Phutnig phdp gidi Todn Htnh hoc theo chuyen de - Nguyen Phu Khdnh, Nguyen Tat Thu Jiuang ddn gidi Goi I la trung diem ciia AD . S AD Ta CO CI = lA = ID = — suy ra AACD vuong tai C => CD 1 AC (1) SA1(ABCD)=>SA1CD (2). Tu (1) va (2) suy ra CD 1SD => ASCD vuong. B C Goi d^;d2 Ian lugt la khoang each tit B,H den mp(SCD) Ta c6: ASAB - ASHA SA SB SH SA^ 2 SH d, 2 ^ 2^ a 2 3 1 SH SA SB SB^ 3' SB dj 3 The tich khoi tu dien SBCD: VSBCD = -SA.iAB.BC = 3 2 6 Ta c6: SC = N/SA^ + AC^ = 2a, CD = Vci^TlD^ = Via SscD = ^SCCD = yfla^ . 3. >/2a3 Ta c6: V< SBCD - g'll-SsCD-^"1 - 2 Vay khoang each tie H den mp(SCD) la dj = 3 Bai 2.3.6. Cho hinh lang try diing ABCAjBiCi c6 AB = a, AC = 2a, AAj = 2aV5 va BAC = 120*'. Gpi M la trung diem cua canh CCj. Chiing minh hai duong thang MB va MAj vuong goc voi nhau. Tinh khoang each tu diem A den m^t phang (AJBM) . Jiuang ddn gidi Taco: BC = VAB^ + AC^ -2AB.AC.eosl20° = ayf? BM = VBC^TCM2 =2>/3a; AjM = ^AjCj^ + C^M^ =3a BAj = 7AB^+AA7 = \/2Ta; BM^ + A^M^ = BA^^ Suy ra MB 1 MAj. Ke CH1AB CH1 (ABAj). Ta CO CH = AC.sin60° = >/3a . 142 Cty TNHH MTV DWH Khang Viet fhe tich kho'i tu di^n M.ABA : V„ ,=icHlAB.AA,=:fi. Qgi K la hinh chieu cua A len mp{AiBM) ta CO: VM.ABA, =^AK.1MB.MAI =>AK = 3 2 MB.MA Vay khoang each tu A den (A^BM) la AK = Bdi 2.3.7. Cho hinh chop S.ABC c6 goc giiia hai mat phang (SBC) va (ABC) bang 60°, eae tam giac ABC va SBC la tam giac deu canh a. Tinh theo a khoang each tu B deh mp(SAC) . Jiuang ddn gidi Gpi M, N Ian lugt la trung diem cua SA, BC . Ke BHlCNtai H suy ra BH la / / \ v : khoang each tu Btoi mp (SAC) Ta CO SNA = 60° la goc giiia hai m|it phing(SBC),(ABC). Tam giac SAN deu c^nh SA = AN = Taco CM = VCA2-AM2 = >/l3a ;MN = VCM2-CN2=^ C Ta c6: BH.CM = MN.BC =:> BH = MN.BC sViSa CM 13 ^^i 2.3.8. Cho lang try diing ABCAiB^Cj c6 tat ca cac canh deu bang a, M la '^ng diem cua AAj. Chung minh BM 1 B^C va tinh khoang each giua hai •^"^ing thang BM va B^C. Jiuang ddn gidi , ' Gpi E la diem tren duong thSng AAj sao cho Aj la trung diem cua ME, taco BM/ZBjE. JiC = V2a,BiE = BM = VBA2+AM2 CE = VCA2+AE2 = ^ y/l3a •i.> • Phumtg phdpgiai Todtt Hinh hgc theo chuyen de- Nguyen Phu Khanh, Nguyen Tai Thu B^C^ + BjE^ = CE^ => BjC 1 BjE => BjC ± BM * Taco BM//(BjCE)^d(BM,BiC) = d(M,(BiCE)) Gpi H la trung diem cua AjCj ta c6 BHl(ACqAi) The tich cua khoi chop Bj .CME : V = -B,H.S. B].CME - 2 •'l^-^CME 1 Vsa 1 VSa^ = —. .—a.a = 3 2 2 12 Goi I la hinh chieu ciia M len mp(BjEC) ta c6 : 3Vp r-k^c TsOa V, = -MI.S 'B,CE >MI = - Bi.CME ^BjCE 10 V^y khoang each giCra hai duong thang BjC,BM la MI = 10 Bai 2.3.9. Cho lang try dung ABC.A'B'C c6 day ABC la tarn giac vuong, AB = BC = a, canh ben AA' = aV2 . Gpi M la trung diem ciia canh BC. Tinh theo a the tich cua khoi lang try ABC.A'B'C va khoang each giiia hai duong thang AM, B'C. Jiuang dan gidi Tu gia thiet suy ra tam giac ABC vuong can tai B. B" . The tich kho'i lang try la: VABCA'SC = AA'.SABC = (<Jvtt). GQi E la trung diem cua BB'. Khi do B'C / /(AME) Suy ra d(AM,B'C) = d(B'C,(AME)) ^ = d(C,(AME)) = d(B,(AME)) Gpi h la khoang each tu B den mat phang (AME). Do tu di?n BAME c6 BA, BM, BE doi mot vuong goc nen: 1 1 1 —- + 1 1 7 . aV? - = ^=>h = h^ BA^ BM^ BE^ h^ 7 V|y khoang each giira hai duong thing AM va B'C la a^/7 Bai 2.3.10. Cho hinh chop tu giac deu S.ABCD c6 day la hinh vuong canh A Gpi E la diem doi xung