T hS. L E H O A N H P H O
e
i
B
H
O
O
I
D
C
L
S
f
I
d
N
N
H
G
,
G
I
O
I
- Ddnh cho HS lop 12 on tap & nang cao klnang lam bdi
Chudn bi cho cdc ki thi qudc gia do Bd GD&DT to chac.
T
O
A
N
i
Boi duBng hoc sinh gioi
Toan Dai so 10-1.
- Boi duQng hoc sinh gioi
Toan Dai so 10-2.
- Boi dii9ng hoc sinh gioi
Toan Hinh hoc 10.
Boi duQng hoc sinh gioi
Toan Dai so 11.
Boi duQng hoc sinh gioi
Toan Hinh hoc 11.
Bp de thi ta luan Toan
hoc.
Phan dang va phirong
phap giai Toan So' phtfc.
Phan dang va phuong
phap giai Toan To hdp va
Xac suat.
1234 Bai tap tu luan
dien hinh Dai so
giai
tfch
1234 Bai tap tii luan
dien hinh Hinh hoc
iu'g'ng giac
ThS. L E H O A N H
Nha gido Uu tu
B
O
I
D
H
O
C
U
S
Q
I
N
N
G
H
PHO
,
G
I
O
I
T
O
A
- Danh cho HS lap 12 on tap & nGng cao kinang lam bai.
- Chudn bi cho cdc ki thi quoc gia do Bo GD&DT td choc.
erjd
©G
1 N6I
NHA XUAT BAN DA! HQC QUOC GtA HA NQI
N
Ldi N6l
D A U
De giup cho hoc sinh lap 12 cd them tai lieu tu boi dudng, nang cao va ren
luyen ki nang gidi todn theo chuong trinh phdn ban moi. Trung tdm sdch gido due
ANPHA xin trdn trong gioi thieu quy ban dong nghiep vd cdc em hoc sinh cuon:
"Bdi dudng hoc sinh gidi todn Hinh hoc 12" nay.
Cuon sdch nay nam trong bd sdch 6 cuon gom:
- Boi duong hoc sinh gidi todn Hinh hoc 10.
- Bdi duong hoc sinh gidi todn Dqi so 10.
- Boi dudng hoc sinh gidi todn Hinh hoc 11.
- Bdi duong hoc sinh gidi todn Dqi so - Gidi tich 11.
- Boi duong hoc sinh gidi todn Hinh hoc 12.
- Boi duong hoc sinh gidi todn Gidi tich 12.
do nhd gido uu tu, Thac si Le Hoanh Phd to chuc bien soqn. Ndi dung sdch
duoc bien soqn theo chuong trinh phdn ban: co bdn vd ndng cao mdi cua Bd
GD & DT, trong dd mdt so van de duoc mo rong vdi cdc dang bai tap hay va khd
dephuc vu cho cdc em yeu thich muon ndng cao todn hoc, cd dieu kien phdt trien
tot nhd't khd nang ciia minh. Cuon sdch la sukethua nhung hieu bii't chuyen mdn
vd kinh nghiem gidng day cua chinh tdc gid trong qua trinh true tiep dung lop boi
duong cho hoc sinh gidi cdc lop chuyen todn.
Vdi mot noi dung sue tich, tdc gid dd cogang sap xep, chon loc cdc bdi todn tieu
bieu cho tirng the loai khdc nhau ung vdi ndi dung ciia SGK. Mot so bai tap cd the
klw nhung cdch giai duoc dim tren nen tdng kien thuc vd ki ndng co ban. Hoc sinh
can tu minh hoan thien cdc ki ndng cung nhu phdt trien tu duy qua viec gidi cdu
bdi tap cd trong sdch trudc khi ddi chie'u vdi loi giai cd trong sdch nay, cd the mot so
ldi gidi cd trong sdch cdn cd dong, hoc sinh cd the'tu minh lam rd hon, chi tiei hon,
cung nhu tu minh dua ra nhirng cdch lap luqn moi hon.
