Một số phương pháp giải toán Hình học theo chuyên đề: Phần 2

Nối tiếp nội dung phần 1 tài liệu Phương pháp giải toán Hình học theo chuyên đề, phần 2 giới thiệu các phương pháp giải toán hình học trong không gian tổng hợp, phương pháp tọa độ trong không gian. Mời các bạn cùng tham khảo nội dung chi tiết.

Phuamg phapgiai Todn Hinh hoc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu

H I N H H Q C K H O N G G I A N T 6 N G HpTP

§ 1 Q U A N Hfi V U O N G G O C

1 Jiai durnig thdng vuong goc

De chu-ng minh hai duong thang AB va CD vuong goc vol nhau, ta c6 cac

each sau

Cach 1; Chung minh AB.CD = 0

Cdch 2;Chung minh c6 mot mat phang (P) chua AB va vuong goc voi CD

Cach 3;Su dung cac ket qua da biet trong hinh hpc phang

2 ^itang thdng vuong goc voi mat phdng

De chung minh duong thang a vuong goc voi mat phang (P) ta thuang di

chung minh duong thing a vuong goc voi hai duong thang cat nhau nam

trong (P) ,

Chu y: Neu duong thSng a vuong goc voi mat phang (P) thi duong thang a

vuong goc voi moi duong thang nam trong mat phang (P)

3.Jiai mdt ptidng vudnggoc

De chung minh hai mat phang vuong goc, ta chiing minh mat phang nay

chua mot duong thang vuong goc voi mat phang kia

Chu y: Neu hai mat phSng cat nhau theo giao tuyen A va vuong goc voi

nhau thi mgi duong thang nam trong mat phang nay ma vuong goc voi A thl

duong thang do se vuong goc voi mat phang kia

Vi du 2.1.1 Cho hinh chop S.ABCD c6 day ABCD la hinh chii nhat voi dp

dai cac canh AB = a, AD = b Canh ben SA vuong goc voi mat phang day va

SA c Gpi H, K Ian lugt la hinh chieu ciia A len cac duong thang SB, SD

1) Chung minh rang eac mat ben ciia hinh chop la nhirng tam giac vuong

Tinh di^n tich xung quanh ciia hinh chop theo a,b,c

2) Chung minh rang SC 1 (AHK)

3) Tinh di^n tich thiet di#n ciia hinh chop cat boi mat phMng (AHK)

JCffigidi

1) Ta CO cac tam giac SAB, SAD la nhung tam giac vuong tai A

Do SA 1 (ABCD) nen suy ra SA 1BC Lai c6 ABCD la hinh chu nhat nen

AB 1 BC

Tir do suy ra BC 1 (SAB), suy ra BC 1 AB hay tam giac SBC vuong tai B

Tuong tu, ta chung minh duoc CD ± (SAD)

nen suy ra tam giac S C O vuong tai D Vay cac mat ben ciia hinh chop deu la nhiJng tam giac vuong

Di^n tich xung quanh ciia hinh chop la:

S = S/iSAB + SASBC + S^sCD + SASDA

= i ( S A A B + SB.BC + S D C D + S A A D )

Cty TNHH MTV DWH Khang Viet

S

2 ac + bVa^ +c^ + aVc^ + b^ + be 2) Ta c6: BCl(SAB) va AHc(SAB) nen A H 1 BC Mat khac A H 1 SB nen ta c6 A H 1 (SBC)

Suy ra A H 1 SC Chung minh tuong tu ta cung c6 A K 1 SC Tir do suy ra SC 1 (AHK) 3) Ggi I la giao diem ciia SC voi mat phang (AHK), ta c6 thiet dien ciia hinh chop cat bai mat phang (AHK) la tu giac AHIK

Ta CO hai tam giac A H I va AKI la hai tam giac vuong tai H va K i

Do A I 1 SC nen SI.SC - SA^ ^ SI = SA^

Phucmg phdp gidi Toiin Hinh hgc theo chuyen de- Nguyen PM Khdnh, Nguyen Tat Thu

Vidu 2.1.2. Cho hinh chop deu S.ABC c6 canh day bang a Goi M, N Ian

lirgt la trung diem cua SA va SC Tim dp dai canh ben cua hinh chop, biet:

,2 7 , 2 Suy ra SA^ - AO^ + OS^ = — +

5 3a^

9' 4 ,2

7 2 6'

2 _ Ar^2 , ^ 2 ^ 3 a ^ 7a^ 23a

4 6 12 6 2) Goi I la trung diem M N , ta c6 A l l M N Mat khac (AMN) 1 (SBC) nen

A I 1 ( A M N ) Suy ra A l l SE => ASAE la tarn giac can nen SA = A i ; = —

Vida 2.2.3 Cho hinK chop S.ABCD c6 day ABCD la hinh vuong Tarn giac

SAD la tarn giac deu va nam trong mat phang vuong goc voi mat day Gpi

M, N Ian lugt la trung diem cua SB, CD Chung minh rang A M 1 B N

Goi H la trung diem canh AD, suy ra SH 1 A D

Ma (SAD) 1 (ABCD) nen S H I (ABCD)

Suy ra SH 1 BN (1)

Taco: BN = BC + CN, C H - C D + D H

1 Suy ra BN.CH = B C D H + CN.CD = - - BC^ + - CD^ = 0

Gpi E la trung diem ciia BC,

ta suy ra dupe (AME) / /(SHC), nen ta c6 dupe BN 1 (AME) Suy ra BN 1 A M (dpcm)

Vi da 2.1.4. Cho tu dien ABCD c6 AB = AC = AD Gpi O la diem thoa man

OA = OB = OC = OD va G la trpng tarn ciia tam giac ACD, gpi E la trung diem cua BG va F la trung diem ciia AE Chung minh OF vuong goc voi

BG khi va chi khi OD vuong goc vol AC

Dat OA = OB = OC = OD = R ( I )

va OA = a,OB = b,OC = c,OD = d

Ta CO AB = AC = A D nen AAOB = AAOC = AAOD (c - c - c)

120F = 6(OA + O E ) = 60A + 3(OB + O G ) = 60A + 30B + 30G

= 60A + 30B + OA + 2 0 M = 70A + 30B + OC + OD = 7a + 3b + c + d (s)

Tu (4) va ( 5 ) ta c6 36BG.OF = (Za + 3b + c + d)(a - 3b + c + d) ,

=7a^ - 9b^ + c % d^ - 18ab + 8ac + 8ad + 2cd Theo (3) ta c6 36BG.OF = 2d(c - a) = 20D.AC

Suy ra BG.OF = 0 OD.AC = 0 hay OF 1 BG o O D 1 AC ,

Phumtg phiipgidi Todn Hinh hoc theo chuyen de- Nguyen Phti Khanh, Nguyen Tat Thu

Vidu 2.1.S.Chotudien ABCD c6 ABJ.CD, AC ± BD

Goi H la true tarn tarn giac BCD

Chung minh rang:

1) A H 1 (BCD) va AD 1 BC

2) AB^ + CD^ = AC^ + BD^ = AD^ + BC^

3) Cac goc xuat phat tu mot dinh ciia hinh chop cung nhon, ciing vuong

hoac cung tu

Xffigidi

1) Vi H la true tarn tam giac BCD nen CD 1 B H ,

lai C O AB 1 CD nen ta suy ra CD 1 (ABH)

Tuong t u ta cung chung minh dugc cho hai dSng thuc con lai

3) Xet tai goc A Ta c6:

A C ^ + A D ^ - C D ^ cosBAC =

Nen AB^ - BC^ = AD^ - C D ^ AC^ - CD^ = AB^ - BD^

Do do: AB.AC cos BAC = A C A D , cos CAD = AD.AB cos DAB

Tu do ta suy ra dugc cosBAC, cos CAD, cos DAB cung duong, cung am

hoac cung tri^t tieu

Tu do ta suy ra dpcm

Cty mHH MTV DWH Khting Viet

du 2.1.6 Cho tam giac deu A B C canh a Goi D la diem doi xung cua A qua

gC Tren duong thang d 1 ( A B C D ) tai A lay diem S sao cho S D = Chung minh ( S A B ) 1 ( S A C )

<l6i mot vuong goc Goi H la hinh chieu cua S len mat phSng (ABC)

^) Chung minh rang H la true tam cua tam giac ABC 2) Tinh SH theo a, b, c

3) Chung minh rang: S^^3^=S^3^3+S^3^+S^3^^ ' • v' '

J-Iuang dan gidi

^) Ta C O SA 1 (SBC) =^ SA 1 BC

Mat khac SH 1 BC nen ta suy ra BC 1 (SAH) A H 1 BC Tuong tu ta cung c6 AB 1 C H , hay H la true tam cua tam giac ABC

Phuang phdp giai Todn Hinh hoc theo chuyen de- Nguyen Phti Khdnh, Nguyen Tat Thu

= i ( a V + b V + c V )

^ASAB ^^ASBC ^^ASAC ^

Bai 2.1.2 Cho hinh hgp chu nhat A B C D A ' B ' C ' D ' c 6 A B = A D = a , A A ' = b

Goi M la trung diem cua C C Xac dinh ti so ^ de hai mat phang ( A ' B D ) va

( M B D ) vuong goc voi nhau

Jlit&ng dan gidi

GQI O la tam cua hinh vuong A B C D

Cty TNHH MTV D W H Khang Vi$t

H a i m a t p h i n g ( A ' B D ) va ( M B D ) vuong goc voi nhau

^ AOMA' vuong tai O <=> OM^ + OA"^ = M A '

' 2 5b^~

a +• « a 2 = b 2 c : > - = l

(A'BD) 1 (MBD) khi ^ = l ( K h i d 6 ABCD.A'B'C'D' la hinh lap phuong) gai 2.1.3 Cho hinh chop deu S.ABC, c6 do dai canh day bang a Goi M, N Ian lu'O't la trung diem ciia cac canh SA, SB Tinh di^n tich tam giac A M N biet

