (ABC; la O.Tinh the tich ciia lang try biet rSng khoi' each tu O den C C la a va 2 m | t ben (AA'C'C) va (BB'C'C) hgp voi nhau n> => H A = HB = H D . Vay H e AC va H la trpng tam ciia A BAD^ goc 90° LOI GIAI ^) Taco: A H = 2 A o = 2 . i : ^ = ^ , 3 3 2 3 Goi H la trung diem cua BC.
AC = 2AO = aVa Goi O = A C o B D , theo de B ' O J. m p ( A B C D ) . a^- V i A B C D h i n h t h o i c6 A = 60° nen A A B D deu c6 T r o n g A H A A ' v u o n g t ^ i H c6: H A ' = V A A ' ^ - A H ^ = , ^ V 3 , BD = a = ^ B O = | , A O =^ = ^ A C = aV3. D i ^ n t i c h h i n h b i n h hanh A C C A ' : S , C C A ' = H A ' . A C = ^ . a V s = a^ Vz . a) . T i m goc h o p b a i canh ben va day. Ta CO O B la h i n h chieu cua BB' tren m a t BDIAC Ta c6: < , ^ BD 1 m p ( A C C V A ' ) => B D 1 A A ' , m a A A ' // BB' . phang ( A B C D ) . Goc g i u a BB' va m a t B D I H A phMng ( A B C D ) la goc O B B ' . Vay: B D 1 BB' T r o n g A OBB' v u o n g tai O c6 : Suy ra t u giac BDD'B' la h i n h v u o n g SgoDB' = = • ^ * a t'Jv*^ t ' ^ ' ' 2 /— cos OBB • = — = 2. = 1 OBB' = 6 0 ° . c) D i ? n tfch cua h i n h t h o i A B C D : S^BCD = ^ A C . B D = -.aV3.a = • BB' a 2 A V i A B C D . A ' B ' C ' D ' la h i n h h p p n e n goc h p p b a i cac canh b e n v a m a t day Tu-^uvu-w^ V HA'Q aV6 a^Va a^Vz v deu bang n h a u va bang 60°. The tich k h o i h p p : V = H A .S^gCD = — —= — — b) T i n h the tich v a t o n g dien tich cac m a t ben ciia h i n h h p p . * '" d) Day c h i n h la bai toan t i n h goc g i u a m a t ben va m a t day, cac ban n h o xem lai p h u o n g p h a p t i m goc 6 bai goc g i i i a hai m a t phang. OB' = OB.tan60° = — , '' ' S''^.*' W"'