Chung tdi hy vong bd sdch nay se la mdt tdi lieu thiet thuc, bo ich cho ngudi
day vd hoc, dqc biet cdc em hoc sinh yeu thich mdn todn vd hoc sinh chuan bi cho
cdc ky thi qudc gia do Bd GD & DT to chuc sap tdi.
Trong qua trinh bien soqn, cudn sdch nay khdng the tranh khdi nhung thieu
sdt, chung tdi ra't mong nhdn duoc gdp y ciia ban doc gan xa debo sdch hoan thien
hon trong lan tdi ban.
Moi y kien dong gop xin lien he:
Trung tam sach giao due Anpha
225C Nguyen Tri Phucmg, P.9, Q.5, Tp. HCM.
- Cong ti sach - thiet bi giao due Anpha
50 Nguyen Van Sang,'Q. Tan Phii, Tp. HCM.
DT: 08. 62676463, 38547464.
Email:
[email protected]
Xin chan thanh cam on!
Tdc gid
C h u o n g
§1. K H O I
I :K H O I
B A D I E N V A T H E
D A D I E N V A P H E P B I E N
T I C H
H I N H
A. KIEN THUC CO BAN
- Hinh da dien gom mot so huu han da giac phang thoa man hai dieu kien:
(1) Hai da giac bat ki hoac khong co diem chung, hoac co mot dinh chung,
hoac co mot canh chung.
(2) M o i canh cua mot da giac la canh chung cua dung hai da giac.
Hinh da dien chia khong gian lam hai phan: phan ben trong va phan ben
ngoai. Hinh da dien ciing voi phan ben trong cua no goi la khoi da dien.
M o i khoi da dien co the phan chia duoc thanh nhirng khoi tii dien.
Phep doi hinh trong khong gian la phep bien hinh bao toan khoang each
giua hai diem bat ki.
Phep tinh tien, phep doi xung true, doi xung tam, phep doi xung qua
mat phang la nhirng phep ddi hinh.
Hai hinh da dien gpi la bang nhau neu co mot phep doi hinh bien hinh
nay thanh hinh kia.
- Phep v i tu tam O ti so k * 0 la phep bien hinh moi diem M thanh diem
M ' sao cho O M ' = k O M .
Hinh H duoc goi la dong dang vdi hinh H ' neu co mot phep v i tu bien
hinh H thanh H i ma hinh H i bang hinh H'.
- Mot khoi da dien duoc goi la khoi da dien loi neu bat k i hai diem A va
B nao cua no thi moi diem cua doan thang A B cung thuoc khoi do.
Khoi da dien deu goi la khoi da dien loi co hai tinh chat sau day:
(1) Cac mat la nhirng da giac deu va co cung so canh;
(2) M o i dinh la dinh chung cua cung mot so canh.
- Co nam loai khoi da dien deu: khoi tu dien deu, khoi lap phuong, khoi
tam mat deu, khoi mudi hai mat deu, khoi hai muoi mat deu.
B. P H A N D A N G T O A N
D A N G 1: K H 6 l O A D l £ N
Hinh H ciing vdi cac diem nam trong H duoc goi la khoi da dien gidi han
bdi hinh H .
M o i da giac cua hinh H duoc goi la mot mat ciia khoi da dien. Cac dinh,
cac canh cua moi mat con goi la dinh, canh cua khoi da dien. Cac diem
nam trong hinh H con goi la diem trong cua khoi da dien.
Khoi da dien duoc goi la khoi chop, khoi chop cut neu no duoc gidi han
bdi mot hinh chop, hinh chop cut. Tuong tu cho kh6i chop n-giac, khoi
chop cut n-giac, khoi chop deu, khoi tii dien,...
Khoi da dien dugc goi la khoi lang tru neu no duoc gioi han boi mpt
hinh lang tru, tuong tu cho khoi hop, khoi hop chu nhat, khoi lap
phuong...
Phan chia va lap ghep cac khoi da dien: M o i khoi chop va khoi lang tru
luon co the phan chia duoc thanh nhung khoi tu dien bang nhieu each
khac nhau.
Chu y: Dac so O-le cua khoi da dien loi:
Doi voi moi khdi da dien loi H . ta ki hieu D la so dinh, C la so canh, M la
so mat ciia H thi so / ( H ) = B - C + M = 2.