=:> I la trung diem ciia SK va M N

Ta CO ASAB = ASAC => hai trung tuyen tuong

ung A M = A N => AAMN can tai A => A I 1 M N

(SBC) 1 ( A M N ) ( S B C) n ( A M N ) = M N

A I e ( A M N )

A I I M N

-A l l (SBC) =^ -A l l SK =^ -AS-AK can tai -A=>S-A = -AK =

B^i 2.1.4 Cho hinh chop S.ABCDco day la hinh chu nhat, AB = a,

= >/2a, SA = a va vuong goc voi mp(ABCD) Goi M, N Ian lugt la trung

* e m cua cac canh AD,SC, Goi I la giao diem cua BM, AC Chung minh '^P(SAC) vuong goc voi mp(SMB) Tinh the tich ciia khoi t u di?n ANIB '

101

Phumig phiip gii'ii Todn Hinh hoc theo chut/en de- Nguyen Phii Khdnh, Nguyen Tat Tttu

J^Iu&ng dan gidi

B a i 2 1 5 Cho hinh hop dung ABCD.A'B'C'D' c6 cac canh AB = AD = a,

AA' = —^ va BAD = 60" Goi M va N Ian lugt la trung diem cua A'D' va

A ' B ' Chung minh A C ' I ( B D M N ) va tinh the tich khoi chop A B D M N

Theo gia thiet ta c6 A B D deu

nen suy ra A C - Vsa

A C ' ^ = A C ^ + CC'^ =3a^ + 2 3a^ 15a^ A

Cty TNHH MTV DWH Khang Viet

EC'^ = B ' E 2 + B ' C ' 2 - 2 B ' E ' B ' C ' C O S 1 2 0 ' ' = —

A E ^ + A C ' ^ = E C ' 2 ^ A E 1 A C ' = > B N 1 A C ' (2)

Tu (1) va (2) suy ra: A C 1 ( B D M N ) Goi I , J Ian lugt la trung diem cua B D , M N va H = A C ' n IJ

Ta CO A H la duong cao ciia hinh chop A B D M N

Tu giac H I C C noi tiep => A H A C = A I A C A H = A C ^ 15a

2 A C 5

Tu giac B D M N la hinh thang can ta c6 : IJ = ^ ^ N ^ -f B D - M N ^ 7l5a

Di?n tich hinh thang BDMN : S = -IJ(BD + MN) = ^^^^^ .1 ;H / i • :•

2 16 The tich khoi chop A B D M N : V = iAH.SBnMN ^ I ' ^ - ^ ^ - ^ T ^ '

Bai 2 1 6 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 cua

A E , N la trung diem cua BC Chung minh M N vuong goc voi BD

Jiu&ng ddn gidi ' '

Goi P la trung diem cua SA ^

Ta CO MP la duong trung binh ciia tarn giac EAD

=>MP//AD=^MP//NC

Va M N = - A D = N C

2 Suy ra MNCP la hinh binh hanh

^ M N / /CP ^ M N / /(SAC)

I Ta de chung minh dugc BD 1 (SAC) => BD 1 M N

ai 2 1 7 Cho duong tron (C) duong kinh ABtrong mat phang ( a ) , mot

<lu6ng thSng d vuong goc voi (a) tai A ; tren d lay diem S ^ A va tren (C) lay diem M (M khac A, B)

1) Chung minh MB 1 (SAM)

2) Dung A H vuong goc voi SB tai H ; A K vuong goc voi S M tai K Chung

minh A K 1 ( S B M ) , S B 1 ( A H M ) i r - : ^ /

l ) T a C O • S A I M B ( l )

(2)

Phuviig phdp giai Todti Hinh hQC theo chuyun lie- Nguyen Phu Khanh, Nguyen Tat Thu

3) Goi I la giao diem cua H K va M B Chung minh A I la tiep tuyen ciia duang

Bai 2.1.8 Cho hinh hop chu nhat ABCD.A,BjCjDj c6 day ABCD la hinh

vuong M di dong tren doan AB ( 0 < A M < AB) Lay N thuoc canh A j D j sao

cho A j N = A M Chiing minh M N luon cat va vuong goc voi mot duong thang

CO dinh khi M thay doi

Jiuang ddn giai Qua M ve M E / / B D ( E e A D ) cat

AC tai F, ta co F la trung diem cua ME

va ME 1 AC

Do AM = A,N=>AE = AiN=i>NE//AAi

Goi I la trung diem cua M N , ve F I

cat A,C, tai Q, ta c6 I la trung diem

doan F Q

Goi K, H Ian luot la trung diem cac

doan thang AAi, CCi suy ra K, I , H

Vay khi M thay doi thi duong thang M N luon vuong goc va cat duong

tharig CO dinh HK

gal 2.1-9' Cho hinh lap phuong ABCD.A'B'C^D' canh a Tren cac canh DC

va BB' lay cac diem M va N sao cho M D = NB = x (0 < x < a) Chung minh rang:

1 ) A C ' 1 B ' D ' 2 ) A C ' 1 M N

Jiu6fng dan giai , : A t i j ; i;

Dat A A ' = a,AB = b,AD = c

1) Taco AC' = a + b + c, B ' D ' = c b nen , f / v

-A C ' B ' D ' = (a + b + c ) ( c - b ) = a ( c - b ) + c^ - b ^ =a^ - a ^ = 0 => -A C ' I B ' D ' 2) M N = A N - A M = (AB + B N ) - ( A D + D M )

Jiuang ddn giai

^) Goi E la trung diem cua canh ABva O la giao diem cua AC va DE thi

•^DCE la hinh vuong c6 tam la O

Ta CO SA 1 (ABCD) => SA 1 O D , them niia O D 1 AC =^ O D 1 (SAC)

Tir do ta c6 OD 1 (SAC) => (SDO) 1 (SAC) •*

105

Phiwug phiip giai Todn Hinh hgc theo chuyen da - Nguyen Phii Khdnh, Nguyen Tat Thu

Vay (SDO) chinh la mat phang ( a )

Thiet dien ciia hinh chop v6i mat phang (a) la tam giac SDE

I Goc giua luii duang thdng cheo nhau ' '

E)e tinh goc giiia hai duong thang cheo nhau a va b ta c6 cac each sau

Cdch l:T\m goc giira hai duong thang a, b bang each chon mot diem O

thich hop (O thuong nam tren mot trong hai duong thang) Tu O dung cac duong thang a b ' Ian lugt song song (c6 the trung neu O nam tren mot trong hai duong thang) voi a va b Goc giua hai duong thang a', b' chinh la goc giij-a hai duong thSng a va b u •]>;(

Chu ^'De tinh goc nay ta thuong su dung dinh l i cosin trong tam giac:

cosA = • b ^ - c ^ - a ^

2 be

Cdch2:Tur[ hai vec to chi phuong u ^ U j ciia hai duong thSng a, b

Khi do goc giua hai duong thang a, b xac dinh boi cos(a,b) U1.U2

Chu y: DG tinh U j U j , U j , U2 ta chon ba vec to a, b, c khong dong

phang ma c6 the tinh duoc do dai va goc giOa ehiing, sau do bieu thj cac vec

to U j , U 2 qua cac vec to a, b, c roi thue hien cac tinh toan

2 Goc giua duang thdng vai mat phang

De xac djnh goc giiia duong thclng a va mat phang (a) ta thuc hien theo cac budc sau:

• Tim giao diem O = a n (a)

• Dung hinh chieu A ' ciia mot diem A e a xuong (a)

• Goc A O A ' = (p chinh la goc

giua duong thang a va ( a )

• De dung hinh chieu A' cua diem A tren (a) ta ehgn mot duong thang

b 1 (a) khi do A A V / b

• De tinh goc (p ta su dung he thuc luQ'ng trong tam giac vuong AOAA'

ll^goai ra neu khong xac djnh goc cp thi ta c6 the tinh goc giira duong thang a

Phucmg fthtip gidi Todn Hhth hoc theo chuyen de - Nguyen Phii Khdnh, Nguyen Tat Thu

va mat phang (a) theo cong thiic sincp =

n la vec ta c6 gia vuong goc vol (a)

3 Goc giua hai mat phang

De tinh goc giua hai mat

phing (a) va (p) ta c6 the thuc

hien theo mot trong cac each sau:

Cdch l : T i m hai duong thang

a, b Ian lugt vuong goc vai hai

mat phang (a) va (p)

Khi do goc gii>a hai duong

thang a,b chinh la goc giiia hai

mat phang (a) va (p)

fa 1 ( a )

u.n trong do u la VTCP ciia a con

(a)

/<y)

b l ( p ) .((a),(p))^(a,b)

Cach 2;Tim hai vec to nj,n2 c6 gia Ian lugt vuong goc vai (a) va (p) khi

do goc giiia hai mat phang (a) va (p) xac dinh boi cos(p =

Cdch S.-Su dung cong thuc hinh chieu S' = Scos(p, tu do de tinh coscp thi ta

can tinh S va S'

Cdch 4: Xac dinh cu the goc giua hai mat phSng roi su dung hf thuc lugng

trong tarn giac de tinh

Ta thuang xac djnh goc giiia hai mat phMng theo mot trong hai each sau:

a) • Tim giao tuyen A = (a) n (p)

• Chgn mat phang (y) 1 A / ( p )

• Tim cac giao tuyen a (y) n (a), b = (y) n (p), / M

Chu ^."Cho hinh chop S.A]A2 A^ c6 duong cao SH De xac dinh goc giua

matphSng (SAjAj) ( i,j e {l,2, ,n}; i j) voi mat day ta lam nhu sau:

• Tu H ve HK 1 A j A j , K e A^A^ , ,,

• Khi do SKH la goc giua hai mat phang (SAjAj) va mat day

Vidu 2.2.1. Cho t i i dien ABCD Ggi M, N Ian lugt la trung diem ciia BC va

AD, biet AB = CD = a,MN = - — Tinh goc giua hai duong thang AB va CD

Xgigidi

Cdch J.-Ggi I la trung diem ciia AC

^^^'^C/CD"^(^H"^)-Dat M I N = a Xet tarn giac I M N c6 AB a CD a ^ aVs

I M = = — , I N = = - , M N = —

2 2 2 2 2 Theo djnh l i cosin, ta c6

' 1

109

Phumig phapgiai foan Hinh hoc theo chmjen dc - Nguyen Pliii Khdnh, Nguyen Td't Thu