ciia D qua trung diem cua SA. M la trung diem ciJ^ 144 ^g, N la trung diem cua BC. Chiing mirJi MN vuong goc vai BD va tinh (theo 3) khoang each giiia hai duong thing MN va AC. Jiuang ddn gidi ? ^ Gpi P la trung diem ciia SA. •-: ; ;, , I Ta CO MP la duong trung binh cua tam giac EAD =>MP//AD=>MP//NC. H Va MN = |AD = NC. I Suy ra MNCP la hinh binh hanh => MN //CP =^ MN //(SAC). Ta de ehung minh dugc BD1(SAC)=^BD1MN Vi MN//(SAC) nen: d(MN,AC) = d(N,(SAC)) = -d(B,(SAC)) = -BD= ^ Vay d(MN,AC) = ^f2a Bai 2.3,11. Cho hinh chop SABC c6 tam giac ABC vuong can tai B, AB = BC = 2a, (SAB) va (SAC) eiing vuong goc voi (ABC). Goi M la trung diem AB, mat phang qua MS song song voi BC cat AC tai N. Biet goc giiia (SBC) va (ABC) bang 60°. Tinh the tich khoi chop S.BCNM va khoang each giiia hai duong thang ABvaSN. Jiuang dan gidi ^ Do hai mat phang (SAB) va (SAC)cat nhau theo giao tuyen SA va cung vuong goc voi (ABC) nen SA 1 (ABC), hay SA la duong jcao cua khoi chop S.BCNM. Ta CO: SBCNM = SABC - SAMN = 2a2 MA.MN = 2a2-ia2=^ ' Do BCIAB (SAB)lBC. BC 1 SA Nen SBA chinh la goe giira hai mat phang (SBC) va (ABC), the thi theo i gia thiet ta c6 SBA = 60°. 145 Trong tam giac vuong SAB ta c6 SA = AB tan 60^ = 2a>/3 . Vay Vs.BCNM =^SA.SBCNM =i.2a>/3.^ = V3a3(dvtt) Goi P la trung diem cua BC thi AB / /NP, AB (2 (SPN) nen AB / /(SPN) do do d (AB, SN) = d (AB; (SPN)) = d (A; (SPN)) Tir A ha AElNP,EePN thi \> PN1 (SAE) ;ha AHISE thi [PNISA V ^ • AH 1 (SPN) =^ d(A;(SPN)) = AH. Taco AE = NP = a;SA-2aN/3=>—^ = -^ + -i- = ^^=>AH = aJ AH^ AS^ AE^ 12a^ V '12 13 Vay d(A;(SPN)) = a^ Bai 2.3.12. Cho lang try ABCD.AiBiCiDi c6 day ABCD la hinh chii nhat, AB = a, AD = a\/3 . Hinh chie'u vuong goc ciia diem Ai tren mat phang (ABCD) trung voi giao diem AC va BD. Goc giiia hai mat phang (ADDiAi) va (ABCD) bang 60°. Tinh the tich khoi lang tru da cho va khoang each tir diem Bi den mSt phSng (AiBD) theo a. Jiit6m.gddngi.di ^, Gpi O = AC n BD, I la trung diem canh AD. Tac6ADl(AOI) • Alio = ((ADDiAi),(ABCD)) = 60° Vi OI = - , nen ta suy ra Ajl = 201 = a ^AiO = OI.tan60° = ^. Do do VABCD.A,B,CID, = AJCSABCD = ^ ^ Gpi Bj la hinh chie'u ciia Bj xuong mgt phSng (ABCD) Do BjC / /AJD =^ BjC / /(AJBD) ^ d(Bi,(AiBD)) = d(C(AiBD)) = CH Trong do CH la duong cao ciia tam giac vuong BCD Ta co: CH = , = — V^y d Bi,(AiBD) = — VCD^+CB^ 2 2 146 Cty TNHH MTV DWH Khang Viet 0di 2.3.13. Cho hinh chop S.ABC c6 day ABC la tam giac vuong tai B, BA = 3a, = 4a; mat phang (SBC) vuong goc voi mat phang (ABC). Bie't SB = 2a73 va ggC= 30" . Tinh the tich khoi chop S.ABC va khoang each tir diem B deh mat phang (SAC) theo a. s " '• ' Jiic&ng dan gidi Goi H la hinh chie'u cua S xuo'ng BC. Vi (SBC)l(ABC) Aen • SHl(ABC).Tac6 SH = aV3. Do do Vs.ABCD = ^SH.S^Bc = 2a373 . Ta CO tam giac SAC vuong tai S I" Vi SA = ar/2T,SC = 2a, AC = 5a Til ••tfyt.li' va S^AC = a^>/21 nen ta c6 duoc d(B,(SAC)) = -j^. ^ASAC v7 Bai 2.3.14. Cho hinh chop S.ABCD c6 day ABCD la hinh vuong canh a. Goi M va N Ian luot la trung diem cua cac canh AB va AD; H la giao diem cua CN va DM. Bie't SH vuong goc voi mat ph5ng (ABCD) va SH = aS. Tinh the tich khoi chop S.CDNM va khoang each giiia hai duong thiing DM va SC theo a. Jiuang ddn gidi Taco: VgcDfji^ =-SH.S^[^DC SMNDC = SABCD - SAAMN - S/^MBC 2 a^ a^ 5a^ = a = . 8 4 8 Nen ^S.CDNM 8 24 (dvtt). Laithay: DM.CN = i(2DA-DC).i(2DC-DA) = DA2-DC^ =0. 