Kiến thức cơ bản và nâng cao hình học 12 (tái bản lần thứ nhất) phần 2

Di tinh khoang each gifia hai ducmg thing cheo nhau A va A' ta lam nhu sau: • Lap phuong trinh mat phang a chfia A va song song \di A'... Cho phuong trinh tham so ciia ducmg thang A

62 Cho hinh tru c6 cac day la hinh tron tarn O vh tarn O', ban kinh ddy bang

chi^u cao va bang a

Tren ducmg tron day tarn O la'y diem A, trdn ducmg tron day tarn O' la'y

diem B sao cho AB = 2a

Tinh the tfch kh6'i tir dien OO'AB

(Trich de thi Dai hoc - Khoi A - 2006)

ChUffng in

PHl/ONG PHAP TOA D O TRONG K H O N G GIAN

I Ht TOA DO TRONG KHONG GIAN

A L t THUYfeT CAN N H 6

1 Toa do cua diem va cua vecta

/ He tog do: Trong khong

gian CO ba true toa d6 vuong goe

vdi nhau doi mot va c6 dinh hudng

Ba true nhu vay duac goi la ht

toa d6 vuong goc trong kh6ng gian Didm O goi la g6'c toa d6

- True hoanh, dinh hudmg

duang X Ox, c6 vecta don vi

o

Hinh 34

i = (l;0;0) True tung, dinh hudng duong y'Oy, eo vecta dan vi

Vi i , } , it la cac vecta dan vi tren true, ma cac true vu6ng goc nhau d6i

ni6t, nen ta cof = 1

va T.J =0; i.k =0; j k = 0

Trong khdng gian vori he toa dp O x y z con dupe gpi la khong gian O x y z

2 Toa dp mot diem

Trong khong gian O x y z cho

m6t diem M tuy y, m6i diem M

hoan toan xac dinh boi vecta OM

V i 3 vecto i , j , k la 3 vecta

khong dong phang nen c6 b6 3 so

duy nha't (x, y, z) sao cho:

3 Toq do cua vecta

Trong khong gian Oxyz, cho vecto u , ta luon c6:

M = a,, i + 3 .2-j +a3 k

Bp ba so (a,, a2, a,) xac dinh duy nhat va gpi la toa dp ciia vecto u, ki

hieu la u (a,, aj, a,)

2 Bieu thirc toa do cua cac phep toan vecto

/ Dinh U Trong kldong gian Oxyz cho cac vec ta a (a,; aa; aj)

S (bi; b2; bj) k h i do

d) a ± b = (a,± b,; a2± bj; a j i b,)

b) k a = (ka,; kaj; k-a,) = k(a,; di^; aj) (k la so thuc)

2 He qua a) a = h « a, = b,, a2 = b2, a, = bj

b) 0 = ( 0 ; 0 ; 0 )

c) a va b ciing phuong c6 mot so k: a, = kb,, aj = kbj, a, = kbj

d) Trong khong gian O x y z c6 A ( a , ; a2; a,) va B(b,; b2; b,) thi 45 = ( b - a , ; b2-a2; bj-aj)

3 T i c h v6 hudmg

1) Bieu thAc toq do

Dinh If Trong khong gian Oxyz tich v6 hudng cua hai vecto a (aj; a.^, a-,;

va A (b,; bj; h^) la m Or sty duac xac dinh bcfi cong thiic:

a b = a,b| + a2b2 + ajbj 2) Ifng dung

* Do dai cua vecto

* Khoang each giua hai diem A A ( X A ; yA; Z A ) B ( X B ; yg; Zg) la

A B = 7(XB - x^f + (ye - yj' + (z^ - z^f

* G p i 9 la goc giiia 2 vecta a va b

a,b, +a2b2 +a3b3

cos 9 =

^ / a f + a f + a ^ -^/bf+bf+b

=> a J b <=> aib] + a2b2 + a3b3 = 0

4 T i c h CO hudng cua hai vecto (hay tich vecto)

/) Dinh nghia: Tich c6 hu6ng (hay tich vecto) cua hai vecta a (ai, a2, a,)

Va b (b„ b2, bj) la mot vecto duac k i hieu \di[a,b] hay a A 6 va c6 toa

d6 duac xac dinh nhu sau:

f a, a, a, a,

V b,b3 3

b3b, >

b,b, ) 2) Tinh chat

[a,b]=0 o a =k.S

{a,b] l a va[a, 6]I.6(ra,b].a = 0; ra,b1.b = 0)

a,b = a sin(a,b)

3)Apdung

J Tick dien tick cua hinh binh hanh va thetich khoi hdp

• ABCD la hinh binh hanh SABCD = AB A D sin A

A B , A D

• A B C D A B C ' D ' la hinh hop

• VABCD.A'B'C'D'= T A B , A D AA

Ba vecta a, c, bdong phang <=> a,b c = 0

Ba vecta a, b, c khdng d6ng phing <» a,b

I

5 Phuong trinh mdt c^u

* Dinh li: Trong khong gian Oxyz,

mat cau (s) tarn I(a;b;c) va ban kinh r

CO phuong trinh:

(x-a)^ + (y-b)2 + (z-c)2 = r2

Nhdn xet:

Phuong trinh

x'+y'+z' + 2Ax + 2By + 2Cz + D = 0v6i: ' Hinh 36

A^ + B^ + - D > 0 la phuong trinh mat c^u tarn I(-A;-B;-C) va

b) Goi M la die'm chia dudng trung tuyen AA, ciia mat phang ABC theo

so 3: 7 ( = — ) Chung minh rang: MA, 7

Cdnh 2: Bien ddi ve phai:

Cho tur dien deu ABCD canh d,

M va N Ian luot la trung di^njL AG '

va BD

1) Tim do dai MN

2) Tim goc giua MN va AB

3) Chiing minh MNl AC, MN 1 BD

U m toa d6 hinh chie'u cua die'm A ( l ; -3; -5) trdn:

1) mp Oxy; 2) mp Oxz 3) mp Oyz;

4) True hoanh; 5) True tung 6) True cao

A 1

5) Tren true tung Oy thi hoanh do x = 0, do cao z = 0 nen toa do hinli

chieu ciia A la A,(0; - 3 ; 0)

6) Tren true cao Oz thi hoanh do x = 0, tung do y = 0 nen toa do hinh

chieu ciia A la A^iO; 0; -5)

Vi du 4: Cho A(-3; 2; -1) Tim toa do diem doi xiing cua A qua gdc toa

d6, qua eac true toa do, qua cac mat phang toa d6

Giai: Qua gdc toa do: toa do diem doi xiJng ciia A la: (3; - 2 ; 1)

Qua true hoanh x' Ox: toa dd di^m ddi xiing ciia A la: (-3; 2; - 1 )

Qua true tung y'Oy: toa do di^m ddi xiing etia A la: (3; 2; 1)

Qua true cao z'Oz: Toa do di^m ddi xiing cua A la: (3; - 2 ; - 1 )

Qua mat phang Oxy: toa do diem ddi xiing ciia A la: (-3; 2; 1)

Qua mp Oyz: toa dd diem ddi xiing cua A la: (3; 2; - 1 )

Qua mp Oxz: toa do di^m ddi xiing ciia A la: (-3; - 2 ; - 1 )

Cho tii dien ABCD c6 A ( l ; - 2 ; -1), B(-5; 10; -1), C(4; 1; 11), D(-8; - 2 ; 2)

ViS't phuang trinh mat ciu ngoai tie'p tii dien ABCD

Khoang each l A = = -^{l-xf +(2 + >;)' + ( l + z ) ' = 9 Mat e^u

ngoai tiep tii dien ABCD c6 tarn I(-2;4;5) va c6 ban kinh r = l A = 9

Vay phuong trinh mat cau la:

(x+2)' +(y-4)' + (y-5)' = 8 1

V i d u 8

Cho a = (3,;2;2)va 6 = (18;-22;-5) Tim cbie't

c tao vdfi true tung goc tu

Chirng minh rling b6'n diem A, B, C, D la 4 dinh ciia hinh tii dien

I Tinh ^AO (O la trong tam ciia mat BCD ciia hinh tii dien)

Giai: a Neu bon di^m A, B, C, D la b6n dinh cua m6t hinh tir diSn thi

ba vector AB, AC, AD khong ddng phang

A B = (-4;4;1), AC = (2;2;2) =2(1;1;1),AD = ( - 3 ; - 5 ; 3 ) Cdch I: Xet bieu thirc [AC,AD].AB

1 1 1 1 1 1

- 4 -5 3 3 -3 -3 -5 + 4 1 1 + 1 1

= -4.8 - 4.6-2 = -58 9^ 0 (dpcm)

Cdch 2 Ta khong tim dugc cap so x, y thoa man A B = xAC + y A D ,

nghia la he sau v6 nghiem:

- 4 = X - 3>'

4 = X - 5y ^ he nay v6 nghiem

1 = X + 3y

b Goi 0(x, y, z) ta c6:

AO = AB + BO

AO = AC + CO AO=AD+DO 3AO = AB + AC + AD-(OB + OC + OD)

Vi O la irong tarn tarn giac BCD ntn OB + OD + OC = 0

Chiing minh di^m O la dilm duy nha't

64 Cho tii dien ABCD Goi A', B', C , D' la cac di^m theo thir tu chia cac

doan thang AB, BC, CD, DA theo ty s6 k:

A ' A B B C C D D

= k

A B B C C D D A

1 CMR vdfi moi di^m O bat ky trong kh6ng gian, ta lu6n c6:

OA + OB + OC + OD = OA' +OB+OC' +OD'

2 Vol gia tri nao ciia k thi b6'n diem A',B',C,D' d6ng phing?

65 Cho a = S,b =l,(a,6) = 30" Tinh goc tao bdi tdng va hifu hai

vecta a,b

66 Tarn giac ABC c6 toa d6 cac dinh A(3; -1;6); B(-l;7; -2), C ( l ; -3;2)

Chiing minh tam giac ABC la tam gidc vu6ng

67 Toa d6 trung diem cac canh cua tam giac ABC la (1;3;2), (0;2;0) (2; -2; 4) Tim toa do cua cac dinh tam giac ABC

68 Tim tren true hoanh mot diem each deu hai diem A(l; -3;7) va B(5;7;- -5)

69 AABC CO A(l;2; -1), B(2; -1;3), C(-4;7;5) Tim d6 dai ducmg phan giac trong BD

70 AABC CO A(-4; -1;2), B(3;5; -10) Tim toa d6 dinh C bie't trung di^m canh AC thu6c true tung,^ trung diem canh BC thu6c mpOxz

71 AABC CO A(6;2;3), goc toa d6 la trung diem canh AC Trong tam G ciia AABC thu6e true tung Tim toa do B, C

72 AABC CO A(-l;2;3), trong tam G trung vdi g6c toa do, Be Ox,

C e mpOyz Tim toa d6 B, C

73 Tim the tich tii dien ABCD biet toa d6 cac dinh A(2; -1;1), B(5;5;4), C(3;2;-1),D(4;1;3)

74 Cho hinh hop ABCD.A'B'C'D', biet A(-1,0,1), B(2;l;2), D(l;l;2), C'(4,-5;l)

a) Tim toa d6 cac di^m eon lai ciia hinh h6p

b) Tim the tich hinh hop tren

75 Cho lang tru diing ABCA,B,C, c6 day ABC la tam giac vudng

AB = A C = a, AA, = aV2 Goi M, N 1^ lugt la trung diem ciia doan

AA, va BC, Chiing minh MN la dirdng vu6ng goc chung cua cac ducmg thing AA,va BQ.Tinh VMA,BC, •

76 Trong khong gian toa d6 Oxyz cho 0(0; 0; 0), B(a; 0; 0), D(0; 1; 0), O' (0,0,a) la bon dinh ciia hinh hop chu nhat OBCD.O'B'CD'

Tim add 'BD L^T:

(Trich d6 thi DHXD, 1999)

77 Trong kh6ng gian toa d6 Oxyz cho hinh tii dien ABCD, bie't toa d6 cac dinh A(2; 3; 1), B(4; 1; -2) C(6; 3; 7), D(-5; -4; 8)

Tinh do dai ducmg cao ciia tii dien xua't phat tur A

(Trieh de thi DH Duoc, 1999)

II M A T P H A N G

A THUYfeT CAN NH6

1 Phuofng trinh long quat cua mat phdng

* Vector phap tuyen: Vecta n^O vu6ng goc vol mat phang («) goi la

vecta phap tuyen cua mat phang (a)

Dinh ly: Trong kh6ng gian Oxyz ne'u m|t phang (a) c6 cap vecto chi

phaang la a (a,; a^, a,) va b (b,; h^, b,) thi ( a ) c6 m6t vecta phap tuy^fn

CO toa do n =

= (ajbj - ajbj-, ajb, - a,b3;a,b2 - a^hi)

Nhuvay n =

Dinh nghia: Phuong trinh c6 dang Ax + By + Cz + D = 0 trong d6 A, B, C

khong d6ng thcri bang 0 (A^+ B^+ 0) dugrc goi la phuong trinh t6ng quat

cua mat phang

* Nhan xet

* Neu mat phang ( a ) c6 phirong trinh t6ng quat laAx + By + Cz + D = ti

thi vecta phap tuyen cua no la n (A,B,C)

* Phuomg trinh mat phang di qua di^m M„(Xo,yo'Zo) nhan vecto

n (A,B,C) 0 lam vecto phap tuySn la:

A(x - Xo) + B(y - yo) + C(z- ZQ) = 0

* Cac truorng hop rieng

Trong khong gian Oxyz cho (or): Ax + By + Cz + D =0 (1)

* Neu D = 0 thi (or) di qua gd'c toa do va ngugfc lai

* Neu trong phuomg trinh (1) khong c6 mat x(A = 0) thi mat phang

tuong ung se song song hoac chiia true Ox

Tuomg tu vdri y va z

* N6'u plijong trinh mat phang c6 dang Cz + D = 0

Kh6ng CO mat x va y (A = 0, B = 0) thi mat phing do song song hoac

Itrung voi mat phang Oxy Tuong tu mat phang Ax +D = 0 song song hoac ttrung vdi mat phang Oyz, mat phang By + D = 0 song song hoac triing v6i

lat phang Oxz

f * Ne'u A,B,C,D khac 0, bang each dat a = -—,b = -—,c = -— ta c6

Phuomg trinh (2) la phuang trinh ciia mat phang theo doan chan

2 Vi trf tuong doi cua hai mat ph^ng

Cho hai mat phang (or,): A,x + B,y + C,z + D, = 0

3 Khoang each tiir mot diem den mot mat phang

* Cho mat phang (a ):Ax + By + Cz + D = 0 va Mo(Xo;yo;Zo) khoang

each tiir diem MQ de'n mat phang a, duac tinh theo c6ng thiic:

d(Mo,a) = ^A'+B'+C

* Khoang each gifia hai mat phang song song la khoang each tiir m6t

diim bat ky cua mat phang nay den mat phang kia

49

B Vf DU

V i du 1: Viet phuong trinh mat phang di qua diem ( 2 ; - l ; - 1 ) va vu6ng

goc vdi true eao

Giai: Mat phang phai tim vu6ng goc vdi true cao nen nhan veeto

k = (0;0;1) lam vecta phap tuydn Vay phuong trinh mat phang phai tim la:

(Trich de thi vao D H LuSt Ha Noi, 1999)

Giai: Veeto phap tuye'n ciia (P): = (1; 0; 0)

Veeto phap tuye'n cua (Q): = (1; 1; -1)

2 Tinh the tich ciia tii dien ABCD

(Trich de thi D H Thuy san, 1999)

nABC = 0 2

0 1 2 -1 1 2 -1 0 2 0

= (0;5;0) = 5(0;1;0) Vay mp(ABC) c6 phuang trinh y + 1 = 0

Khoang each tir D den mp (ABC) la:

2 + 1 d(D, (ABC)) =

The tfch tir didn DABC la:

80 Viet phuang trinh mat phang qua true tung va di^m A(l;4;-3)

81 Viet phuong trinh mat phang qua hai diem A(7;2;-3), B(5;6;-4) va song song vdi true hoanh

82 Tim dien tich tam gidc do mp (P): 5x - 6y + 3z +120 =0 cat mp Oxy

83 Tim the tich tii dien do mp(P): 2x - 3y + 6z - 12 = 0 cat cac mat phang

88 Xet xem diem A ( 2 ; - l ; l ) va goc toa do O c6 nkm cung phia, hay khac

phia dd'i vdi mp.(P), mp(Q)

(P): 3 x - 2 y + 2 z - 7 =0 ( Q ) : 5 x - 3 y + z - 1 8 =0

89 Xet xem dia'm A(2; - 1 ; 3) va goc toa do O ciing n^m trong goc ciia nhi

ditn (P; Q) hay nam trong hai goc ke nhau cua nhi dien nay?

1 (P): 2 x - y + 3 z - 5 = 0 2 (P): 2x + 3 y - 5 z - 1 5 =0

(Q): 3x + 2 y - z + 3 =0 (Q): 5 x - y - 3 z - 7 = 0

90 Viet PT mp phan giac ciia goc nhi dien (P; Q) biet P:

2x -14y +6z - 1 = 0, Q : 3x+5y -5z +3 = 0 va g6c toa do thudc goc nhi

didn CO mp phan giac

91 Tim m de hai mp c6 PT: 3x -5y + mz - 3 = 0 va x + 3y + 2z + 5 = 0

vuong goc voi nhau

92 Trong khong gian Oxyz cho hinh lang tru diing ABC.A,B|C, v6i

A(0;-3;0), B(4;0;0); C(0;3;0); B,(4;0;4)

a) Tim toa do cac dinh A, va C, Viet phuofng trinh mat c^u c6 tarn la A

va tiep xiic vdi mat phang (BCC,B,)

b) Goi M la trung diem ciia A,B, Viet phuong trinh mat phang (P) di

qua hai diem A, M va song song vdi BC,

III PHUONG TRINH O U O N G T H A N G

A L t T H U Y ^ T C A N N H 6

1 Phirong trinh tong quat cua^ircmg thang

Ta da biet giao tuyen cua hai mat phang phan biet cat nhau la mot dudng thang Vay trong khdng gian Oxyz ta xem ducmg thang (d) la giao cua hai

mat phang phan biet (a) va (a') wdi:

( a ) : Ax + By + Cz + D = 0 (a'): A'x + B'y + C'z + D = 0 'Ax + By + Cz + D = 0

A'x + B'y + C'z + D'=0 (1)

A'+B'+C^^ 0, A ' +B'' +C'' ^ Ova A : B : C ^ A' : B': C

He phuong trinh (1) la phuong trinh tdng quat ciia ducmg thang

2 Phuong trinh tham so cua ducmg thdng

• Dinh ly: Trong khong gian Oxyz cho ducmg thang A di qua diem M() (xo; Yo; Z()) va nhan vecto a (a; b; c) 0 lam vecta chi phucmg Di^u

kien can va du de diim M (x; y; z) nam tren A la c6 mot s61 sao cho:

Khi do A di qua diem (XQ; yo; z^) va vecto chi phuong la a (a, b, c)

3 Phucmg trinh chinh tic cua ducmg thang

Tit (2) va neu a, b, c deu khac 0, khii t d cac phuong trinh tren ta c6:

^ - ^ 0 y-yo 2 - ^ 0 (3)

a b c

Dinh nghla: Phuomg trinh (3) v6i a, b, c 0 duoc goi la phuomg trinh

chinh tic cua ducmg thang

4 Vi trf tuong doi gifia ducmg thang va mat phdng

Cho duomg thing A:

x = XQ+at

y = yo+bt

Z = ZQ+Ct

va mat phang ( « ) : Ax + By + Cz + D = 0

De tim so giao di^m cua A va (a) ta giai he phuong trinh g6m cac

phuong trinh cua A va (a) va cho ta phuomg trinh

(Aa + Bb + Cc) t + Axo + Byo + CZo + D = 0 (*) M6i nghiSm ciia phuong trinh(*) ^n t umg vdi 1 giao di^m cua A va (a):

1) Ne'u Aa + Bb + Cc 7i 0 thi phuong trinh (*) c6 nghiem duy nha't =>

Trong Ichong gian Oxyz cho ducmg thang A di qua M va c6 vecto chi

phuong a , duomg thang A' di qua diem M' va c6 vecto chi phuomg a', ta c6:

1) A va A' cheonhau o

2) A va A' cat nhau o •

a,a a,a MM' =0 a,a ^ 0

3) A// A' <=>

4) A = A' ^

a,a a,MM a,a

= 0

^0

a,MM' = 0

6 Khoang each

1) Khoang tic mot diem din mot ducmg thang

Cach 1: Mu6'n tim khoang each tut mdt diem M de'n duomg thang (A) ta lam nhu sau:

• Viet pt mat phang (a) qua M va vu6ng goc vdi duomg thing A

• Tim toa do giao di^m H ciia (A) va (a)

• Tinh d6 dai MH, do chinh la khoang each tir M de'n A ky hieu la d(M, A)

Cdch 2: Sit dung c6ng thirc d(M, A) = M H = (Mo € ( A), a vecto chi phuomg ciia (A)

2) Khoang cdch giita dudng thing vd mat phang song song

De tinh khoang each gifia duomg thing A va mat phing (a) ta lam nhu sau:

• La'y m6t diim tuy y MQ e A

• Tinh khoang each d(Mo, a) tiif Mo den A Khoang each nay chinh la khoang each giua A va (a) va duoc ky hieu la d(A, a)

3) Khoang cdch giQa hai ducmg thang cheo nhau Cdch I Di tinh khoang each gifia hai ducmg thing cheo nhau A va A' ta

lam nhu sau:

• Lap phuong trinh mat phang a chfia A va song song \di A'

• Lay m6t di^m M tuy y trdn A' r6i tfnh khoang each tiir M d6n a

Khoang each nay chmh la khoang each gifla hai ducmg thang cheo nhau A va

A' ky hidu la d(A A')

a,d MM'

a,a Cdch 2: S\x dung cong thiic: d( A ;A') = h =

(M e A, a la vecta chi phuofng ciia A M ' e A', a'la vecto chi phuong

l 2 x - ; ; + 5 z - 4 = 0

Giai: Ta c6:

+

x + y - z + 3 = 0 2x - y + 5z - 4 = 0

(1) (2)