Vidu 2.2.2 Cho hlnh chop S.ABCD c6 day ABCD la hinh thang vuong tai A

va B, AB = BC = a, AD = 3a Hinh chieu cua S len mat phang day trung voi

trung diem canh AD Mat phang (SCD) tao voi day mot goc 60" Goi M la

trung diem doan CD Tinh c6 sin cua goc giua hai duang thang AM va SC

£gl gidi

Goi H la trung diem cua AD, ta c6 SH 1 (ABCD)

Ta CO ABCH la hinh vuong canh a

Goi I la hinh chieu cua C len AD

Do tam giac CID dong

dang voi tam giac HKD nen

Gpi E, F Ian \ugt la trung diem cua

cac doan thing SD va HD

A F = 1 A D = ^

4 4

Taco: M F l AD=*MF = l c i = - , suy ra AM = N / A F ^ T M F ^ =

2 2 10

Cty TNHH MTVDWH Khang Viet

Ap dving djnh li Co sin cho tam giac A M E , ta c6:

A M ^ + M E ^ - A E ^ 57 cos AME = • 2.MA.ME

57

474181 Vay cos(AM,SC) = ^ - ^

X>i dii 2.2.3 Cho hinh chop S.ABC c6 SA 1 (ABC), SA = a, AB = a,

5 A C = 120° Tinh c6 sin ciia goc giiia hai mat phang:

7 7

V T O

Suy ra cos SKA = A K V S O

SK 10 2) Gpi M la hlnh chieu cua B len AC, ta c6 B M 1 (SAC) Suy ra tam giac SMC la hinh chieu cua tam giac SBC len mat phSng (SAC)

S (Theo cong thuc hinh chieu, ta c6: coscp = - ^ ^ ^ voi (p la goc giua hai m^t tiang (SAC) va (SBC)

T a c 6 : S ^ B C = - S K B c 4 ^ a 7 ^ =

-'ASBC

Phumtgphapgiai Todn Hinh hgc theo chuyen de- Nguyen Phii KItdnh, Nguyen Tat Thu

Do BAM = 60" => A M = ABcos60° = ^ S^^MP = ^SA.MC =

3

2 • -ASMC 2 4

Vhy cos(p =

27To '

W 2 2 4 Cho langtru dung A B C A ' B ' C c6 AB = a; AC = 2a; A A ' = 2aV5

va BAC = 120" Goi M la trung diem cua canh C C Tinh c6 sin cua goc giug

hai mat phang (BMA') vai (ABC)

A'B^ = A'A^ + AB^ = 21a2 = A ' M ^ + BM^

Suy ra tam giac A ' M B vuong tai M ,

suy ra S^^'MB = ^ M A ' M B - Sa^Vs

Ta CO tam giac ABC la hinh chieu cua tam giac BMA' len mat phang (ABC),

nen ap dyng cong thuc hinh chieu ta c6:

COSCP = | M B C ^ ^ 1 ^ = ( ( i K i X ^ ^ ) )

^ABMA' 3a^V3 ^

Vi du 2.2.5 Cho hinh chop S.ABCD c6 day la hinh vuong canh a, O la tam

cua day, SO 1 ( A B C D ) ; M , N Ian lugt la trung diem cua SA, CD Biet goc

giua M N voi ( A B C D ) bang 60° Tinh goc giira M N va (SBD)

suy ra N H la hinh chieu aia M N tren ( A B C D )

=> M N H chinh la goc giiia duong thSng M N voi ( A B C D )

2

I

/ i > a

Vaygoc giira M N va (SBD) la (p = a r c t a n i Cdch2;TaCO M N = i(sC + A B ) = |(sO + OC + A O + O B ) - ~ ( S O

Suy ra MN^ = ifSO^ + AC^ + OB^) =

Taco (p la goc giua M N va (SBD) nen sin(p =

Jong goc voi (SBD) )

I'jiianin pluipgiai Todn Hinh hoc theo chuyett de- Nguyen Phu Khdnh, Nguyen Tat Thu

Do goc giua duang thang M N va ( A B C D ) bang 60*^ nen

250^ = ISa^

Thay vao (*) suy ra sincp = -i=r =>(p = arcsin-j=

Vay goc giua M N va ( S B D ) la 9 = arcsin-^

v5

Vi du 2.2.6 Cho hinh hop chu nhat ABCD.AjBjCjDi c6 day ABCD la hinh

vuong Tim goc Ion nhat giira duong thang BDj va mat phang ( B D C ^ )

Xgigidi Cdchl:

Goi I = AC n B D , 0 la trung diem cua BDj thi O € ( C A A j C i )

Goi H la hinh chieu cua C tren C j l thi C H 1 C j l

Cty TNHHMTV UVVII Khang Vi?t

va C H l B D = > C n i ( B D C i ) C,H CjH.CiI CC? b^ _

Vi du 2.2.7 Cho hinh chop S.ABCD c6 day A B C D la nua luc giac deu npi

tiep trong duong tron duong kinh AB = 2a; canh ben SA vuong goc vm

dayvaSA = a V 3

1) Tinh goc giCra hai mat phang ( S A D ) va ( S B C ) 2) Tinh goc giua hai mat phSng (SBC) va ( S C D )

Phuaitg phiipgiiii loan Iliiili lioc Ihco chuyen de - Nguyen Vhii Khiinh, Nguyen Tat Thu

Dung D E 1 SI, E e SI khi do ( B D E ) 1 SI

Do do BED la goc giQ-a hai mat phang ( S A D ) va (SBC)

Do day ABCD la nua luc giac deu nen lAB = IBA = 60° => AIBA deu

Vi vay A I = A B = 2a, SI = TSA^TAF = J(aSf + {laf = a77

Ap2 AS^ AH^ - + •

D l thay ASAC vuong can tai A nen AQ = ^ S C = = ^ ,

Do A P 1 (SCD) ^ A P 1 P Q

Trong AAPQ c6 cosAPQ = AP _ ^is^^ VlO

AQ aVe 5 APQ = arccos

Vio

116

Cty TNHH MTV DWH Khang Viet

Vay ((SBC),(SCD)) = arccos Vio

X?i du 2.2.8 Cho tam giac ABC c6 AB = 3a , duong cao CH = a va A H = a

nam trong mat phang (P) Tren cac duong thang vuong goc voi (P) ke tu A,B,C Ian luot lay cac diem A',B',C' tuong ung nam ve mot phia ciia (p) sao cho AAj = 3a,BBj = 2a,CCj = a Tinh dien tich tam giac A ' B ' C

2

C C = i A A - = ^ C J = l A C = ^

3 2 2 Xet ABCH ta c6 BC^ = BH^ + CH^ = Sa^ ^ BC = aVs Mat khac AB^ = CA^ + CB^ - 2CA.ABcosC

= > C 0 S C = : - C A ^ + C B ^ - A B ^ 1

Xet AICJ ta c6 IJ^ = C P +CJ^ -2CI.CJcosICJ = 26a'

ft.'!,-Ke duong cao C K cua A I C K , do C C 1 (iCj) nen C ' K 1 I J Vay C " ^ chinh la goc giua hai mat phang ( A B C ) va ( A ' B ' C ' ) n e n

Phuwig phap giai Toan Hinh hgc theo chuyen de- Nguyen Phu Khdnh, Nguyen TS^ Thu

Bai 2.2.1 Cho hinh chop S.ABC c6 day ABC la tarn giac can tai A, AB = a

va SA 1 (ABC) Mat phang (SBC) tao voi day mot goc 60" Goi M , N Ian luot

la trung diem cua SB va AC Tinh :

1) Co sin ciia goc giiia hai mat phang (SAC) va (SBC)

2) Co sin ciia goc giiia hai duong thang A M va B N

Jiicang dan giai

Gpi K l a trung diem BC, suy ra A K 1 BC B C l ( S A K )

Do do SKA la goc giiia hai mat phSng (SBC) va (ABC), hay SKA = 60°

Ta c6: B A K = 60° A K = AB.cos60° = - , B K = AB.sin60° =

2 Trong tam giac S A K , ta c6:

S A = A K t a n 6 0 ° = ^

2

= > S K = V S A ' + A K 2 =a

1) Goi B ' la hinh chieu ciia B len

A C , suy ra B B ' l ( S A C ) nen tam

giac S B ' C la hinh chieu cua tam

giac S B C len mat phang ( S A C )

1V3 BC-aVs

Do do coscp = SASB'C

'ASBC Taco: S,,SBC =^SK.BC = ^ ,

Do BAB' = 60° =^ AB' = ABcos60° = - ^ CB' = —

2 2

— Vay cos(p = - 2) Trong mat phSng (ABC) dung hinh binh hanh ANBE, suy ra AE / /BN

2.AM.AE

Bai 2.2.2 Cho hinh chop S.ABC c6 SA = SB = SC = a va BC = aV2 Tinh goc

giiia hai duong thing AB va SC

Jlicong ddn giai ' ! v

Goi M, N , P Ian luot la trung diem cua SA,SB, A C , khi do M N / / AB ^

nen ( A B , S C ) = ( M N , S C )