2V— / 2' V^y CN 1 DM h:r do SC 1 DM bai vay: d(SC;DM) = d(H;SC)=^^'^"SC-^"-^^ SH.CH SC SC VsH^+CH^ 147 Phumig phiifigidi Tiu'ui Ilinh hoc theo chuyen de - Nguyen Phti Khiinh, Nguyen TA't Thu DM '^is Laico: CH = ^^^^ DM Hay ta c6 khoang each can tinh la: ^ay— • Bdi 2.3.15. Cho hinh lap phuong ABCD.A'B'C'D' c6 canh b^ng a . Goi M,N Ian lugt la trung diem cua AB va B'C. Tinh khoang each giCra hai duang thing AN va DM. JIuang ddn gidi 1.1 Gpi E la trung diein ciia BC . Dethay AADM = ABAE nen AMD = AEB, ma AEB + BAE = 90' ,0 D :^ AMD + BAE = 90" => DM 1 AE. Lai CO EN 1 (ABCD) => EN 1 DM dodo (AEN)lDM tai I. f / 7 \ E \ \ \ \ ; \ \ \ \ \ • \ \ N D' Xet phep chieu vuong goc len (ANE), ta c6 AN chinh la hinh chieu cua no nen d(DM,AN) = d(l,AN) Goi K la hinh chieu cua I tren AN thi d(l,AN) = IK. Taco AAKI-AAEN,suyra i|^-^r^IK = ^^^ (l) EN AN AN^ = AE^ + EN^ = AB^ + BE^ + EN^ = 1 AN AN = —. 4 2 1 1 AI^ AD^ AM^ a^ Thay vao (l) ta duac IK - 14 5 aS + — = — => AI = 15 Vay d(DM,AN) = 2aS 15 148 Cty rNIUI MTV DWII Khang Viet § 4. THH TIC:H KHOI DA DlfiN , . De tinh the tich riui mot khoi dn dii'n (lang try va hinh chop) ta thuang thuc jiien theo cac cacii snii Qdch //Tinh trirc tiep , ^^^•''"^'^ Su dung cac cong thlie: /iO :. < , • , The tich khoi chop: V = -h.S^^, trong do h la chieu cao, Sj la dien tich day. 3 Dacbiet: Ne'u hinh chop S.ABC c6 SA,SB,SC doi mot vuong gck thi: Vs.ABC=^SA.SB.SC. •^•-^'^^ • The tich khoi lang tru: V = h.Sj, trong do h la chieu cao cua lang try, la dien tich day. Dacbiet: +) Hinh hop chi> nhat ba canh a,b,e : V = abe +) Hinh hop lap phuong canh a: V = a'' /lift,* ' ' Cdch 2. Tinh gian tiep. • Ne'u hinh H duoc taeh thanh hai hinh roi nhau Hj, H2 thi V,]^ = Vj^ ~^H2 • Tren cac duong thang SA, SB, SC ciia hinh chop S.ABC ta lay Ian lugt cac . , „, ^, , SA'.SB'.SC',, diem A ,B ,C . Ta co: VS.A'WC = g^SBSC ^'^"^ " Chijy:Khi xet ti so the tich ciia hai khoi chop thi ta thuong tim each chuyen ve hai khoi chop c6 ehung mat phang day. Vidu 2.4.1.Cho hinh chop tu giac deu S.ABCD . Tinh the tich khoi chop biet 1) Canh ben bang a\/5 va mat ben tao vol day mot goc 60" 2) Duong cao cua hinh chop tao voi day mot goc 45" va khoang each giira hai duong thang AB va SC b^ng 2a. Goi O la tarn cua day, ta c6 SO 1 (ABCD) suy ra : Vg yi^g,;^ = -^SCSy^gco J[ffi gidi. L (ABCD; 1) Goi M la trung diem CD, ta c6: CD 1 (SMO) Do do goc SMO la goc giua mat ben voi mat day, nen SMO = 60° I Dat AB = 2x => MO = x,OC = xV2 Trong cac tam giac vuong SOC,SOM ta c6: A=:;Ar^ ii; SO^ = SC^ - OC^ - 5a2 - 2x^ SO = OM. tan 60" =xS , ,, • ., 149 Phuang phdp gidi Todn Hinh hoc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tai Thu Nen ta c6 phuong trinh: 5a^ - 2x^ = 3x^ =^ x = a Vay Vs.^3CD 4'<^-(2x)^ =^x^ =^a3. 3 3 2) Goi K la hinh chieu ciia O len AM, ta CO OK ± (SCD) nen OSK la goc giiia duong cao SO vol mat ben nen OSK = 45° . Goi N la trung diem AB. Do AB//(SCD) =^ d(AB,SC) = d(AB,(SCD)) = d(N,(SCD)) = NH = 2a Trong do HN //OK => OK = ^NH = a 1 2 Cac tarn giac SKO,SOM la cac tam giac vuong can nen ta c6 SO = OKV2 =a>/2, OM = SO = a72 VayVs.