1 4 3x + 4 z - 1 = 0 <:> x = - - - z

3 3

1 4 Datz = t,thayvao(1): - - - t + y - t + 3 = 0<::> y =

10 7 + - r

^ Giai: Mat phang Oxy c6 vecto phap tuyen k = (0; 0; 1) va di qua goc

toa d6 nen c6 phuorng trinh: z = 0

Vay pt ducmg thing phai tim la:

f 5 x - 7 y + 2 z - 3 = 0

t z = 0

V f d u S Viet phuong trinh ducmg thang song song voi hai mp:

Giai: Ducmg tiiang (A) piiai tim la giao tuye'n cua liai mp (P) ^'a mp(Q)

M p (P) cluia (d,): x + 5 _ y-3 _ z + 1 di qua di^m (-5; 3; -1) va c6 vecto

2 - 4 3

ciii phuong = (2; -4; 3) V i (A) la giao tuye'n ciia mp (P) va mp (Q),

ma (P) // (P'), (Q) // (Q') nen (P) c6 cung vecto phap tuyen vdi (P);

3x + 12y - 3z - 5 = 0 ^ = (3; 12; -3) = 3(1; 4; -1), va (Q) c6 cung

vecto phap tuye'n vdfi ( Q ) : 3x - 4y + 9z + 7 = 0

^ = ( 3 ; - 4 ; 9 ) Vecto chi phuong u^ ciia (A) la tich c6 hudng ciia n, va n ^

=^ Vecto chi phuong

X + 3;; + 2z + 1 = 0 va song song v6i true Ox

Giai, M p phai tim thudc chum mp:

m(2x - 3y + z - 3) + n(x + 3y + 2z + 1) = 0, (m^ + n^^0)

ci> (2m + n)x + (-3m+3n)y + (m + 2n)z - 3m + n = 0

f V i mp phai tim song song vdri true Ox c6 vecto ehi phuong T = (1; 0; 0)

nen ta c6: 2m + n = 0 <=> 2m = - n Chpn m = - 1 , n = 2, ta duoc FT mp phai

Chon m = n = 1, ta C O mp phai tim la: 3x - 2y + 6z + 21 = 0

Chpn m = 19, n =85, ta c6 mp phai tim la: 189x + 28y + 48z - 591 = 0

Vf du 8 Tim toa do giao didm cua ducmg thang:

va mp 3 x - y + 2 z - 5 = 0

Giai: PT ducmg thartg c6 thi vie't dudri dang tham s6:

h^7 + 5t

y = 4 + t

z = 5 + 4t

Thay chung vao PT mp: 3(7 + 5 t ) - (4 + t) + 2(5 + 4t) - 5 = 0 o t = - l

Thay lai vao PT dudng thang, ta dugc toa do giao di^m (2; 3; 1)

Vi du 9

Tim diim Q doi xiJng vdi P(4; 1; 6) qua dudng thang:

'x-y-4z + l2 = 0 2x + y-2z + 3 = 0

Giai: Vie't PT ducmg thang (d) da cho

du6i dang tham so:

u = (2; - 2 ; 1) Viet PT mp (a) qua P(4;l;6) va vu6ng goc v6i (d) nSn

nhan u ciia (d) lam vecta phap tuydn

V a y P T m p ( a ) l a : 2 ( x - 4 ) - 2 ( y - l ) + z - 6 = 0 o 2 x - 2 y + z - 1 2 = 0

Tim toa d6 giao diem A cua (d) va ( « ) bang each thay PT tham s6' ciia

(d) vao PT ( a ): 2(-5 + 2t) - 2(7 - 2t) + 1 - 12 = 0

t = 4 Thay lai vao PT tham s6' ciia (d) dugc toa d6 A(3; - 1 ; 4)

Theo tinh chat doi xiing thi A la trung die'm PQ, de dang c6:

^A = T ( 2 / ' + Z O )

3 = ^ ( 4 + ^ , )

4 = ^ ( 6 + Z e )

Vi du 10 Tim diem B doi xiing v6i A ( l ; 3; - 4 ) qua mp: 3x + y - 2z = 0

Giai: Trudc het, tim PT ducmg thang

y^B Theo tinh chat d6'i xiing, ducmg thang AB di qua A va vuong goc vdi (P) ndn nhan vecta phap tuyen

n = (3; 1; - 2 ) ciia (P) lam vecto chi phucng, suy ra phuong trinh ducmg thang

x = \ 3t

AB la: • y = 3 + t Toa do giao di^m I

z = - 4 - 2 / ciia AB va (P) la nghiem ciia he PT:

x = l + 3t

y = 3 + t

z = - 4 - 2 t • 3x + y - 2 z = 0

-2x + 4y + 8z - 1 = 0

'-2x + 4y + 8z-l = 0

FT hinh chieu cua ducmg thang phai tim la:

2x-y + z-l = 0

V i du 12 (Dai hoc su pham thanh pho Ho Chi Minh - A, B - 20(X))

Trong khSng gian vdi he true toa d6 Oxyz cho cae ducmg thing:

• 1 " 2 " 3 '

x + 2y-3 = 0 2x-y + 3z-5 = 0

Tinh khoang each giiJa (D,) va (Dj)

Giai M , e D„ M , = (1; 2; 3) vecta chi phuong ciia (D,): ui = (1; 2; 3)

2) Chiing minh (d) tao vdi true Oz m6t goe kh6pg phu thu6e a

3) Viet PT hinh ehi^i (d') ciia (d) trdn mp Oxyj 4) Chiing minh vdd moi gia tri a ducmg thing (d') lu6n tiS'p xuc vdi mdt dudng tr5n e6' dinh thu6e mp Oxy

' x + h.-k = ^

94 Dudng thing (d,) CO pt:

[(1 -k)x-ky = ^

ring khi k thay ddi dudng thing d^ ludn:

1) D i qua 1 diem cd' dinh

2) Thudc mdt mp cd dinh

vdi k 0, bat ky Chiing minh

95 Viet PT mat phang chiia ducmg thang

v6i mat phang: x - 2y | f z + 5 = 0

x-2z^0 3x-2y + z-3^0

Cling thuoc mot mp, vijet PT mp do

97 Tim PT hinh chieu cua ducmg thang ^ ^ ^

100 Tim tap hop cac dien'^ M trong khong gian each deu ba diem A ( l ; 1; 1),

thang — ^ = - Y ducmg thang •

2) Cho A ( l ; 2; - 1 ) , B('[7; - 2 ; 3) va duomg thang (d) c6 PT:

x^\\_y-2 _z-2

3 '~ -2 ~ 2 ^ •

a) Chiing minh rangi ducmg thang (d) va dudng thang A B cung nam

trong mot mp |

b) Tim diem I e (d) sao cho A I + BI nho nha't

102 Cho mat phang (P): 2x + y + z - 1 = 0, va du6ng thang (d): = y =

Viet PT ctia ducmg, thang qua giao diem ciia (d) va (P), vu6ng goc vdi

2) Tinh khoang each gifla chiing

3) Vie't PT ducmg thang qua M(2; 3; 1) va cat (d,), (dz)

5 Cho hai ducmg thing : (d,)

1) Chiing to rang (d,), (d2) la hai ducmg thang cheo nhau

2) Tinh khoang each gifla (dj) va (d2)

(Dai hoc tong hop Ha Noi, khoi A, nam 1994)

/ Trong khong gian v6i hd toa do Oxyz cho diem M ( l ; 2; -1) va ducmg

(Dai hoc Bach khoa Ha Noi, nam 1997)

Trong khong gian, cho hinh binh hanh ABCD c6 hai dinh C(-2; 3; -5),

7 D(0; 4; - 7 ) va giao diem hai duomg cheo M ( l ; 2; - - ) 1) Viet PT ducmg thing chiia canh AB

2) Tinh khoang each tiir goc toa do den mp chiia hinh binh hanh

(Dai hoc dan lap Dong D6 Ha Noi, khdi A, 1997)