, M N ^ + M P ^ - N P ^ / X

I Dat (p = N M P , trong tam giac M N P c6: coscp = 2MN M P

Ta CO M N = M P = - , A B ^ + A C ^ = BC^ => A A B C vuong tai A , vi vay

P B 2 A P 2 + A C 2 = ^ , P S 2 = ^

4 4 ^ Trong tam giac PBS theo cong thuc tinh duong trung tuyen ta c6

5a^ 3a^

pj^j2 _ P B ^ + P S ^ SB^ ^ 4 ^ 4 a^^3a^

1 'Thay M N , M P , N P vao (l)tadugc cos(p = = 120°

Phuoitg phcip giiii Todii Hinh hoc theo chuyen de - Nguyen Phii Khdnh, Nguyen Tat Thu

mgt goc 30*^ Tinh the tich khoi lang try A B C A ' B ' C va c6 sin cua goc giiia

hai duang thang A M va C ' N » L

Jiicong dan gidi

Bdl 2.2.4 Cho hinh chop S.ABCD c6 day ABCD la hinh chii nhat AB = a,

A D = 2a Gpi O la tam ciia day, SO = SD Mat phang (SBD) vuong goc voi mat

phang day, mat phSng (SAB) tao voi day mot goc 6 0 ° Gpi M la trung diem

doan BC T i n h c6 sin cua goc giiia hai duang thang A M va SB

Jiuang dan gidi

Gpi H la trung diem cua O D , suy ra SH 1 BD => SH 1 (ABCD)

Ve H K X AB, K 6 AB

120

Cty TNHH MTV DWH Khang Vi?t

suy ra AB ± (SKH) => A K H la goc ijja hai mat phSng (SAB) va mat day

^gnSKH = 60°

Taco: H K = | A B = y

, S H = H K t a n 6 0 ° = 3a V3

Gpi I la giao diem cua A M voi BC

Ve IJ / /SB, J e SD, ta CO (AM, SB) = (AI, IJ)

Vi I la trpng tam tam giac ABC

1) Chung minh cos^ a + cos^ P + cos^ y = 1 2) Tinh SBCD theo khi a - 30°, p = 45°,y = 60°

Jiuang dan gidi

1) Cach 1

121

Phucmg phdp giai Todn Hinh hoc theo chuyen de- Nguyen Phii Khdnh, Nguyen Tat Thu

Vay ( A B H ) I C D va C D la giao tuyen B

cua hai mat phang ( A C D ) va ( B C D )

Cach 2. G(?i H la hinh chieu cua A tren (BCD) va I la trung diem cua CD

Dat AB = b,AC = c , A D - d = b = d

De thay A H 1 ( B C D ) va

BH.BI = BA2=b2 b^fc^+dM IH.IB = l = c^d^

c^+d^ I H c^d^ = k

1, Suy ra A H = — A B + A I ,

b V + c ^ d ^ + d V

c^d^

b V + c ^ d ^ + d V

AB + - b V + d V bV+c^d^+d^b

2) Su dung cong thuc hinh chieu Gpi H la hinh chieu cua A tren ( B C D ) Truoc tien ta chung minh tam giac BCD nhon , , ^ Khong giam tong quat, gia su B Ion nhat B 4 / -1 I

Taco C D ^ - A C ^ + A D ^ ^c^+d^ \l Tuong hf CB^ = b^ + c^DB^ = b^ + d^

Ap d\ing djnh l i cosin cho ABCD ta c6:

Do do SpcD = SHBC + SHBD + SHCD = ^ABC ^osy + S^go cosp + S^CD

l u ^nO l j r O 1 J onO bc + 72bd + Vscd

= -bccos60" +-bdcos45" +-cdcos30" = —•

fi 2.2.6 Cho hinh chop S.ABCD c6 day la hinh vuong canh a, tam O va

1 ( A B C D ) Mat phSng (a) di qua A va vuong goc voi SC cat hinh chop

*heo mpt thiet dien c6 di^n tich S^^ = -ậ Tinh goc giCra duong thang SC va

•^^t phing (ABCD) .aixmiiffrn IM! níuig "yog "JJ min nmi-i

Phumtg phdp giai Todn Hinh hoc theo chuyen de- Nguyen Phu Khdnh, NguySn Tat Thu

Jiuang dan gidi

Gia su (a) cat cac canh SB,SC,SD Ian lugt tai cac diem H , J, K

Do 1 ^ ^ - ^ ^ ° =:^BD1(SAC)^BD1SC ma (a) 1SC => (a)//BD

Taco AJ = ACsin(p = aV2sin(p; SO = OCtan(p = ^ ^ t a n ( p

ASOC~ASJI ^sTj = SCO = (p=>AIO = SU = (p

• K H = BD(l-cot2 9 ) = aN/2(l-cot2(p) D

Vay S^HJK = ^ H K A I = a^/2sin(p.a^/2|l-cot^ (pj = 2a^ sin(p|l-cot^9J

Tie gia thiet suy ra 2a^ sincp^l - cot^ tpj = ia^ o 4sin^ 9 - sincp - 2 = 0

8

Bdi 2.2.7 Cho hinh lang tru dung ABC.A'B'C c6 day ABC la tam giac can

tai A voi AB = AC = a va goc 6 X 0 = 120", canh ben BB' = a Goi I la trung

diem cua C C Chung minh rang tam giac AB'I vuong tai A Tinh c6 sin cii^

goc giira hai mat phang (ABC) va (AB'I)

124

Cty TNHH MTV DWH Khang Viet JIu&ng ddn gidi

Goi H la trung diem ciia BC

Do AABC can tai A nen A H 1 BC va ACH = 30"

Ta c6: A H = AC sin ACH = a sin 30° = |-, B

BC = 2BH = 2.acos30'' = aVs

=>IB'2==IC'2+B'C'2=^ + 3a2=.13^'

4 4 AA'B'B la hinh vuong canh a

nen A B ' = aVi, A I ^ = I C ^ + A C ^ = 5a ^

A l 2 + A B ' 2 = ^ + 2 a 2= l ^ = IB'2

• A A B' I vuong tai A

a^x/To Taco: S ^ B , = - A I A B ' = ^

GQI 9 = ( ( A B C ) , ( A B ' I ) ) C O S 9 = =

10

Bai 2.2.8. Cho lang tru A B C A ' B ' C , c6 day ABC la tam giac deu canh a, va

A'A = A'B = A ' C = a ^ Tinh the tich khoi lang tru ABC.A'B'C theo a va

goc giua hai mat phing (ABB' A') va (ABC)

Jiu6mg dan gidi

Gpi H la hinh chieu cua A tren (ABC)

•

Vi A ' A = A'B = A'Cnen HA = HB = HC Suy ra H la tam tam giac deu ABC

Gpi I , J Ian lugt la trung diem cua BC,AB

Phumtg phdp gidi Toan Hinh hgc theo chuySn dJ-Nguyht Phu Khdnh, Nguyen Tat Thu

a a^Vs a^^/3 The tich khoi lang tru A B C A ' B ' C la: V = A ' R S ^ B C = 2 • 4

Vay goc giira hai mat phang (ABB'A') va ( A B C ) bang 60°

Bai 2,2.9 Cho lang tru ABC.A'B'C c6 dQ dai canh ben bang 2a, day ABC la

tam giac vuong tai A, AB = a, AC = aV3 va hinh chieu vuong goc cua dinh A'

tren mat phang (ABC) la trung diem ciia canh BC Tinh theo a the tich khoi

chop A'.ABC va tinh cosin cua goc giiia hai duong thang AA', B ' C

Jiuang dan gidi

G(?i H la trung diem BC A ' H 1 (ABC)

thi: (p = B'BH Vay cos(p = ;

Bdi 2.2.10 Cho hinh chop S.ABCD c6 day ABCD la hinh vuong canh 2a,

SA = a, SB = aVs va mat phSng (SAB) vuong goc vol mat phSng day Goi M/

N Ian lu9t la trung diem ciia cac canh AB, BC Tinh theo a the tich cua khoi

chop S.BMDN va tinh cosin cua goc giiia hai duong thang SM, D N

Cty TNHH MTV DWH Khang Vi?t

Jiicong dan gidi

Goi H la hinh chieu cua S tren AB, suy ra SH ± (ABCD) Do do SH la (jaong cao ciia hinh chop S.BMDN Ta c6: SA^ + SB^ = a^ + 3a^ = AB^

Bai 2.2.11 Cho t u dien ABCD c6 cac mat (ABC) va (ABD) la cac tam giac deu

c^nh a, cac mat (ACD) va (BCD) vuong goc voi nhau Hay tinh theo a the tich khoi t u dien ABCD va tinh so do cua goc giiia hai duong thang AD va BC

Jiuang dan gidi

Gpi M , N , I Ian lug't la cac trung diem cac c^h CD, AB, BD

Phuang phdp gidi Todn Hinh hgc theo chuyen de- Nguyen Phi't Khdnh, Nguyen Tat Thu

=> D M = V N D ^ - = - ^ = ^ ^ S^CND = M N M D = ^

Vay the tich tii di§n A B C D : VABCD = 2VANCD = 2 | A N - S ^ C D = •

Ta CO AMNE la tarn giac deu => MEN = 60*^

Do ^''^^ (AD^BC) = ( M J M ) = 60°

l l M Z / B C

Bai 2.2.12 Cho t u dien ABCD c6 AC = AD = aN/2 , BC = BD = a, khoang each

tir B den mat phang (ACD) bang Tinh goc giua hai mat phang (ACD) va

(BCD) Biet the cuakhoitiidi^n ABCD bang

Jiuang d&n gidi

Goi E la trung diem cua CD, ke B H 1 A E

Ta CO AACD can tai A nen CD 1 AE Tuong tu CD ± BE

Suy ra CD 1 (ABE) =^ CD 1 BH

Ma BH 1 AE BH 1 (ACD) BH = Goi a = ((ACD),(BCD)

1 a^JlE

The tich ciia khoi tu di|n ABCD la V = - B H S ^ c D =

Mat khac: AE^ + DE^ = 2a^ => AE^,DE^ la hai nghiem ciia phuang trinh :

Vay goc giua hai mp(ACD) va (BCD) la a = 45°

128

Cty TNHH MTV DWH Khang Viet

§ 3 K H O A N G C A C H

^ <finh khoangaichtum^t diem (Mnm > , r i c ,

De tinh dugc khoang t u diem M den mat phSng (a) ta c6 cac each sau:

Cdch l:Xac djnh hinh chieu vuong goc H ciia M len (a)