^3CD=^aV^(2aV^f=^. Vi du 2.4.2. Cho hinh chop S.ABC c6 day ABC la tam giac vuong AB = a, AC = aVs , SA 1 (ABC). Tinh the tich cua khoi chop S.ABC cac truong hop sau 1) Mat phang (SBC) tao vai day mot goc 60° 2) A each mat phang (SBC) mot khoang bang 4 tai A, trong Xgigidi. ,2 a'Vs Ta CO BC = 2a, S^pc = - AB.AC = ^ va V< S.ABC = isA.S AABC SA 1) Goi K la hinh chieu cua A len BC, taco BCl(SAK). Suy ra SKA = ((SBC), (ABC)) = 60°. Taco: AK='^^^ = ^ BC 6 nen SA = AK. tan 60° = |. Vay Vg ^BC = ^2 2) Gpi H la hinh chieu cua A len SK, ta c6 AH L (SBC) 150 Cty TNHH MTV DWH Khang Viet Trong tam giac SAK, ta c6: 1 1 AH^ SA^ AK' .,3 Vay Vs,ABC = • SA^ AH-^ AK^ a-^76 12 2 fit vcnoVT' X^i du 2.4.3. Cho hinh chop S.ABCD c6 day ABCD la hinh thang vuong tai va B, AB = BC = a, AD = 4a. Tam giac SAD la tam giac deu va nam trong 0iat phang vuong goc voi day. Mat phang (SCD) tao vai day mot goc 60° . Tinh the tich cua kho'i chop S.ABCD theo a . Xffigidi. Goi H la trung diem doan AD, ta CO SH1 AD ^ SH1 (ABCD) Goi K la hinh chieu,cua H len CD, ta CO CD 1 (SKH). Suy ra SKH la goc giii-a mat phang (SCD) voi mat day, do do SKH = 60° Goi E la hinh chieu aia C len AD, suy ra ABCD la hinh vuong canh a. Ta c6: CD = VCE2+ED2 =aylw Do ACED ~ AHKD nen ta c6: HK KD CE DE JVTO =*HK = KD.CE DE 3a Suy ra SH = HKtan60° - Vay the tich khoi chop la: VS.ABCD = ^SH.SABCD ^ ^^-^ • du. 2.4.4. Cho hinh chop S.ABCD c6 day ABCD la hinh vuong canh a, SA vuong goc voi day. Mat phang (SBD) tao voi day mot goc 60° . Goi M, 1^ Ian lugt la hinh chieu cua A len SB, SD. Mat phang (AMN) cat SC tai P. Tmh the tich khoi chop S.AMPN . JCsngidi. Goi O la tam ciia day, ta c6 BD1 (SOA) Phtfcmg phdpgiiii Toan Hinh hoc theo chuyen de- Nguyen Phu Khanh, Nguyen Tat Thu suy ra goc SOA la goc giiia hai mat 5 phing (SBD) va mat day nen SOA = 60°. /Vs. Trong tam vuong SAO ta c6: / j \Vr\,. SA:.AO.tan60" ==^.73 = ^. / ]^»\ BC ± AB / A ' ' ^ ^ \ BC ± AM => AM 1 (SBC) ri> AM 1 SC / ~'X^=zz-\ V-' Tuong tu: AN 1 (SCD) => AN 1SC, '- '"'o'^^ tu do suy ra; SC1 (AMN) ^ C Nen AP la duong cao cua hinh chop S.AMPN Suy ra: Vg^^^p^ = - AP.S^^^,,^, Ap dung he thiic lugng trong tam giac vuong SAC ta c6: SP ^ SP.SC ^ SA^ ^ 3 _ gp^ 3g^^ 3aVT4 SC" SC^ SA^+AC^ 7^ 7 14 ^ r-AT, SA.AB as!\5 Trong tam giac vuong SAB ta co: AM = = SB 5 Acnr^ .cn^^ MP SP SP.BC SaTsS Do ASBC ~ ASPM => = — => MP = = BC SB SB 35 a2V2T Suy ra: S^MPN = '^^.^AMV = AM.MP = 35 1 3aVl4 3^721 3a^V6 3 14 35 70 Vi du 2.4.5. Cho hinh chop SABCD c6 day ABCD la hinh vuong tam O, SA vuong goc voi (ABCD), AB = a,SA = aV2. Gpi H,K Ian lupt la hinh chieu vuong goc cua A tren SB, SD. Chung minh: SC 1 (AHK) va tinh the tich cua khoi chop OHAK theo a . . JCffigidi . Ta CO : BC 1 (SAB) =^ BCl AH ma AH 1 SB ^ AH 1 (SBC) AH 1 SC . Tuong tu AK1 SC SCI(AHK). SH SH.SB SA^ 2a^ 2 SK SK.SD SA^ 2a^ 2 Do SB SB2 SB^ 3a2 3 SD SD^ SD^ 3a^ SH SK „^ / /no > 2 „^ 2 /r 2^f2a — = — => HK / /BD va HK = - BD = -av2 = . SB SD 3 3 3 Cty TNIIIl MTV DWII Khaug Viet Gpi G la giao diem cua SO vh KH thi G la trung diem ciia KH, ma AH = AK = ^a ^ AG 1 HK . De thay G la trpng tam cua ASAC: nen AG = - AM = IsC = — 3 3 2 3 (M la trung diem cua SC). Vay SAHK=^AG.HK ^ 1 2a 272a ^ 2V2a^ 2' 3 • 3 ~ 9 -f-s u! < .H-,:;:, Gpi I la trung diem ciia AM, ta c6 OI / /CM => OI1 (AHK) ' ' ' ' . CM SC a c 1 la 2723^ 723^ va OI = — = — = Suy ra VQ.AHK = 3 OI-S^HK = 3' 2= • Cdch 2: Gpi E la hinh chieu ciia A tren SO thi AE 1 (OHK) nen AE la duong cao cua hinh chop A.OHK Ta c6: 1 1 - + • 1 2 5 - + —= • AE^ AS^ ' AO^ 2a^ ' a^ 23^ AE = a.,- 'SHK SgoH _ BH.BO 1 BH.BS 1 BA^ 1 'SBD BS.BD 2' BS^ 2BS2 6^^^°" 1 2 Tuong