108 A ABC CO A ( l ; 2; 5) va PT hai trung tuye'n 1^:

x-3 _ y-6 _ z - 1 x-4 ^ y-2 ^ z-2

- 2 2 1 1) Viet PT chinh tac cac canh ciia AABC

2) Viet PT chinh tdc duomg phan giac trong goc A

(Hoc vien Ky thuat quan su B6 Qu6'c phong)

109 Trong khong gian vdi he toa do Decac vu6ng goc Oxyz cho hai'ducmg

thang:

Ix-2y+z-4=0

[x + 2y-2z + 4 = 0 va A, y = 2 + t z = \ 2t

a) Vie't phuong trinh mat phang (P) chiia ducmg thang A, va song song

vdi ducfng thang A2

b) Cho diem M (2; 1; 4) Tim toa d6 diem H thu6c ducmg thang A2 sao

cho doan thang M H c6 do dai nho nha't

(Trich de thi vao dai hoc khoi A - 2002)

110 Trong khong gian vdi he toa do Oxyz cho hinh chop S.ABCD c6 day

ABCD la hinh thoi, AC cat BD tai goc toa do O Biet A(2; 0; 0), B(0; 1; 0),

S(0; 0; 2 \/2 ) Goi M la trung diem ciia canh SC

a) Tinh goc va khoang each gifla hai ducmg thang SA,

BM-b) Gia sir mat phang (ABM) cat ducmg thang SD tai di^m N Tinh the'

tich khoi chop S.ABMN

(Trich de thi vao dai hoc khoi A - 2004)

111 Trong khong gian Oxyz cho 2 du5ng thing:

'x = -l + 2t vad.,: \ = \ t

1) Chung minh d, va 02 cheo nhau

2) Viet phucmg trinh dudng thang d vu6ng goc vdi mat phang (P):

7x + y - 4z = 0 va cat hai ducmg thang d, va d2

(Trich de thi vao dai hoc khoi A - 2007)

112 Trong khong gian vdi h6 tea d6 Oxyz, cho mat cSu

(S): + y^ + - 2x + 4y + 2z - 3 = 0 va mat phSng

(P): 2x - y + 2z - 14 = 0 1) Vie't phuong trinh mat phang (Q) chiia true Ox va cat (S) theo mot ducmg tron c6 ban kinh bang 3

2) Tim toa do diem M thuoc mat cSu (S) sao cho khoang each tiir M den

mat phang (P) Idn nha't

(Trich d^ thi vao dai hoc khoi B - 2007)

ON TAP CHl/ONG III

1. CAU HOI T R A C NGHlfiM

113 Trong khong gian Oxyz, toa do cua vecto a = 3 i + 2 k la

A Hinh binh hanh B Hinh vuong

C ffinhthoi D Hinh chu nhat

116 Tim tam va ban kinh hinh ciu c6 phuong trinh la:

+ + + 4x - 2y - 20 = 0 A.(1;-2;0);R = 5, B (-2; 0; 1); R = 5,

118 Tinh Idioang each tir diem M ( l ; - 1 ; 2) den mat phing (P) c6 phuong

trinh lOx + lOy + 5z + 2 = 0

120 Cho phuong trinh tham so ciia ducmg thang ( A )

phuong trinh chinh tSc cua dudng thang (A) la:

>1 Cho 3 diem M(3; 6; -7), N(-5; 2; 3), P(4; -7; -2) The thi phuong trinh

^ • duomg thang QP, Q la trung diem cua MN la:

A Cat nhau; B Cheo nhau; C Song song; D Triing nhau

II BAI TAP

5 Cho tii dien ABCD E, F, I theo thu tu la trung diem ciia AB, CD, EF

I a) Chiing minh lA + IB + IC + ID = 0

b) Vdi diem M bat ky trong khong gian, hay chiing minh:

4 MI = MA + MB + MC + MD

'124 Cho tii dien ABCD ma M la diem di dpng trong khong gian, G,, G2 Ian

lugt la trong tam tii dien va trong tam tam giac BCD 1) Chung minh G,C +G,5 +G,D = 0

2) Chvoig minli GA + GB + GC+ GD = 0

Ghi chu: Trong tarn cua tiJ dien la giao didm cac ducmg ndi m6i dinh ciia

tii didn tdi trong tftm cij a mat doi dien

3) Tim tap hop diem M thoa man he thiic:

MA+ MB + MC + MD = 4MB + MC + MD

125 Trong khong gian vdi he toa do Oxyz, cho tii dien ABCD voi

A(3; 2; 6;), B(3, -1, 0), C(0, -7, 3), D(-2, 1, -1)

a) Chiing minh tur dien c 6 cac cap canh d6'i vuong goc v6i nhau

b) Tim goc giua dirong ithang (d) di qua hai diem A, B va mp (a) di qua

badiemA,B,C

c) Thiet lap PT mat ciu ingoai tiep tii didn

(Dai hoc Bach khoa Ha ]Noi, nam 1996)

126 Cho mat ciu (S) c6 PT: (x - 1)^ + (y - 1)^ + z^ = 6 va hai duofng thing:

(di): X = 1 + 2t, y = 3 - 2t, z = 1 + 2t (dj): X = 1 - t, y = 2 + 2t, z = 1 - 3t

Viet PT mp tidp xuc mat c^u (S) dong thdi song song vdi (dj) va (d2)

127 Trong kh6ng gian vcti l?^ toa do Oxyz cho ba diim A(l; 0; 0),

B(0; 2; 0) va C(0; 0; 3)

1) Viet 0iuong tnnh tong quat cua cac mp (OAB), (OBQ, (OCA) va (ABQ

2) Xac dinh toa d6 tam I ciia mat ciu n6i tiep tii dien OABC

3) Tim toa d6 diem J d6'i xiing vdi I qua mat phang ABC

(Dai hoc Hue'-2000)

128 Trong khdng gian vdi he toa d6 Oxyz, cho diem A(l; 2; 1) va ducmg

thang (d) CO PT: - = — = z + 3 3 4

1) Vict PT mp di qua A va chiJa ducmg thang (d)

2) Tinh khoang each tilt die'm A da'n ducmg thang (d)

(Dai hoc Kien true Ha Noi, nam 1997)

19 Trong khdng gian v6i ht toa d6 Oxyz cho ba diem H -;0;0

K 0;1;0 ,1 4 a) Viet PT giao tuya'n ciia mp (KHI) va mp x + z = 0 of dang chinh tac b) Tinh cosin cia goc phang tao bdi mp (KHI) va mp Oxy

(Dai hoc Giao thdng Van tai Ha N6i, nam 1997)

T30 Cho hai ducmg thkg c6 PT: (d) j ^ ^ ~ ^

1; (Dai hoc Xay dung Ha Ndi (HS chua phan ban), nam 1997)

r31.,.Viet phuong trinh dudng thang di qua dilm A(3; -2; -4), song song vdi

X — 2 v + 4 z — \

mp 3x - 2y -3z -7 = 0, ddng thdi cat dudng th^g = =

(Dai hoc Thuy Igi Ha Ndi nam 1997)

132 Viet phuong trinh mp chiia gdc toa dd va vudng gdc vdi hai mp cd PT:

X - y + z + 7 = 0 va 3x + 2y - 12z + 5 = 0

(ViSn Dai hoc Md Ha Ndi, khdi A, nam 1997)

133 Cho hai didm A(0; 0; -3), B(2; 0; -1) va mat phang (P) cd phuong trinh la: 3x-8y + 7 z - l =0

1) Tim toa dd giao di^m ciia dudng thang di qua hai di^m A, B vdi mat phang (P)