De xac dinh dugc vi tri hinh chieu H ta c6 mpt so' luu y sau: ^

• Chpn (p) chua diem M va (P) 1 (a), roi xac djnh giao tuyen A = ( a ) n (p) Trong (P) dung M H 1 A M H 1 ( a ) (h.2)

• Neu trong (a) c6 hai diem A, B sao cho M A = MB thi trong (a) ke duong

trung true d cua doan AB, roi trong mp(M,d) dyng M H 1 d

Khido M H l ( a ) (h.3) That vay, gpi I la trung diem ciia AB Do M A = MB nen AMAB can tai

M M I 1 AB c ( a ) Lai c6 AB 1 d => AB 1 mp(M,d) => AB 1 M H

V a y | ^ " ^ ^ ^ ^ M H l ( a ) [ M H l d ^ '

• Neu trong (a) c6 cac diem A^, A2, ,Aj, (n > 3)

Ma M A j = M A j = = MA„ hoac cac duong thang MAi,MA2, ,MA„ tao

129

Phitang phdp gidi Todn Hinh liQC theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu

voi (a) cac goc bang nhau thi hinh chie'u ciia M tren (a) chinh la tarn duong

tron ngoai tiep da giac AjAj.-.A^

• Neu trong (a) c6 cac diem A j , A2, , A„ (n > 3) ma cac mat phang

( M A I A 2 ) , ( M A 2 A 3 ) , , ( M A J , A J ) thi hinh chie'u ciia M la tam duong tron noi

tiep da giac A j A j - A ^

• Mot ket qua c6 nhieu ung dung de tinh khoang each tir mpt diem den mat

phang doi voi t u dien vuong (tuang t u nhu he thuc lug-ng trong tam giac

Cdch 3: Chuyen viec tinh khoang each tir M ve tinh khoang each t u diem N de

tinh hon bang each su dung eae ket qua sau:

• Neu MN//(a) thi d(M,(a)) = d ( N , ( a ) )

• N e u M N n ( a ) = {l} thid(M,(a)) = ^ d ( N , ( a ) )

2 JChodng cdch giua hai duong thdng cheo nhau

De tinh khoang each giiia hai duong thing cheo nhau ta c6 the dung mpt

trong cac each sau:

Cdch i : D y n g doan vuong goc chung M N cua a va b Khi do d(a,b) = M N

Chii y:Neu a l b thi ta dung doan vuong goc chung cua a va b nhu sau

• Dyng mat phang (a) ehua b va vuong goc voi a

• Tim giao diem O = a n (a)

, Dung OH 1 b Doan O H chinh la doan vuong goc chung ciia a va b

Qfich 2;Dung mat phang (a) di qua a va song song voi b , khi do : d(a,b) = d(a,(a)) = d(M,(a)) voi M la diem bat ki thuoc (a)

Q^h 3;Dung hai mat phang (a) di qua a va song song voi b, (P) di qua b va song song voi a Khi do: d(a,b) = d((a),((3))

Cck;/i 4;Phuong phap vec to:

M N la doan vuong goc chung ciia AB va CD khi va chi khi

A M = xAB

CN = yCD MN.AB = 0 MN.CD = 0

Vi du 2.3.1. Cho hinh chop S.ABC c6 S A l ( A B C ) ; AB = a, AC = 2a, BAC = 120° Mat phang (SBC) tao voi mat phang (ABC) mot goc 60° Tinh: 1) Khoang each tu A den mat phing (SBC)

2) Khoang each tu B den (SAC)

JUgigidi

Ap dyng dinh If c6 sin, ta eo:

fsC^ = AB^ +AC^ -2.AB.AC.cosBAC = 7a2

BC = a77 ipi K la hinh chie'u ciia A len BC suy ra B C l ( S A K ) ne n SKA la goc giiia hai mat phang (SBC) va (ABC)

Suy ra SKA = 60°

Ta c6:

SAABC = | A K B C - 1 ABAC, sin 120° =

=>AK = >SA = AK.tan60°=

^) Gpi H la hinh chie'u ciia A len SK

Do BC 1 (SAK) r:> BC 1 A H => A H 1 (SBC)

SA.AK 3aV7

Suy ra d(A,(SBC)) = A H =

V S A 2 + A K 2 1 4

Phucmgphdp gidi Todtt Hinh hqc theo chuyen de- Nguyen Phu Khanh, Nguyen Tai Thu Cty TNHH MTV DWH Khang Viet

Vidu 2.3.2 Cho hinh chop S.ABCD c6 day ABCD la hinh thang vuong tai A

va B, AB = BC = a, AD = 2a Tarn giac SAD la tarn giac deu va nam trong

mat phang vuong goc voi day Goi H la hinh chieu vuong goc ciia A len SB

Tinh khoang each tu H den mat phSng (SCD)

G(?i H la trung diem ciia A D ,

suy ra S H 1 A D ^ S H 1 ( A B C D )

GQI E la giao diem ciia A B va C D ,

suy ra B la trung diem ciia A E

Ta c6:

HS d(H,(SCD))-

Vi du 2.3.3 Cho hinh lap phuong ABCD.A'B'C'D' canh a Tinh khoang

each giija hai duong thang AD' va BD ^

GQI MN la doan vuong goc chung ciia AD' va BD voi M e A D ' , N e B D

Dat AB = x,AD = y,AA' = z=> x = y = z = a,xy = yz = zx = 0 i '

AD^ = y + z ^ AM = kAD^ = k (y + z), DB = X - y => DN = m (X - y j T^co MN = A N - A M = AD + D N - A M = mx + ( l - k - m ) y ^

Vi M N l D B : ^ M N D B = O o ( m x + ( l - k - m ) y + k z ) ( x - y ) = 0

I o 2m + k -1 = 0 [Tuong tu MN.AD' = 0 o 1 - m - 2 k = 0, tu do ta c6 h^

2m + k = 1 1

<:i>m = k = -

m + 2k = l 3 Eyay MN = - x + - v — z

3 3^ 3 ' => MN = MN [x +y +z J = iV3

Cdch3.(Hinh 2) Chon ( D C B ' A ' ) vuong goc voi AD'

Strung diem O ciia A D ' Goi I la tam ciia hinh vuong BCC'B'

B I I C B ' va B I I C D nen B I l ( D C B ' A ' ) tu do D I la hinh

*ie'ucua DB len (DCB'A')

Hinh 2

133

Phumig yihdp giai Toan Hinh hgc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu

Trong ( D C B ' A ' ) ke O H 1 D I , tu H dung duong thang song song voi A D '

cat B D tai M , t u M dung duong thang song song voi O H cat O A tai N thi M N

la doan vuong goc chung ciia cua A D ' va B D do do d ( A D ' , B D ) = M N

Ta CO O H M N la hinh chii nhat nen M N — O H , mat khac O H la duong cao

trong tarn giac vuong O D I nen

Vi du 2.3.4. Cho hinh chop S.ABCD c6 day ABCD la hinh thoi canh a,

SD = ayfl, SA = SB = SC = a Goi E la trung diem canh CD Ti'nh :

1) Khoang each giua hai duong thang AC va SB

2) Khoang each giCra hai duong thang AE va SB

i j p r gidi

Vi SA = SB = SC nen hinh chieu ciia

S len mat phang (ABCD) la tarn duong

tron ngoai tiep tam giac ABC Gpi H la

tam duong tron ngoai tiep tam giac

Ma AS = AB = A D = a nen ta c6 O la tam duong tron ngoai tiep tam giac

SBD, ma O la trung diem BD nen ta suy ra tam giac SBD vuong tai S, suy ra

BD = TSB^TSD^ = a V i => BO = —

Suy ra tam giac A B C deu nen B H = ^ ^ S H W S B 2 - B H 2 = ^

3 3 1) Ve duong cao OF cua tam giac SOB, ta c6 OF 1 SB

Mat khac AC 1 (SBD) => AC 1 OF OF la duong vuong goc chung cua hai

duong thang AC va SB Hay d(AC,SB) = OF

Cty TNHH MTV DWH Khang Viet

Ta c6: OF.SB = SH.BO => FO = SH.OB a72

SB Vay d(AC,SB) =

2) Gpi K la giao diem cua AE voi 3D, K la trong tam tam giac ADC

Tu K ve duong thang song song voi SB, cat SD tai M Suy ra SB//(AME)

^ , SM SK ^ D N D M 1

Ta co: = = 2, = = —

D M D K D H DS 3 Suy ra d(SB, AE) = - d ( N , ( A M E ) )

3 9 DE KD 3 3 6 Suy ra NQ = N M N P

\ / N M ^ + N P ^

aV22 a722 - Vay d(SB,AE) = -

Pi du 2.3.5.Cho lang try dung A B C A ' B ' C c6 day la tam giac deu canh a, chieu cao bang h Tinh goc va khoang each giua hai duong thang AC va BC

Xgigiai

Taco mat phang (BA'C) chua B C

va song song voi AC nen

Phuomg phlip giiii Todn Hiitlt hoc theo chuyen de - Nguyen Phu Khdnh, Nguyen Tat Thu

va BH = A ' H ^ = Ja^ + - — = -^JAh^ +

V 4 2

Suy ra S ^ ^ B C = - ^ H A ' C ' = - a V 4 h ^ T 3 a ^ Do vay d = '

2 4 V4h2+3a2

Taco A'C7/AC nen (AC,BC') = (A'C',BC') = A'C'B

Trong tam giac vuong B C ' H , ta c6:

a

Vay (AC, BC') =

arccos-Chil y: Ta cung c6 the sir dung phep chie'u

vuong goc de tinh khoang each giua hai duong

thang cheo nhau:

Cho hai duong thang cheo nhau AB va CD

Xet mat phang (a) vuong goc vai CD tai /a\

diem O Goi IJ la doan vuong goc chung ciia

AB va CD ( I e A B , j G C D )

Xet phep chie'u vuong goc len (a), Goi A',B',r la hinh chie'u cua A , B , I thi

I J - O r , t i r d 6 d(AB,CD) = d ( 0 , A ' B ' )

Vaydeti'nh IJ ta qui ve tinh OI' trong mat phang ( a )