tu SpoK ~ ^ ^OHK = SgBo - (SgHK + SBOH SDOK ' Ma SsBD - -SO.BD = i VAS^TAO^" .BD = -,23^ + —a72 = SBD 2 2 2V 2 2 , : =*S OHK U^c 1/2 a^Vs a^72 ^ • V3y V.oHK = 3 AE.S0HK = 3^5-— = 3^75 Pi du 2.4.6. Cho hinh chop S.ABC c6 C3c canh day AB = 53, BC = 63, AC = 7a. Cac m3t ben t3o voi dsy mot goc bang nhau V3 bSng 60°. Tinh the tich khoi chop S.ABC V3 tinh khosng each tir A deh mat phSng (SBC). Bie't hinh chieu cua dinh S thupc mien trong t3m gidc ABC. ____ 153 Phuattg phapgiai Todn Hinh hoc theo chuyen de- Nguyen Phii Khanh, Nguyen Tat Thu GQ'I I la hinh chieu vuong goc ciia S tren (ABC), A',B',C' Ian lugt la hinli chieu cua I tren BC,CA,AB . Tu gia thie't suy ra SAl = SB^I = SCI = 60° . Cac tam giac vuong SIA',SIB',SIC' bang nhau nen lA' = IB' = IC => I la tam du6n» tron noi tiep tam giac ABC . Goi p la nua chu vi tam giac ABC 5a + 6a + 7a =^p = - = 9a SAABC = VP(P-BC)(P-AC)(P-AB) = 79a(9a -6a)(9a -7a)(9a -5a) = sSa^ Goi r la ban kinh duang tron noi tiep tam giac ABC, ta c6 : ^ r _ SABC ^ 6V6a^ ^ 2^a p 9a 3 276 a SABC = Pr • =^IA' = r = Taco: SI-IA'tan60° =^^^3 =272a 3 Suy ra Vg^gc = ISLS^BC = ^2V2a.6^/6a = sVSa^ . Vi du 2.4.7. Cho hinh chop S.ABCD c6 day ABCD la hinh thoi canh a, SA = SB = SC = a. Tinh SD theo a de khoi chop S.ABCD c6 the tich Ion nha't. Xffigidi. Goi H la hinh chieu ciia S len mat day, ta suy ra H la tam duang tron ngoai tiep tam giac ABC nen H thuQC BD. Mat khac •g^^^^^ACl(SBD)^0 = BDnAC la hinh chieu cua A len mat phang (SBD), ma AS = AB = AD = a => O la tam duang tron ngoai tiep tam giac SBD =i> ASBD vuong tai S. Dat SD = x Ta c6: SH.BD = SB.SD => SH = va S^BCD = | ACBD Nen V. S.ABCD JACBD = iAB.SD.OA MaOA^=AB^-:52l = a^-^ 3a2-x2 154 Cty TNHH MTV DWH Khang Vigt Dodo: Vs^ABCD = —.a.x.Vsa^ - x^ y\ dung bdt Co si ta c6: Suy ra: V< S.ABCD ^ g -^^ 2 x2+3a^-x^ Sa^ 1 3a' Dang thiic xay ra <=> x = 3a - x <=> x = Vay ABCD ^^^^ <=> SD = 't v ' Vi dii 2.4.8. Cho hinh chop S.ABCD c6 day ABCD la hinh thang vuong tai A va D, tam giac SAD deu c6 canh bang 2a, BC = 3a. Cac mat ben tao vai day cac goc bang nhau. Tinh the tich cua khoi chop S.ABCD . J!gi gidi. GQI I la hinh chieu vuong goc cua S tren(ABCD), tuong tu nhu <i du tren ta Cling CO I la tam dudng tron npi tiep hinh thang ABCD. Vi tu giac ABCD ngoai tiep nen AB + DC = AD + BC = 5a Dien tich hinh thang ABCD la S = i( AB + DC) AD = 5a.2a = 5a^ 2 2 ^ Goi, p la nua chu vi va r la ban kinh duong tron noi tiep cua hinh thang ABCD thi AB+DC+AD+BC /" /'I /'I /'I J'' 1 I' 1 V 10a ^ . _ S = 5a va S = pr => r = — = 2 ^ p 5a^ 5a \ Tam giac SAD deu va c6 canh 2a nen SK = ^ = aV3^SI = VsK2-IK2=V3a2-a2=aV2 5V2a3 = a^IK = r = a i' |: Vay V = |SI.SABCD4"^-^''= 3 ^idy 2.4.9. Cho hinh chop S.ABC c6 SA = SB = SC = a va ASB = CSA = y. Tinh the tich khoi chop S.ABC theo a, a, (3, y. = a, BSC = P, 155 Phuonig phapgini Todn Hinh hoc theo chuyeit tic - Nguyen Phil Khanh, Nguyen Tat Thu JCgrigidi. V Ap dung dinh h' ham so'co sin cho tarn giac SAB ta co: AB^ =SA^ +56^ -2SA.SB.cosa = 2a^ (1 -cosa) = 4a^ cos^ ^ =>AB = 2acos— 2 B V Tuong tu: BC = 2a cos ^, CA = 2a cos ^ A Goi H la hinh chieu ciia S len mat phang day (ABC), ta CO H la tarn duong tron ngoai tiep Aor^ XT- ATT AB.BC.CA _ „ tarn giac ABC. Nen AH = , -5 = b 4S ^^^^^^^ Ctif TNini MTV DWli Khaug Viet I—; Jiea^S^-(AB.BC.CA) Suy ra SH = VSA^ - AH^ = ^— ^ Ha AHlBC=i>AH±(BCC'B'), Vi