2) Tim toa dd diem C nam trfen mp(P) sao cho tam giac ABC deu

(Dai hoc Qudc gia Ha Ndi - A - 2000)

7 ^

ON TAP CUOl N A M

134 Mot hinh h6p chu nhat c6 do dai dudng cheo d, no tao vdri day goc a

va mat ben \dn goc p Chiing minh the tich hinh hop bario

d^'sin a sin /? -yjcosia + P) cos(a - P)

135 Day hinh chop la tam giac vuong c6 canh huyen a va goc nhpn a Mat

ben qua canh huyen vuong goc v6i day, hai mat con lai tao vdi day goc

B Chiing minh the tich hinh chop bang — Q sin latgP

24V2sin(a + 45°)

136 (Dai hoc Quoc gia thanh pho Ho Chi M i n h A - 2000)

Cho tam giac din ABC canh a Tren dudng thdng d vuong goc vdi m;ii

phang (ABC) tai A lay diem M Goi H la true tam tam giac ABC, K It

true tam tam giac BCM

1) Chiing minh rang M C 1 (BHK) va H K 1 (BMC)

2) K h i M thay doi tren d, tim gia tri Idn nhat cua the tich tii dien KABC

137 Cho hinh chop tii giac deu S.ABCD vdfi day la hinh vuong ABCD co

canh bang a Mat ben tao vdi day mot goc 60"

Mat phang (P) chiia canh A B va cat SC, SD l^n luot tai M va N Cho biet

goc tao bai mat phang (P) va mat day hinh chop la 30"

1) T i i giac A B M N la hinh gi? Tinh dien tich tii giac A B M N theo a

2) Tinh the tich hinh chop S.ABMN theo a

138 Cho goc tam dien dinh O, cac goc b dinh deu bang 60" Tren cac canh

Ox, Oy, Oz ta lay cac didm A, B, C sao cho OA = a, OB = b, OC = c

1) Cho a = b = c, thi hinh chop OABC co gi dac biet? Tinh khoang each

tir O den mp (ABC) va tinh the tich ciia hinh chop nay

2) V 6 i a 5^ b 9i c, tinh cac canh cua AABC theo a, b, c Chiing minh dien

kien can va du de BAC = I v la be + 2a^ = a (b + c)

3) Cho biet a va b + c = d, BAC = Iv Tinh the tich cua hinh chop the(

a va d Lap phuong trinh de tinh b, c trong truomg hop nay Tim diei

kien de tinh dupe b, c

4) Chiing minh rang nS'u di^u kien tren duac nghiSm diing thi mot trong hai so b, c nho hem a, s6' con lai 16n ban 2a

j39 Trong mp (?) cho du6ng thang d co dinh va mot diem c6' dinh O g d, mot goc vuPng Oxy quay quanh O, Ox va Oy cat d tai A va B Cho d' ± P tai O Lay S € d' thoa man SO = ^ , SA = |oA Khoang each t i i O

d€n d bang a va OAB = a

a) Tinh a

b) Ke OE 1 SA, OF 1 SB T i m quy tich E, F khi xOy quay quanh O c) Gpi G la trpng tam A SAB, I la tam mat cau ngoai tiep tii dien SOAB Chiing minh O, G, I thang hang

Cho hinh c^u (O, R) tie'p xiic v6i mat phang (P) Cho hinh non (nam

cdng phia vdi hinh cin doi vdfi (P), day thupc (P), dudng cao h, ban kinh

day bang R Cat hai hinh bang mp (Q) // (P), each nhau mot khoang bang X

a) Cho X < 2R va X < h Tinh t6ng dien tich S cua hai thiet dien Bieu

thiic tim dupe co con thich hop kh6ng n€\x h < x < 2R (keo dai cac

i duofng sinh ciia hinh non dd chiing cat (Q)

\h) Khao sat sir bieh thien va ve do thi S (x la doi so) Bien luan cac

trucmg hpp

141 Cho hinh chop S.ABC co day ABC la tam giac deu canh bang a, SA

vuong goc vori mat phang (ABC); SA = a; I la trung diem ciia BC

a) Tinh khoang each tir A den mat phang (SBC)

b) Viet phuong trinh mat ci\x ngoai tiep tii dien SAIC

•2 Trong khong gian vdri he toa dp Oxyz cho hai diem: A(0; - 2 ; 0), B(2; 1;

4) va mat phang (a): x + y - z + 5 = 0

a) Viet phuong trinh tham so ciia dudng thang d diilqua A va B

b) T i m tren dudng thang d diem M , sao cho khogng each tir M den mat phang (a) bang 2 V3

c) Viet phuong trinh mat cin (S) co dudng kinh AB Xet vi t r i tuong doi

giiia mat c^u (S) va mat phang (a)

nl

143 Trong khdng gian vori he tea d6 Oxyz cho 4 di^m: S(2; 2; 6), A(4; 0; Oj

B(4; 4; 0), C(0; 4; 0)

a) Chung minh rSng hinh chop SABCO la hinh chop Hi giac d^u

b) Tmh th^ tich cua khoi chop SABCO

c) Vie 't phuang trinh mat ciu ngoai tie 'p hinh chop S.ABCO

144 Trong khong gian v6i he toa do Oxyz cho hai ducmg thang c6 phuang

trinh Mn luat la:

A,: \2x-y + 3z-5 = 0 [ x + 2y-z = 0 A , : 2x-2y-3z-n = 0 va diem A(3; 2; 5)

A , : • < va A,: f x + y + z-4 = 0 ^ ^ x-l 2x-y + 5z-2 = 0 -2

2x-y-2z-3 = 0

a) Tim toa do diem A' doi xiing vdi diem A qua ducmg thang A2

b) Lap phuang trinh mat phang di qua ducmg thang Aj va song song vui

ducmg thang A2

c) Tinh khoang each giua hai ducmg thang Aj va A2

145 Trong khong gian vdi he toa do Oxyz cho 2 duotng thing:

y_z-2

3 1

a) Xet vi tri tuang doi cua A, va A 2

b) Cho diem A(0; 1; 3) Tim diem M trtn A 2 sao cho doan AM ng '

nha't

146 Trong khOng gian vdfi he toa d6 De cac vu6ng goc Oxyz cho hai die

A(l; 2; 1), B(2; 1; 3) va mat phang (P): x - 3y + 2z - 6 = 0

a) Viet phuang trinh mat phang (Q) di qua A, B va vuong goc v6i m.'

phang (P)

b) Goi ducmg thang A la giao tuye'n ciia hai mat phang (P) va (Q) Hay

vie't phuang trinh chinh tac cua dudng thang A

c) Goi H la hinh chieu vu6ng goc cua A tren mat phang (P) Tim toa do

la) Vie't phucmg trinh mat phang (P) chiia A, va song song v6i A2

lb) Tinh khoang each giiia A, va Aj

|c) Viet phuang trinh ducmg thang A3 di qua M(2; 3; 1) va cat ca A, fva A 2

l48 Trong khong gian Oxyz cho 2 mat phang:

(«):2x-y + 2 z - l = 0 (/?):x + 6y + 2z + 5 = 0 a) Chiing minh rang ( or) va (/?) vu6ng goc voi nhau

b) Lap phuang trinh t6ng quat cua mat phang (P) di qua goc toa do O va chiia giao tuyen ciia hai mat phang ( a ) va (/?)