Vidu 2.3.6. Cho tu dien deu ABCD canh a Gpi M, N Ian luot la trung diem

cua AB va CD Tinh khoang each giua hai duong thang BN va CM

JCffi gidi

Goi H la tam ciia tam giac deu BCD thi

A H 1 (BCD) Goi (a) la mat phSng di

qua N va song song voi A H thi (a) 1 BN

Xet phep chie'u vuong goc len (a), ggi

A',B',C',D',H',M',N' Ian luot la anh cua

rang M N 1 IJ va M N bi IJ chia thanh hai phan bang nhau

3) O la mot diem thupc doan thang IJ Chung minh rang O each deu mat phang (ABC), (ABD), O each deu hai mat phang (ACD) va (BCD)

JCgi gidi

» T a c 6 : | " ^ ' ^ « :

C D I A B AB 1 (CID) => AB 1 CI, AB 1 D I Suy ra cac tam giac ACB, ADB la nhiing tam giac can tai C va D

Tu do suy ra AC = BC, AD = BD (1)

Chung minh tuang hx thi cac tam giac CAD, CBD can tai A va B

Suy ra AC = A D , BC = BD (2)

Tu (1) va (2) ta c6: AC = AD = BD = BC

^) Npi tiep tu dien bang hinh hop AC'BD'.A'CB'D Tu gia thiet bai toan, ta

L^^y ra dupe hinh hgp AC'BD'.A'CB'D la hinh hop dung va day la hinh thoi 137

Phuomg phap giai Todn Hinh hqc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tai Thu

Mat phang di qua IJ, cat hinh hop theo thiet dien la hinh chu nhat PQRS

Vi I la tarn cua hinh thoi A'DB'C nen ta c6:

CS = DR =^ ACSM = ADEN =^ MS = NR

Suy ra M N / /CR , nen IJ vuong goc va chia doi M N

3) Got H la hinh chieu ciia O len BJ, K la hinh chieu ciia O len AJ

Do CD 1 (AJB) =^ CD 1 O H => OH 1 (BCD) d(0,(BCD)) = O H

Tuong tu d(0,(ACD)) = OK

Vi JB = JA nen tarn giac AJB can tai J va JI la duong phan giac trong goc AJB

Suy ra O H = OK => O each deu hai mat phang (ACD) va (BCD)

Chung minh tuong t u ta cung c6 duoc O each deu hai mat phang (ABC) va

ABD)

Vi du 2.3.8 Cho hinh lap phuong ABCD.A'B'C'D' canh a Lay diem M

thupc doan AD', N thuoc BD sao cho A M = D N = x (0 < x < aVi)

1 ) Tim X theo a de M N ngan nhat Khi do hay chung minh: M N la duong

vuong goc chung ciia hai duong thang AD', BD va M N // A'C

2) Chung minh rang khi x thay doi thi M N luon song song vai mpt mat

-Tu do, ta CO M N nho nhat khi va chi khi x =

Cty TNHHMTV DVVII Khang Viet

Ca BAI TAP gdi 2 3 1 Cho hinh chop S A B C c6 day A B C la mot tarn giac deu canh a, c^nh S A vuong goc voi ( A B C ) va S A = h Tinh khoang each tu A den (SBC) theo a va h

Jiix&ng dan giai ,

A I 1 BC Goi I la trung diem cua BC, ta eo

•AH = - ahVs V4h2+3a2

Hayd(A,(SBC)) =

-V4h2+3a2

Bai 2.3,2. Cho hinh chop S A B C D c6 day A B C D la hinh thang vuong tai A

va B , B A = B C = a, A D = 2a Canh ben SA vuong goc voi day va SA = a^l •

Gpi H la hinh chieu vuong goc cua A tren SB Tinh khoang each t u H den inatphing ( S C D )

Jiuang dan giai

Trong (ABCD) goi M = A B n C D , trong ( S A M ) goi K = A H n S M , ke

1 S C tai E va goi N la trung diem ciia A D

D l thay ABCN la hinh vuong nen NC = AB = a

Do do N A = NC = ND = a => AACD vuong tai C CD 1 AC Lai CO C D 1 S A => C D 1 ( S A C ) => ( S A C ) 1 ( S C D )

AE = a

Vay =^d(H,(SCD)) = HF = -!-AE = -

3 3 Bai 2.3,3. Cho hinh hpp chu nhat A B C D A ' B ' C ' D ' c6 ba kich thuoc A B = a,

A D = b , A A ' = c Tinh khoang each t u A deh mat p h i n g ( D A ' C )

JIuong dan giai

Goi I la tarn cua hinh binh hanh A D D ' A '

thi I la trung diem cua A D '

^ ^ ^ , d ( A , ( D A X ' ) )

d ( D ' , ( D A ' C ' ) ) I D '

r i d ( A , ( D A ' C ' ) ) = d ( D ' , ( D A ' C ' ) )

Mat khac ta c6 t u d i ^ n D ' A D C c6 cac

canh D ' D, D ' A', D ' C' doi mot vuong goc nen

B6i 2.3.4. Cho hinh chop S A B C D c6 day A B C D la hinh thang vuong tai A

va D , tarn giac S A D deu va c6 canh bang 2a , B C = 3a cac mat ben tao vol

(Jay cac goc bang nhau Tinh khoang each t u S den mat phang ( A B C D )

Jiu6ng ddn giai •

G Q I I la hinh chieu vuong goc cua S t r e n ( A B C D ) ;•

Gpi I j , 12,13,14 Ian lugt la hinh chieu cua I tren cac canh A B , B C , C D , D A

thi cac goc 11(8(1 = 1,4) la goc giiia cac mat ben va mat day do do chung bang nhau,suy ra cac tam giac vuong SIIj,SIl2,SIl3,SIl4 bang nhau nen

IIj = I I j = I I 3 = II4 => I la tam ducmg tron ngi tie'p hinh thang A B C D

V i tit giac A B C D ngoai tie'p nen AB + DC = A D + BC = 5a Di^n ti'ch hinh thang A B C D la

Bal 2.3.5 Cho hinh chop S.ABCDday la hinh thang, Aic = ABC = 90°,

BA = BC = a, A D = 2a Canh ben SA vuong goc vol day va SA = V2a Gpi H Ife hinh chieu cua A len SB Chung minh tam giac SCD vuong va tinh (theo a)

khoang each t u H den mp(SCD) '

141

Phutnig phdp gidi Todn Htnh hoc theo chuyen de - Nguyen Phu Khdnh, Nguyen Tat Thu

Goi d^;d2 Ian lugt la khoang each tit B,H den mp(SCD)

Ta c6: ASAB - ASHA S A S B S H SA^ 2 S H d, 2 ^ 2^ a 2 3 1

AAj = 2aV5 va BAC = 120*' Gpi M la trung diem cua canh C C j Chiing minh

hai duong thang MB va MAj vuong goc voi nhau Tinh khoang each tu diem

A den m^t phang ( A J B M )

Jiuang ddn gidi

Taco: BC = V A B ^ + AC^ -2AB.AC.eosl20° = ayf?

BM = V B C ^ T C M 2 =2>/3a; AjM = ^AjCj^ + C^M^ =3a BAj = 7 A B ^ + A A 7 = \/2Ta; BM^ + A^M^ = BA^^

Suy ra MB 1 MAj Ke C H 1 A B C H 1 (ABAj)

Ta CO CH = AC.sin60° = >/3a

Cty TNHH MTV DWH Khang Viet

fhe tich kho'i tu di^n M A B A :

Jiuang ddn gidi

Gpi M, N Ian lugt la trung diem cua SA, BC

Ke B H l C N t a i 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 = A N =

Gpi E la diem tren duong thSng AAj sao cho Aj la trung diem cua

ME, taco BM/ZBjE

JiC = V2a,BiE = BM = V B A 2 + A M 2 C E = V C A 2 + A E 2 = ^ 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 ± B M

* T a c o B M / / ( B j C E ) ^ d ( B M , B i C ) = d(M,(BiCE))

Gpi H la t r u n g d i e m cua A j C j ta c6

B H l ( A C q A i )

The tich cua khoi chop Bj CME :

V B ] C M E -= - B , H S 2 • ' l ^ - ^ C M E = — .—a.a = 1 Vsa 1 VSa^

3 2 2 12 Goi I la hinh chieu ciia M len mp(BjEC) ta c6 :

Bai 2.3.9. Cho lang t r y d u n g ABC.A'B'C c6 day A B C la tarn giac vuong,

A B = BC = a , canh ben A A ' = aV2 Gpi M la trung d i e m ciia canh BC Tinh

theo a the tich cua khoi lang t r y ABC.A'B'C va khoang each giiia hai duong

thang A M , B'C

Jiuang dan gidi

T u gia thiet suy ra tam giac ABC v u o n g can tai B B"

The tich kho'i lang try la:

D o t u di?n B A M E c6 BA, B M , BE doi mot v u o n g goc nen:

— - +

- = ^ = > h = h^ BA^ B M ^ BE^ h^ 7

V|y khoang each giira hai d u o n g t h i n g A M va B'C la a^/7

Bai 2.3.10. Cho h i n h chop t u giac deu S.ABCD c6 day la h i n h v u o n g canh A

Gpi E la d i e m d o i x u n g ciia D qua trung d i e m cua SA M la trung d i e m ciJ^

144

^ g , N la trung d i e m cua BC C h i i n g mirJi M N vuong goc v a i BD va tinh (theo

3 ) khoang each giiia hai d u o n g t h i n g M N va AC

Jiuang ddn gidi ? ^

Gpi P la t r u n g d i e m ciia SA •-: ; ;, ,

I Ta CO M P la d u o n g trung binh cua tam giac E A D

Bai 2.3,11 Cho h i n h chop SABC c6 tam giac ABC vuong can tai B, AB = BC = 2a,

(SAB) va (SAC) eiing v u o n g goc v o i (ABC) Goi M la trung diem AB, mat phang qua MS song song v o i BC cat A C tai N Biet goc giiia (SBC) va (ABC) bang 60° T i n h the tich khoi chop S.BCNM va khoang each giiia hai duong thang A B v a S N