AA7/(13CC'B') ^d(AA',(BCC'B')) = d(A,(BCC'B')) = AH = a Ha CKl AC, vi AB± AC va ABIAA' =>AB1(ACCA') =>AB1CK =:>CKl(ABC) =>CK = d(C,(ABC)) = b. Ta CO ABl(ACCA')=i>CA2' la goc giira hai mat phang (ABC) va (ABC) => CAC = cp. Do do: VsABC = ^SH.S^ABC = Goi p la nira chu vi tarn giac ABC, ta co: p = a Nen - p(p - AB)(p - BC)(p - CA) 16a2s2-64a^os2^cos2Pcos2l a p y cos—+ cos- + cos- 2 2 2; smtp smcp cos 9 coscp 1111 sin^cp b2-a^sin^(p AB^ AH^ AC^ b^ a^b^ = a Vay V. SABC a p cos—+ COS — 2 2 a^k vol -cos^I 2 cos^I- 2 a p cos COS — 2 2 j^VAABC ^R'he tic 1 1 AB.AC = - ab AB = ab^ ab a^ sin^ (p ^Vb^-a^sin^cp'si"^ 2sin(pVb2-a^ sin^ cp le tich lang try ABC.A' B' C la: .1 '^^l: ab^ ab^ ^ 2 sin (px/b^-a^sin^cp sin lifyjb^-a^ sin^cp I.' 2 a 2 P 2 7 -4cos — COS -cos r Vidu 2.4.10.Cho lang try dung ABCA'B'C, co day ABC la tam giac vuong tai A. Khoang each tir AA'deh (BCCB')bang a, khoang each tir C den (ABC) bang b, goc giiia hai mat phSng (ABC) va (ABC) bang cp . 1) Tinh the tich kho'i lang try ABCA'B'C theo a,b va (p. 2) Khi a = b khong doi, hay xac dinh cp de the tich kho'i lang try ABCA'B'C nho nhat. 156 2)Khia = b=>V = —. s| 2sin(pcos cp Do ^sincpcos^ (pj =i.2sin^(p.cos^(p.cos^(p<^ 2 ^273 ,,^33^73 |, =>sin(pcos (p<—^=>V>—-—. 2sin (p + 2cos (p 3 , 1 1 Dang thiic xay ra khi 2 sin'' 9 = cos'' cp <=> tan (p = o 9 = arctan -j=. Vay khi (p = arctan-^ thi V dat gia tri nho nhat. ,;; V2 157 Phucmg phdp gidi Toan Hinh hgc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu Vidu 2.4.1-0.Cho lang try dung ABCD.A'B'C'D' c6 day ABCD la hinh thoi canh 2a . Mat phang (B'AC) tao voi day mot goc 2>QP , khoang each tu B de'n mat phang (D' AC) bang |. Tinh the tich khoi hi di^n ACB' D'. JUsfi gidi. Gpi O la giao cua hai duong cheo AC va BD, ta c6 AC1(B'OB)=>B^ = 30" GQI H la hinh chieu cua B len B'O, suy ra: BH = d(B,(B'AC)) = d(B,(D'AC))-| ^' ^' Dodo: BO = n = a sin 30° . OC = N/BC^ - BO^ = a^/3, BB' = BOtan300=^ •SABCD = 2^^.60 = 2BO.CO = 2a2V3 B 2a^V3- 2a^ M$t khac Vg.^Bc - VQ^^^D - ^CB'C'D' - ^AA'B'C - ^^ABCD.A'B'CD' 1 23^ Nen suy ra V^CBD' = gVABCD.ABC'D' = —• Vi du 2.4.11. Cho hinh hpp ABCD.A'B'C'D' c6 cac mat ben va mat (A'BD) hgp voi day goc 60", biet goc B^ = 60°,AB = 2a,BD = a\/7. Tinh Va^^-B'Ciy • Xgigidi. Gpi H la hinh chieu cua A' tren (ABD), J, K la hinh chieu cua H tren AB, AD. Ap dyng djnh li cosin cho AABD: BD^ = AB^ + AD^ - 2AB.AD.cosBAD => AD^ - 2a.AD - Sa^ = 0 o AD = 3a S^^BD = ^AB.AD.sinBAD - ^ Tu gia thiet suy ra hir\ chop A' .ABD c6 cac mat ben hqip day goc 60" Nen H la each deu cac canh cua AABD 158 1: Neu H nam trong AABD thi H la tam duong tron npi tiep AABD . Goc giiia mat ben (ABB'A') va day bang A^H = 6O''. Gpi r la ban kinh duong tron npi tjg'p AABD thi: SAABD 3\/3a Cty TNHH MTV I) VVII Khang Vi^t r = 5 + V7 =>A'H = r.tan60" = - / / \/5V/7 _ 5 + V7 D Tudo, 1 VABCD.A'B'C'D' -6VA'.ABD =6-O'^'^-SAABD ^ 1^- ^ 5 + V7 TH 2: Neu H nam ngoai AABD thi H la tam duong tron bang tiep goc BAD ciia AABD. Gpi Tg la ban kinh duong tron bang tiep AABD tuong ling thi: p-BD 5_77 Tudo, V^BCD.A'B'C'D' =6V^,ABD -6.iA'H.S,A3i, =?^^ ^du 2.4.12. Cho lang tru ABC.A'B'C c6 the tich bang the tich khoi lap Phuang canh a. Tren cac canh AA',BB' lay M,N sao cho — = ^ = AA' BB' 3 Gpi E,F Ian lupt la giao diem cua CM vai C A' va CN vai C'B'. 