HLfdNG D A N G I A I - D A P S O

1 a) Trong khoi da dien m6i

canh la canh chung ciia dung

2 mat

b) Cung sir dung tinh chat

tren

2 Chia khoi lap phuofng thanh 6

khoi tii dien

3 Cho tir dien deu ABCD Tarn

ctia cac mat ABC, BCD,

ACD, ADB la M, N, P, Q Hinh 43

Xet khoi tii dien MNPQ Dung dinh ly Talet ta chimg minh M N = AD

NP = ^ ; M Q = ^ D o d 6 M N = NP = NQ = MQ = MP = PQ= ^

Khoi tii dien deu canh a c6 the tich ^ ~ ^

Khoi tii dien MNPQ canh - c6 M ti'ch V =

4 a) AEDF - AEFC - BEDF - BEFC

b) VAEDF = VAEFC vi S AADF = S A^pcva ciing chidu cao EH Chiing minh

tuong tu ta suy ra:

^AEDF - ^AEFC " ^BEDF " V

BEFC-c) Neu ABCD la tii dien deu khi do ta chimg minl> (EDC) la mat phang

doi xiJng cua hinh, (ABF) cung la mat phang dd'i xiing cua hinh Khi do

ta chimg minh dugc 4 khd'i til dien AEDF, AEFC, BEDF va BEFC bang

nhau theo nghia c6 phep ddi hinh (doi xiing qua mat phang) bien khoi

nay thanh Lhoi kia

c) Cho S.AB'C CO chieu cao SC, day la A AB'C vuong b B' vi

(AB' 1 (SBC) nen AB' 1 B'C) Ta c6 AB' = ^ va A C SC = a.AC

chop CO cung chieu cao Ta dh dang chiing minh

-i AC ED "

b) BC 1 A H , BC 1 A H => BC 1 ( A A ' H ) => BC 1 A A '

=> BC 1 BB' => BCC'B' la hinh chG nhat

c) A ' A H = 60° => A A ' H = 30° A A ' = 2 A H = 2a => SBCC-B' = 2a'

Tiir E ve EK 1 AA' => BKC la thiet diSn thing cua lang tm Ta tinh duoc

Vay = a l Vl3 + 2a' = a'(2 + Vo ) don vi dien tich

Ve chop S.ABC TiT B va B' ha BH 1 (SAC), B'H' ± (SAC) K h i do

=> A C i= A B cotg30°' = AB> /3 ma AB = AC.cotg30° = b V3 => A C = 3b

b) ^ c.Vc = S A A B C C C =

= ( l b b V 3 ) V % ^ = ^ b ^ ^ = b l V 6

81

Hinh 49

10 Ha A , H 1 A C ( H e AC) A A , B D can ( do A , B = A,D)

BD X A , 0 Mat khac BD 1 A C BD 1 (A,AO) BD 1 A , H

Dan den A , H 1 (ABCD) Dat A, AC = c(? ta c6 he thiic:

V = a.b.x = abc ^ (don vi the tich)

Goi 0 0 , la giao tuyen ciia 2 mat cheo ( A , C i C A ) va ( B , D i D B ) Qua

l € 0 0 , ke lin lugt 2 ducfng thang

K E va M H deu vuong goc v6i

0 0 ] K h i do a la goc gifia M H va

: E va M E H K la thiet dien thang

:iia h6p Dat K E = x, M H = y thi

=> khoang each ttr A den (SBC) chinh la khoang each tir E de'n (SBC) Ke EK 1 SH

EK 1 (SBC) => E K = 2a

Do BC ± SH, BC J - O H => SHO la goc gifla (SBC) va day

Dat S H O = x t a c 6 B H = ^ , O H - — , SO = O H tgx =

• sinx smx cosjx:

^ ^ y ^ « = i ^ ^ ° = 3 c o s x s i n ^ x

-V,h6p dat gia tri nho nhat khi m(x) = cosx sin^x dat gia tri Ion nha't Ta

C O m'(x) = - sin^x + 2sinx cos^x = sinx (2cos^x - sin^x)

= sinx (2 - 3sin^x) = 3sinx ' 1 2 ^ ^ Sinx

12

V i X € (0; 90°) nen m ( x ) = 0 khi sinx = ^ - va qua gia tri do thi m'(x)

[2

doi dau tir (+) sang (-) nen m(x) dat gia tri Idn nhat khi sinx = ^ -

Ket luan: V^^^p dat gia tri nho nha't khi sinx = sinSHO = ^ -

14 Cho tii dien ABCD Dung qua A B

va CD cap mp // duy nhat, m6i mat

chiia 1 du5ng thang Tuomg tu vdi

A D va BC; A C va BD ta cung dung ^

2 cap mp nhu vay

Ta dung duac 1 hinh hop ngoai

tiep tii didn do la AEBF MDNC

Chidu cao cua hop la khoang each

gitta A B va CD

Theo bai V tii dien = VABEF.MDNC = >

the tich bon khoi t i i dien A M C D ,

BNCD, ABFC, ABED ma m6i khoi

tii dien = - khoi hop

Do APQ, APR, ASR la vuong =>

AP" + A Q ' = 4c'

A Q ' + A R ' = 4a' AP' + A R ' = 4b' AP' + A Q ' + A R ' = 2(a' + b ' + c') Vay A P ' = 2(b' + c' - a'); A Q ' = 2(a' + c' - b'); A R ' = 2(a' + b' - c')

6 a) Keo dai EP cat A'D' tai N, cat A'B' tai M

; A N cat DD' tai P, A M cat BB' tai Q

J Vay thi€t dien c^n tim khi cat lap

phuong bai (AEF) la ngu giac APEFQ

b) Coi thi tich lap phucfng la V

V , la th^ tich khoi ABCD.QEC'FD

V2 la thi tich kh6'i AQEFD.B'A'D'

V 3 la the tich tii dien A M N A '

V 4 la th^ tich tii dien PEDN V5 la the tich tir dien QMB'E'

17 Theo dSu bai ta tha'y A C , B'D' va SO ( O = A C n BD) d6ng quy tai I ^

I la trung diem SO

-18. Khoi ABCD dugc phan chia

thanh b6'n khoi tiir dien OBCD,

OCAD, OABD, OABC

SABC V A.BSC AA,

Chung minh tuofng tu ta c6 M.CAS

Vdri A', B', C la giao diem ciia ba mat ben: SBC; SCA; SAB voi cac

duong thang qua M va // vdi SA; SB; SC, M 6 day ABC

De tha'y cac tam giac ABC, ^ ACD, ABD, BCD deu c6 dien

tich bang nhau va bang nufa dien tich S cua hinh binh hanh ABCD Cac hinh chop S.ABC, S.ACD, S.BCD, S.ABD CO cung chieu cao. vay VsABC = VSACD = VSABD

goc doi vuong n6n la t i i giac

n6i tie'p du&ng tron dudng

v a y cac tarn g i a c ABC, ADC, AB'C, AO'C, AC'C deu la tarn g i a c v u o n g

C O C h u n g canh h u y^ n A C c6 d i n h O s u y ra OA = OC = OD' = OB' = O C

= OD = OB v a y 7 6iim A , B, C, D, B', C, U e a c h deu didm O co dinh

AC vdi khoang khdng d6i c) Do BC ± (SAB) nen SB la hinh chieu ciia SC tren (SAB) va do do BSC = X Ta c6:

3 sinx ' 3sinxcos^x cos^2x

22 a) Do H la true tarn A BCD nen B H 1 CD

Mat khac A H 1 (BCD) A H 1 CD Vay CD 1 (ABH) => CD J AB

Gia thiet cho A C 1 A B => A B 1 (ACD) A B 1 A D b) Tiif A B = A C = A D HB = HC = HD, nghla la H la tarn ducmg tron ngoai tia'p A d^u BCD

Xet tarn giac vu6ng A H D c6 — ^ = — ^ + • ^

H J ' AH^ HD^ '