Jiuang dan gidi ^

Do hai mat phang ( S A B ) va ( S A C ) c a t nhau theo giao tuyen S A va cung v u o n g goc voi ( A B C ) nen S A 1 ( A B C ) , hay S A la d u o n g

jcao cua khoi chop S B C N M

Trong tam giac vuong SAB ta c6 S A = AB tan 60^ = 2a>/3

Vay V s B C N M = ^ S A S B C N M = i 2 a > / 3 ^ = V3a3(dvtt)

Goi P la trung diem cua BC thi AB / /NP, AB (2 ( S P N ) nen AB / / ( S P N ) do

do d (AB, SN) = d (AB; ( S P N ) ) = d (A; ( S P N ) )

Tir A ha A E l N P , E e P N thi \> P N 1 ( S A E ) ;ha A H I S E thi

Bai 2.3.12. Cho lang try ABCD.AiBiCiDi c6 day ABCD la hinh chii nhat,

AB = a, A D = a\/3 Hinh chie'u vuong goc ciia diem A i 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

Trong do C H la duong cao ciia tam giac vuong B C D

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

Bai 2.3.14. Cho hinh chop S A B C D c6 day A B C D la hinh vuong canh a Goi

M va N Ian luot la trung diem cua cac canh AB va A D ; H la giao diem cua C N

va D M Bie't SH vuong goc voi mat ph5ng ( A B C D ) va SH = aS Tinh the tich

khoi chop S C D N M va khoang each giiia hai duong thiing D M va SC theo a

Jiuang ddn gidi

Taco: VgcDfji^ = - S H S ^ [ ^ D C SMNDC = SABCD - SAAMN - S/^MBC

2 a^ a^ 5a^

= a =

8 4 8 Nen

^ S C D N M 8 24 (dvtt)

Laithay: DM.CN = i ( 2 D A - D C ) i ( 2 D C - D A ) = D A 2 - D C ^ = 0

2V— / 2' V^y CN 1 D M h:r do SC 1 D M bai vay:

d(SC;DM) = d(H;SC)=^^'^"SC-^"-^^ SH.CH

SC SC VsH^+CH^

Phumig phiifigidi Tiu'ui Ilinh hoc theo chuyen de - Nguyen Phti Khiinh, Nguyen TA't Thu

Laico: CH = ^ ^ ^ ^

D M 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 A N va D M

Gpi E la trung diein ciia BC

Dethay AADM = ABAE

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:

• Ne'u hinh H duoc taeh thanh hai hinh roi nhau H j , H2 thi V,]^ = Vj^ ~ ^ H 2

• 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 C D , ta c6: C D 1 (SMO)

Do do goc S M O la goc giua mat ben voi mat day, nen S M O = 60°

I Dat A B = 2x => M O = x,OC = xV2 Trong cac tam giac vuong SOC,SOM ta c6: A = : ; A r ^ i i ;

SO^ = SC^ - OC^ - 5a2 - 2 x ^ SO = O M 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 ^ = ^ a 3

3 3 2) Goi K la hinh chieu ciia O len AM,

ta CO O K ± (SCD) nen OSK la goc giiia

duong cao SO vol mat ben nen OSK = 45°

Goi N la trung diem AB

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:

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

Suy ra SKH la goc giii-a mat phang

(SCD) voi mat day, do do S K H = 60°

Goi E la hinh chieu aia C len AD, suy ra ABCD la hinh vuong canh a

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° / V s

Trong tam vuong SAO ta c6: / j \ V r \ ,

SA:.AO.tan60" = = ^ 7 3 = ^ / ] ^ » \

BC ± AB / A ' ' ^ ^ \

BC ± A M => A M 1 (SBC) ri> A M 1 SC / ~'X^=zz-\ V-'

Tuong tu: A N 1 (SCD) => A N 1SC, '- ' " ' o ' ^ ^

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:

Vi du 2.4.5. Cho hinh chop S A B C D c6 day A B C D la hinh vuong tam O, SA

vuong goc voi ( A B C D ) , A B = a,SA = aV2 Gpi H , K Ian lupt la hinh chieu

vuong goc cua A tren SB, SD Chung minh: SC 1 ( A H K ) va tinh the tich cua

khoi chop O H A K theo a

Cty TNIIIl MTV DWII Khaug Viet

Gpi G la giao diem cua S O vh K H

thi G la trung diem ciia K H , ma

va OI = — = — = - Suy ra VQ.AHK = 3 OI-S^HK = 3 ' 2 = •

Cdch 2: Gpi E la hinh chieu ciia A tren SO thi A E 1 ( O H K ) nen A E la duong

cao cua hinh chop A O H K

Pi du 2.4.6 Cho hinh chop S.ABC c6 C3c canh day AB = 5 3 , BC = 6 3 , AC = 7a

Cac m3t ben t3o voi dsy mot goc bang nhau V3 bSng 60° Tinh the tich khoi chop S.ABC V 3 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 ( A B C ) , A ' , B ' , C ' Ian lugt la hinli

chieu cua I tren BC,CA,AB Tu gia thie't suy ra S A l = SB^I = S C I = 60° Cac

tam giac vuong SIA',SIB',SIC' bang nhau nen l A ' = IB' = I C => I la tam du6n»

tron noi tiep tam giac ABC

Goi p la nua chu vi tam giac ABC

5a + 6a + 7a

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

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 ^ ^ ^ ^ ^ A C l ( S B D ) ^ 0 = B D n A C 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 = | A C B D

Nen V S.ABCD J A C B D = iAB.SD.OA

Cty TNHH MTV DWH Khang Vigt

Dodo: Vs^ABCD = —.a.x.Vsa^ - x^

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

^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

Tuong tu: B C = 2a cos ^ , C A = 2a cos ^ A

Goi H la hinh chieu ciia S len mat phang day

( A B C ) , ta CO H la tarn duong tron ngoai tiep

la goc giira hai mat phang ( A B C ) va ( A B C ) => C A C = cp

Do do: VsABC = ^SH.S^ABC =

Goi p la nira chu vi tarn giac A B C , ta co: p = a

Vidu 2.4.10.Cho lang try dung A B C A ' B ' C , co day A B C la tam giac vuong

tai A Khoang each tir A A ' d e h ( B C C B ' ) b a n g a, khoang each tir C den

( A B C ) bang b , goc giiia hai mat phSng ( A B C ) va ( A B C ) bang cp

1) Tinh the tich kho'i lang try A B C A ' B ' C theo a,b va (p

2) K h i a = b khong doi, hay xac dinh cp de the tich kho'i lang try A B C A ' B ' C

Dang thiic xay ra khi 2 sin'' 9 = cos'' cp <=> tan (p = o 9 = arctan -j=

Vay khi (p = a r c t a n - ^ 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 h i di^n ACB' D '

JUsfi gidi

Gpi O la giao cua hai duong cheo AC va BD, ta c6 A C 1 ( B ' O B ) = > B ^ = 30"

GQI H la hinh chieu cua B len B'O, suy ra:

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 V a ^ ^ - B ' C i y •

Xgigidi

Gpi H la hinh chieu cua A' tren (ABD), J, K la hinh chieu cua H tren AB,

AD

Ap dyng djnh l i cosin cho AABD: BD^ = AB^ + AD^ - 2AB.AD.cosBAD

=> AD^ 2a.AD Sa^ = 0 o A D = 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

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:

^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

Phumig pUiipgidi Toan With hoc titeo chmien de- Nguyen Phii Kluinit, Nguyen Tti't Thu

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 ( M N E )

.Cgi gidi

Duong th3ng M N 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

la duang trung binh ^

cua ASOC nen EH = —

Suy ra = VBCDNMQEP = ^ECKL -fV,

GQ'I V 2 la phan the tich SEQMANP ta c6:

KBMQ + ^ L D N p l - 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)

Taco ASC = 90° nen SA = AC cos a = 72.a cos 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

^ t a m giac ABD la tam giac deu Goi M , N Ian lugt la trung diem cua cac

161

Phuang phdp gidi Todn Hinh liQC theo chuyen de- Nguyen Pku Khdnh, Nguyen Tat Thu

canh B C C ' D ' Tinh khoang each tif D den mat phang (AMN) biet rang

/ ^ / / \ ' ' ^ - \

Goi H la trung diem cua DC thi N H 1 (ABCD),NH = — a nen

re

' ^ D A M N - ^ N A M D - ^ N H S ^ M D " 24 a3

Ke HK 1 A M ta c6 NK i A M Theo djnh li ham so cosin

AM^ = BA^ + BC^ - 2BA.BC.cosl20° - ^a^ => A M - ^ a

22 Vay khoang each tu diem D den mat phang (AMN) la ——a

Cty TNHH MTV DWH Khang Viet

p Bai tap

BOi 2.4.1, Cho hinh chop S.ABCD c6 day ABCD la hinh thoi va AB = BD = a,

SA = a>/3 , SA 1 (ABCD) Ggi M la diem tren canh SB sao choBM = - SB, gia

3

sCr N la diem di dong tren canh AD Tim vi tri ciia diem N de BN 1 D M va

Ichi do tinh the tich cua khoi tii dien BDMN

V M B D N - ^ M L S ^ B N D 1 2a73 a^V3 _2a^

•3" 3 - _ 5 15 '^<^i 2.4.2 Cho hai khoi chop S A B C D va 5 ' A B C D c6 chung day A B C D la f»Ot hinh vuong canh a (S va S' nam ve cimg mpt phia cua ( A B C D ) ) Goi H, K

lugt la trung diem cua A D va B C , biet SH = S'K = h va SH,S'K ciing

Phucmg phdp gidi To&n Hinh hgc theo chuySn AJ- Nguyen Phu Khdnh, Nguyen Tat Thu

vuong goc vai ( A B C D ) Ti'nh the tich phan chung ciia hai khoi chop S A B C D

va S ' A B C D theo a va h

Jiudrng ddn gidi

Tir gia thiet de bai, ta suy ra cac t i i S

giac S D C S S A B S ' la cac hinh binh hanh

Gpi E, F Ian luc^t la tam ciia cac hinh binh

hanh SDCS', S A B S ' Ta c6 phan chung

cua hai khoi chop S A B C D va S ' A B C D

Suy ra VS.BCEF = ^s.gcF + ^ s x E F = • ^ay VABCDEF = Sa^h

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 ^

Jiudrng ddn gidi

Goi H la hinh chieu cua S len BC; E,F Ian lugt

la hinh chieu cua H len AB, AC suy ra S H 1 (ABC)

va HE = HF nen A H la phan giac ciia goc BAC

W = I AB.AC = a2 Vay Vs.A3C - •

Bdi 2.4.4. Cho hinh chop tam giac deu S.ABC Tinh the tich khoi chop S.ABC biet:

1) Canh day bang a va mat ben tao voi day mpt goc 60°

2) Canh ben bang 2a va SA 1 B M , voi M la trung diem SC

164

Cty TNHH MTV DWH Khang Viet Jiitong ddn gidi

Goi O la tam ciia day, I la trung diem B C

1) Ta CO B C 1 (SIO) => slo = ((SBC),(AB"C)) = 60°,

lO = i A I = SO = IOtan60° = - ,

3 6 2 'AABC

^r" w Icr^c l a a^V3 a^Vs

Vay VsABC - 3SO.SAABC = 3 • 2 = '

2) Goi E , F , P Ian lugt la trung diem ciia A B , BS, S M , ta c6:

Tam giac E F P vuong tai F nen EP^ = EF^ + FP^ o - 8a^ x = 2aV2

f'Bdi 2,4.5. Cho lang tru tam giac ABCAjB^Ci c6 ta't ca cac canh bang a, goc tao boi canh ben va mat phang day bang 30° Hinh chieu H ciia diem A tren mat phang ( A i B j C i ) thugc doan thSng BjCj Tinh the tich khoi lang tru ABC.AjBjCj va khoang each giiia hai duong thang A A j va BjCj theo a

Phuontg phdp gidi Todn Hinh hoc theo chuyen de- Nguyen Phii Khdnh, Nguyen Td't Thu

A H = A A j s i n 3 0 " = | , A j H = AApCosSO" =

Ma tarn giac A j B j C j deu

nen H la t r u n g diem ciia B[C|

The tich kho'i lang t r u la:

phang (a) l u o n song song v o i A B va C D T i m v i t r i cua (a) de (a) chia t i i dien

thanh hai phan c6 the tich bang nhau

Jiu&ng dan gidi A

V i cac mat cua t u di?n c6 dien tich bang

nhau nen cac d u o n g cao bang nhau

Mat phang (a) song song v o i A B va CD

A D

A H A D 2 - 2 A H 2

o M a t phang a d i qua trung diem cua canh A D

CtyTNHH MTV DWH Khang Viet

gdi 2.4.7 Cho hinh chop S.ABCD c6 day ABCD la hinh chii nhat AB = a, A D = 2a,

c^nh SA v u o n g goc v o i day, canh SB tao v o i mat phang day m g t goc 60" Tren c^nh SA lay diem M sao cho A M = ^ M a t phSng ( B C M ) cat canh SD tai N Tinh the tich khoi chop S B C M N ,^

JIuang dan gidi

Ha SH 1 B M =^ SH 1 ( B C M N ) : ^ SH

la d u o n g cao cua khoi chop S.BCMN

Do A M H S ~ A M A B nen suy ra:

Bai 2,4.8 Cho t i i dien ABCD c6 A C = A D = aV2 , BC = BD - a, khoang each t u

B den mat phSng (ACD) bang Tinh goc giiia hai mat phang (ACD) va

V3 (BCD) Biet the tich cua khoi t u dien A B C D bang a^Vl5

Phuongphiipgiai Todn Hinh hoc theo chuyen de- Nguyen Phu Khdnit, Nguyen Tat Thu

Mat khac: A E ^ + DE^ = 2a^ =^ AE^, DE^

la hai nghiem cua phuong trinh

Xet A B E D vuong tai E nen B E = N / B D ^ - D E ^ = —

Xet A B H E vuong tai H nen sin a = BH 1

• a = 45' 0

BE ^ Vay goc giua hai mp(ACD) va (BCD) la a = 45°

Bdi 2.4.9 Cho hinh chop S.ABCD c6 day ABCD la hinh thoi canh a, BD = a

Tren canh AB lay M sao cho BM = 2AM Goi I la giao diem ctia AC va D M SI

vuong goc voi mat phang day va mat ben (SAB) tao voi day mot goc 60" Tinh

the tich cua khol chop S.IMBC

Jluongddngidi

Goi H la hinh chieu cua I len AB, suy ra AB 1 (SIH) => SHI la goc giCra mat

ben (SAB) va mat day Do do SHI = 60°

sin60'^ sin BDM MD V7 7 ->tanBDM =

(jiem AO

- ^ : ^ - l = ^ = > O I = OD.tanBDM = ^ ^ I la trung cos^BDM 2 4 "

- A A T T T A A r ^ D A I AI.OB aS

Ta CO AAHI - AAOB = => IH = • Suy ra SI = IHtan60° =

OB AB AB 3a

8 SAMI _ A M A I _ 1 1 1 _ 1 _ _ a \ / |

^ Suy ra ((SAB), (ABCD)) = ( S K J I K ) = SKH = 60°

Gpi O la tam ciia day, ta CO AAKH - AAOB ,^ „ ^

Phuang phdp gidi Toiin Hinh hoc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu

Bai 2.4.11 Cho hinh chop S.ABCD c6 day A B C D la hinh vuong tarn ,)

A B = a G o i M , N Ian lugt la trung d i e m cua cac canh OB, SD M a t phark>

( N A C ) tao v 6 i mat phang day mot goc 60°; hai mat phang (SAM) va (SCl^^j

ciing v u o n g goc v o l mat phang day T i n h the tich cua kho'i chop S.ACN

Jiixang ddn gidi ^

V i hai m a t phang (SAM) va (SCM) cung

v u o n g goc v o i mat phang day nen giao tuyen

S M ciia hai mat phang do v u o n g goc v o i day

Trong tam giac v u o n g N H O , ta c6: N H = O H tan 60° = S H =

•^NACD *'SACD ^ S A C N ^SACD -SH.S A A C D 1 aV6 a^

6 4 • 2

N6_

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Bai 2.4.12 Cho h i n h chop S.ABC c6 mat phang (SAC) vuong goc v o i mat

phang (ABC), SA = A B = a, A C = 2a va ASC = ABC = 90° T i n h the tich khoi

chop S.ABC va cosin ciia goc giua hai mat phang (SAB), (SBC)

Goi M la t r u n g diem SB v^ (pla goc

giua hai mat phang (SAB) va (SBC) ^

Ta c6: SA = A B = a, SC = BC = aS

cos A M C

A M 1 SB va C M 1 SB, suy ra coscp =

ax/3 ASAC = ABAC => SH - B H = • SB = Jx/6

A M la t r u n g tuyen ASAB nen:

Cty T'iVHH MTV DWH Khang Viet

gjjj 2.4.13 Cho lang t r y A B C A ' B ' C c6 day A B C la t a m giac can

^B = A C = a,BAC = 120°va A B ' vuong goc v o i day ( A ' B ' C ) G o i M , N Ian

\^xQt la t r u n g diem cac canh C C va A ' B ' , mat phang ( A A ' C ) tao v a i mat

phang (ABC) mot goc 30° Tinh the tich khoi lang t r u A B C A ' B ' C va c6 sin cua goc giira hai d u o n g thang A M va C ' N

V = AB'.S AABC - g • Goi E la t r u n g d i e m cua A B ' , suy ra M E / / C ' N , nen ( C ' N , A M ) = ( E M , A M )

Phuong phap gidi Todn Hinh hgc theo chuyen de - Nguyen Phu Khdnh, Nguyen Tat Thu

Bai 2.4.14 Cho hinh lang tru deu ABC.A'B'C, M la trung diem cua canh CC

Mat phSng (A'B'M) tao voi mat phang (ABC) mpt goc 60" va tam giac A'MB'

di^n tich bang

13 Tinh the tich khoi chop A M A ' B '

Jivcang dan gidi

Gpi N la trung diem cua A'B', ta c6

C N I A ' B '

Matkhac: A ' M = B'N ^ M N 1 A'B',

suy ra A ' B ' l ( M N C ' )

Do do M N C la goc giiia mat phSng

(A'MN) voi (A'B'C)

chieu cua S len mat day trimg voi diem H la trung diem ciia AO Mat phang

(SAD) tao voi day mpt goc 60° va SC = a Tinh Vg^Bj^D va d(AB,SC)

J^it&ng ddn gidi

D a t A B = x,x>0 ^

Ve H K I A D , suy ra A D 1 ( S H K ) ^ S K H la goc giua mat ben (SAD) va

mat day nen SKH = 60°

Taco: HC = - A C = ^ ^ ; HK = - D C = —

4 4 4 4 Trong tam giac vuong SHC, ta c6: SH^ = SC^ - HC^ =a^-^^

4a 3>/5 ' 16Vl5a

675

Bdi 2.4.16 Cho hinh chop S.ABC c6 day ABC la tam giac vuong can tai B,

AB = BC = 2a; hai mat phang (SAB) va (SAC) ciing vuong goc voi mat phang (ABC) Gpi M la trung diem cua AB; mat phang SM va song song voi BC, cat

AC tai N Biet goc giiia hai mat phSng (SBC) va (ABC) bSng 60° Tinh the tich khoi chop S.BCNM va khoang each giiia hai duang thang AB va S N theo a

Jiicang dan gidi

Do hai mat phang ( S A B ) va (SAC) cat nhau theo giao tuye'n SA va cung vuong goc voi ( A B C ) nen SA 1 ( A B C ) , hay SA la duong cao ciia khoi chop S.BCNM

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