1) Mat phang (CMN) chia khoi lang try thanh hai phan. Tinh ti so the tich hai phan do. j)Tmh the tich khoi chop C'CEF . JCgrigidL ^) Do \ 1 \ c T7 2,, 2a' ABC = 3 VABCA'BC - Suy ra VC.ABBA' = 3 VABC.A'B'C =^ M^t khac S 0:; ABNM •SA'B'NM V - ^ V ^ ^C'A'B'NM "2 ^C'ABB'A' = 153 Phumig pUiipgidi Toan With hoc titeo chmien de- Nguyen Phii Kluinit, Nguyen Tti't Thu Suy ra VCCABNM = 2a- Vay ^C'A'BNM ^C'CABNM 2) Taco: 1 '\CF.F _ 2 CE.CF.sinECF 1 ACAB 'CA.CB.sinACB EA MA 2 CE „ BC CC 3 CB 2 S ^ACAB 9 3a Suy ra CCCEF = 2^CXAB = Pi du 2.4./3. Cho hinh chop deu S.ABCD c6 M,N,E Ian luat la trung diem cac canh AB, AD, SC. Tinh ty so the tich hai phan cua hinh chop dugc cat boi mat phang (MNE) . .Cgi gidi Duong th3ng MN cat BC va CD tai K va L; EL cat SD tai P; EK cat SB tai Q. Mat phang (MNE) cat hinh chop theo mat cat la ngu giac NMPEQ. Dat AB = a,SO = h. Ta CO KB = DL - 2 Ha EH//SO=>EH la duang trung binh ^ cua ASOC nen EH = — . 2 'ACKL = -CK.CL = 4- 1 3a 3a 9a 2 2 2 2 Ta CO Q la trung diem ciia EK nen VECKL-3EH.SCKL-3-2- 8 1 h 9a^ Sa^h 16 V, KCEL KB.KQ.KM _ 1 1 1 ^J_^v KC.KE.KL • • " KCEL 96 160 Cty TNHH MTV DWH Khang Viet Tuong tu VLNDP = a^ 96 Suy ra = VBCDNMQEP = ^ECKL -fV, GQ'I V2 la phan the tich SEQMANP ta c6: KBMQ + ^LDNpl- Sa^h a^h a^h 16 48 Suy ra V2 = VSABCD - Vj = a^h a^h a^h X>i dijL 2.4.14. Cho hinh chop S.ABCD c6 day la hinh vuong canh a, ASC = 90°,SA lap voi day goc a (0° < a < 90°) va mat phang (SAC) vuong ^6c voi mat phang (ABC). Tinh khoang each tu A den (SBC). J[gigidi. Taco VA.sBc=^d(A,(SBC)).SBcs nen d(A,(SBC)) = SBCS Vi (SAC) 1 (ABC) nengpi H la hinh chieu ciia S tren canh AC thi SH1 (ABC), hinh chieu cua SA tren mat phang (ABCD) la AH nen (SA,(ABCfD)) = SAH = a. Taco ASC = 90° nen SA = AC. cos a = 72.a. cos a. Do do SH = SA.sina = \/2.a.cosasina f l -JT. Nen Vs^B(; =-SH.S^B( =—.a^.cosasina. C 3 '^"^6 ^ GQI K la trung diem ciia SC thi OK la duong trung binh ciia tam giac SAC nen OK / /SA ^ OK 1 SC. Ma BD1 (SAC) ^ BD1 SC nen BK1 SC. Taco SC = AC. sin a = 72.a. sin a nen BK = VBC^^^CK^ =; 2 - sin^ a 1 2 sin^a . Vay khoang each can tim la: d(A,(SBC)) = 3. — .a . cos a sm a. yl2.a. cos a -a^ sina.\/2-sin2a V2-sin^a ^idvL 2.4.15. Cho hinh hop dung ABCD.A'B'C'D' c6 day la hinh thoi canh ^tam giac ABD la tam giac deu. Goi M,N Ian lugt la trung diem cua cac 161 [...]... cosl20°=-o^ a AB 2x -21 1 Vx^+13.V38 = — 2 = x< ^21 x< — 2 2(2x -21 )2 =19(x^+13) 21 2 o x= llx2+168x-635 = 0 -84 ± V14041 11 o 5 ( n + 2) 2 -64(n + 2) -21 1 = 0 n +2 = n +2 = 32 + 3 V 2 3 T 5 32- 3 723 1 22 + 3N /23 T n =• 5 22 -3^ /23 T n =• 189 + 6 7 2 3 1 m =- 5 189 - 6 7 2 3 1 ' m = l%y C O hai bp thoa yeu cau bai toan: M, loSc M^ '189 + 6 7 2 3 1 15 f 189 - 6 7 2 3 1 15 ;0;0 ;0;0 " ^^ ^22 + 3 723 1^^1 0; 22 -3 723 1... + ( - 4 ) 2 = (-3 )2 + (n + 2) ^ + a = (2; 3;l), b = (-3; -2; 0), c = (x ;2; -3) 1) Tim X de a A b vuong goc voi c 2( n + 2) + 16 3 2( n + 2) + 16n2 = (n + 2) 2+5 m-3 = 2) Tim X de goc giua hai vec to a A b va c bang 120 " (-4)^ (1) (2) Jiuang dan gidi Ta c6: (2) o 4(n + if + 64(n + 2) + 25 6 = 9(n + if + 45 Ta CO a A b = (2; -3;5) 1) a A b l c o ( a A b ) c = 0c>2x-3 .2 + 5.