Goi X la canh day tii dien ABCD thi BE ^ cV5 3dh

Goi I la giao diem ciia CD vdi A , ve IJ ± A B thi IJ la duofng vuong g( c

Chung cua A B va CD Ta c6 SACAB = SADAB C K = D L trong do

CK 1 AB, D L 1 AB

Goi L' va K' la hinh chieu ciia L

va K tren {a), khi do L' va

K' € A va L L ' = K K '

A DLL' = A C K K ' DL' = CK'

^ AIDL' = AICK'

Vay I D = IC => dpcm

23 a) Cho chop deu day la da giac

deu n canh Chia hinh chop deu

thanh n hinh chop c6 chieu cao

SO CO day la A can AOB Biet

canh A B = a, goc giiia canh ben

va day la a = SBO = SAO

Vay chop deu n giac la ^ cot g —

^ 4cosP n

24 Lam tuong tu bai 23 chii y cong thiic tinh dien tich xung quanh va the

tich chop cut la:

S = ^ chu vi 2 day x trung doan

V = ^ h ( B + B ' + ylW )

25 Theo gt: SA la dudng cao hinh

chop

a) mp S A D va mat phang thid't di6n

chijfa A D // B C nen chiing cit nhau

theo giao tuyen M N // A D // B C

=> thiet dien la hinh thang

*ABCD-Chung minh tuong tu cho cac hinh chop khac theo d^u bai => dpcm

27 a) De cho gon, ta dat SA = a, SB = b, SC = c K e S H J , mp A B C Coi A

la dinh, SBC la day thi the tich tir dien la: V = - abc (1)

6 • J^u coi S la dinh, A B C la day thi the tich tir dien la:

Thay vao (3) cho: SABC= ^ +6'c' + c ' a ' (4)

« V S SABC ^ SsAB + SsBc + SsAc (dpciTi)

b) De tha'y: V = - SA.SB.SC (xem (1) eua eau a) => V = ^ ax(k - x)

cheo day: AH = 2^^^ ^~2~' ^""^ '^^^ Pitago vao tarn giac

vuong SHA cho SH =

Vay V = -dt ABCD.SH=

3 6 b) Goi J la trung diem BC thi SJ

Gia tri Ion nha't, nho nha't ciia MN:

- Neu a = Iv thi cos a = 0 va

MN' = d' + k Vm, n thoa man

m' + n' = k => gia tri 16n nha't cua

MN bang gia tri nho nha't ciia

MN= -Jd^+k

- Neu 0 < a <lv thi cos a > 0 va 2nin < m' + n' = k ( dang thiic xay

ra khi m = n = J— (3) n&n c/' +k-kcosa < d' + k - 2mncosa <

mp(BAN) va A BAN vudng a B

Vay V = - dt A BAN.MA = - - BA.BN.MA = - mnd < 3 3 2 6

32 a) Goi d la cac du&ng thang vu6ng goc mp ABC tai A Vi AABC can

dinh A nen A MBC can dinh M Ke MI 1 BC thi I la trung diem BC

=> AI J BC MI, AI vijra la ducmg cao, dudng trung tuyen ciia AMBC

va AABC =:> G, H la trong tarn, true tarn, AMBC thuoc MI Hieo tinh chat

trong tam AMBC: Gl_ MI 3 I IG =- IM 3 G la anh cua M trong

phep vi tu tam I, ti s6' ^ => M chay tren d thi G chay tren ducmg thang d' la anh ciia ducmg thang d trong phep vi tu tam I, ti s6' - : d' 1 (ABC) tai trong tam AABC, dV/d

Quy tich true tam H: (HS tu lam)

^ b) Do d 1 mp ABC nfin mp

f MAI 1 mp ABC, ke HK 1 AI

thi HK 1 mp ABC nen HK la

ducmg cao hinh chop HOBC, c6

Hinh 69

V = | d t (ABOC).HK

Nhimg A HOC CO dinh nen V Idn nha't khi HK dai nha't HK = ^ OI

(OI la dudng kinh dudng tron ngoai tiep A HOI (theo cau tren)

33 a) Dap so: S^p = a'(l + V3 )

V =

6 b) Goi M la trung diem SA, theo gia thiet cac mat ben la cac tam giac

d^u SA 1 B M , SA ± D M => B M D la goc phang ciia nhi dien

34 Goi F, E \in lugt la trung di^m

cua cac canh AB, CD K la hinh

chieu ciia F tren SE, de tha'y

DC 1 (SEF) FK 1 DC nen

FK 1 (SDC) => (ABK) la mat

phang qua AB va vuong goc (SDC)

va D'C = (ABK) n (SDC) Thiet

dien ABCD' la hinh thang can

De dang chiing minh SEF la tam

giac deu (SF = EF = SE = 2a)

D'C = a, FK = a V 3 , ta c6

AB = 2a (h 70)

3

1 Vay dien tich thiet dien ABCD': dtABC'D' = - (AB + D'C)

b) De dang chiing minh duac SK ± (ABCD')

V3a^

Goi O la tam day, khi do S O la duong cao ciia hinh chop d^u S A B C D

va S O cung la ducmg cao ciia tam giac d^u S E F canh bang 2a, nen

I S O = a V3 vay VS^BCD = - 2a.2a.a V3 = - V3 a'

=> M O H = a

Du5ng thing d 1 mp (P) tai Oc6'dinh,d l O H

vay ( luon tao vdri d m6t

goc 90" - a khong ddi va di

qua di^m O c6' dinh e d =>

C thudc mat non tron xoay

CO true la d va goc b dinh

P = 2(d; i) = 1 8 0 ° - 2a,

va dinh ciia mat non la O

36 Ne'u SAO = 30° =i> SO = — =2,5

Hinh 71

Ilk

vay dien tich thiet dien qua true la:

38 OA la hinh chieu cua ducmg sinh 5

SA tren day nen SAO la goc tao

bofi dudng sinh va day

Trong tam giac vuong SAO:

b Theo cong thiic tinh dien tich xung quanh hinh non: S^q = Ttrl

6'day r = OA = 3m, 1 = SA = 5m Vay S,p = TT 3 5 = ISTI (m^)

c Theo c6ng thiJc tinh the tich kh6'i non: V = Tr.r'h

CJ day r = OA =3m, h = SO = 4m rz> V = n.3'.4

V= 3671 (m')

39 a) S.ABCD c6 day la hinh vuong ABCD c6 cac canh ben SA = SB = SC = SD (vi la cac ducmg sinh cua hinh non) suy ra ducmg cao SO cua hinh non cung la ducmg cao SO cua hinh chop, ma O la tam cua ducmg tron day cung la tam cua hinh vu6ng ABCD Vay S.ABCD la hinh chop deu

b) SAB la tam giac d6u canh a

(vi SAB la tam giac can va SAB = 60"(gt)

Goi H la trung die'm cua A B thi

SH = ^ (SH la ducmg cao cua tam giac SAB deu canh a)

De dang c6 O H = ^ Trong tam giac vuong SOH c6:

Theo c6ng thiic tinh the tich hinh chop: V = ^ Bh

6 day B = SABCD = a'; h = S O = a4i

v a y the tich hinh chop S.ABCD la V = ^ a' ^ c) Theo c6ng thiic tinh the tich hinh non:

hinh chie'u cua SA tren (ABC),

vay S A O la goc tao hdi canh

ben S A va day => SA6= <p

Goi M la trung di^m cua

De dang tinh duoc SO = OA tg ^ => SO = tgcp

Trong tarn giac vu6ng SOM ta c6 SM^ = SO^ + OM^ =^tg^^ + ^

V, 108 • 12 " ~ 9 ~