(-3) = 0 « x = y 2) Yeu cau bai toan... vec to - • 1 AB,AC 3 -2 = (10; 12; 2) 0 Do do, ta c6 A, B, C, D khong dong phang _ ^ BC,BD = - V l O ^ + 12^ +2^ = 2^ 62 2 V i SABCD = ^ B H C D nen suy ra B H = - ^ ^ ^ ^ ^ D _2M V 42 CD _ 27 651 21 35 3)Tac6:VABCD = - B A BC,BD Gpi h = d(A,(BCD)),thethi h = ^ ^ = - ^ ^ABCD 2V 62 x = 4 + 5t 4) Ta CO phuang trinh A B : y = 2 + 2 t , s u y r a M ( 4 + 5t ;2 + 2 t ; l - 2 t ) z = l-2t • Di?n tich tam giac:... a^h VS'CEF - g Vg.ABCD 24 Sa^h • ^ay VABCDEF = Ta c6: E F - a, BM 2 2(BS2 + BC2)-SC2 x^+2a^ ,FP = ' 3 X ! + S A 2 - A E 2 ^ -SC^ BM 4a2+x2 ^J^ ^2_ 2(EC^+ES^)-SC^ Bdi 2, 4.3, Cho hinh chop S.ABC c6 day la tam giac vuong tai A, AB = a, AC = 2a Mat phSng (SBC) vuong goc voi day, hai mat phang (SAB) va (SAC) cimg tao voi mat phang day goc 60" Tinh the tich kho'i chop S.ABC theo a gp2 ^ 2 ( S E ^ + E M ^ ) -... - 2 "2 2 2) Ta CO- JABCD - 2 AA'+ BB'+CC'I cos 9 = cos -3 - 2 -2 Dodo: C M = (2 + 5t;4 + 2 t ; l - 2 t ) = > CM,CD = ( 1 0 t ; 5 t - 7 ; 3 0 t + 28 ) 1 Suy ra S^MCD = ^ L ^ ^ ' ^ ^ = -Vl 025 t2+1610t + 833 The tich: 2 + ) H i n h h p p : The tich hinh hop A B C D A ' B ' C ' D ' dugc tinh boi cong thuc VABCD.A'B'C'D'=|[AB,AD].AA +) T u di^n: The tich t u dien A B C D dug-c tinh boi cong thiic Do 1 025 t2... phuang trinh ' ^- ^ , ^'^ 20 1 Phuamg phap gidi Todn Hinh hQc theo chuyen de- Nguyen Phu Khanh, Nguyht Tat Thu (x - af + ( y - hf + (z - cf = Cty TNHH MTV DWH A (1) B C Khang Vift D P h u o n g t r i n h (1) c6 the d u g c b i e u dien each khac n h u sau + + - 2ax - 2by - 2cz + d = 0 A B C D ( P ) ( Q ) « _ =- =- = - (2) , , , , fa^+b2+c^-d>0 Voi d = a2+b2+c2-R2=^^ R = Va2+b2+c2-d , (P) 1 (Q) 4, JChodng... - ^NH.S^MD " 24 nen A M ^ = BA^ + BC^ - 2BA.BC.cosl20° - ^a^ => A M - Nen H K = 3yf^ SABCD " ( S A D H + ^ C H M + ^ A B M ) = g SABCD = AM 28 1 Suy ra 3^^,^ = ^ N K A M = ^ a ^ , 2 ^ 14 do do d(D,(AMN)) = 22 Vay khoang each tu diem D den mat phang ( A M N ) la ——a 1 62 2 BNlDMoBNlDI 3 1 2 2 AB.AD 5 2 a 2 =—a^x + 1 a ; - '1 D o d o D I l B N < » D I B N = Oc^x = i : r > A N = i a 5 5 2 Taco: M I =... ^ / / ' ' nen M N - MC + C C + C'N hay / / C in ii\ 11 ^ !'\ 1 1 ^ i \ 11 1 1 ^- \ 5 Theo bai ra M N 1 B'D nen MN.DB' = 0 ,2 o x ^2 + - 1 a^ 2 2 1, 2 - z + X 1 —y 2^ =0 1 a2^ — a2^ +1 a = 0 M I l ( A B C D ) : ^ M I l B N D o d 6 = ^ S A... ciia mat cau npi tiep hinh chop Ban kinh 2 Tam ciia hai hinh cau trung nhau < > R + r = SH = aVcosa 4\/cosa.sin^ aVcosa 2( sin^+cos^) 2sin^ cos a a a sin + c o s 2 2 1 cos a a sin 2 a 2sin2 a cos a a + c o s — sin — = 2 2 a sin- SH MS + M H SH.MH a IH = — a ^^^^ a ; SM = - c o t — , M H = - MS + M H 2 2 aVcosa 2 Vay ban kinh mat cau npi tiep la: r = I H = MH 2( sm 3V IS- vuong tao voi day hinh tru... 4;M 02 : ^ r | - M A ^ = 1 Ta CO t u giac MO1OO2 la t u giac n o i tiep COSO1MO2 = OS.ON 1) Gpi V la the tich t i i dien A B M N + y^ - xy = 21 (1) AB'' Ta t i n h dupe V = D o t u giac M O j O O j n o i tiep nen M O i OO2 + MO2 OOi = M O O j O j o 4x + y = N/2T.Z (2) O M 2 = M O ^ + O j 0 2 = M O ^ + 0 2 0 ^ < » z 2 = 1 6 + y 2 = l + x2 3V doi, m a r = — 12 la gia t r i k h o n g (3) dat gia t r j n . 5 Theo bai ra MN 1 B'D nen MN.DB' = 0 Do do (x + y - z) ,2 1,. 1. -z + X — y 2 2^ = 0 2 1 a^ 1 2 1 2 1 a n ox^+ .a^—.a^+ —= 0 <»x = —a. 2 2 2 2 2 2 2 Taco: . FP . Dat AB = x Ta c6: EF - a, BM ^J^ ^2_ 2(EC^+ES^)-SC^ 2 2( BS2 + BC2)-SC2 x^+2a^ ,FP = '3X! + SA2-AE2^ -SC^ BM 4a2+x2 gp2 ^ 2( SE^+EM^)-SM^ _9a^ / , , , 4 ~ 16 Tam . SCI(AHK). SH SH.SB SA^ 2a^ 2 SK SK.SD SA^ 2a^ 2 Do SB SB2 SB^ 3a2 3 SD SD^ SD^ 3a^ SH SK „^ / /no > 2 „^ 2 /r 2^ f2a — = — => HK / /BD va HK = - BD = -av2 = . SB SD 3 3 3 Cty