V i du 1: Chung minh rang neu khoi da dien co cac mat la tam giac thi so mat
phai la so chan. Hay chi ra nhirng khoi da dien nhu the voi so mat bang
4. 6. 8. 10.
Giai
Gpi s6 canh cua khoi da dien la C. so mat la M . V i moi mat co ba canh
va moi canh lai chung cho hai mat nen 3M = 2C. Suy ra M la so chan.
Sau day la mpt so khoi da dien co so cac mat tam giac la 4, 6. 8. 10.
V i du 2: Chiing minh rang moi dinh ciia mpt hinh da dien la dinh chung ciia
it nhat ba canh va la dinh chung ciia it nhat ba mat.
Giai
Ta dirng phan chiing. Neu xuat phat tir mpt dinh nao do chi co hai canh. thi
moi canh nhu thi la canh ciia chi mpt da giac. trai voi dieu kien trong djnh
nghia ciia hinh da dien.
Vay moi dinh phai la dinh chung ciia it nhat la ba canh. va vi vay no
ciing phai la dinh chung ciia ba mat.
V i du 3: Chiing minh rang n i u khoi da dien co moi dinh la dinh chung cua
ba canh thi so dinh phai la so chan.
6
Vi
Giai
Gia sir khoi da dien co C canh va co D dinh. V i moi dinh la dinh chung
cua ba canh va moi canh co hai dinh nen 3D = 2C. Vay D phai la so
chan.
du 4: Chung minh rang n i u khdi da dien co cac mat la tam giac va moi
dinh la dinh chung cua ba canh thi do la khoi tti dien.
Giai
Goi A la mot dinh ciia khoi da dien. Theo
gia thiet, dinh A la dinh chung cho ba canh,
ta goi ba canh do la A B , AC, A D . Canh A B
phai la canh chung ciia hai mat tam giac, do
B
la hai mat ABC va A D B (vi qua dinh A chi
co 3 canh).
Tuong tu, ta co cac mat tam giac ACD va BCD. Vay khoi da dien do
chinh la khoi t i i dien ABCD.
du 5: Chiing minh rang, so goc ciia tat ca cac mat gap doi so canh ciia
khoi da dien. Suy ra so goc chan.
Giai
Goi so goc la G va so canh ciia khoi da dien la C. Trong moi mat la da
giac thi so goc bang so canh, ma so canh duoc tinh 2 lan nen G = 2C, do
do G chan.
du 6: Chiing minh khong ton tai khoi da dien co mot so le mat va moi
mat lai co mot so le canh.
Giai
Gia su ton tai khoi da dien co so mat la M le va moi mat chiia so le canh
C „ i = 1,2,...',M.
Ta co so goc ciia khoi da dien: G = Ci + C + ... + C => G le: vo ly.
Vay khong ton tai khoi da dien thoa de bai.
du 7: Chiing minh dac so O-le ciia khoi da dien loi: Doi voi moi khoi da
dien loi H , ta ki hieu D la so dinh, C la so canh, M la so mat ciia H thi so
X(H) = D - C + M = 2.
Giai
Ta chiing minh quy nap theo so dinh D > 4.
Khi D = 4 thi khdi da dien la tii dien co D = 4. C = 6, M = 4 nen
A
Vi
Vi
2
Vi
M
D - C + M = 4 - 6 + 4 = 2: diing.
Gia sir khang dinh diing voi so dinh D : D - C + M = 2.
Xet khoi da dien co D' = D + 1 dinh. Goi A la mot dinh va mat A1A2...A,, la
mot mat ciia khoi da dien sao cho mat phang chiia mat nay chia khong gian
lam 2 phan, mot phan chiia dinh A va phan kia chiia khoi da dien loi co D
dinh con lai, ta co D - C + M = 2.
So dinh D' = D + 1. so canh C = C + n, s6 mat M ' = M + n - 1
Do do: B' - C + M' = (B + 1) - (C + n) + (M + n -1) = B - C + M = 2.
7
Vay x(H) = D - C + M = 2.
Cach khac: Diing phep chieu tir mot diem S khong thuoc bat ky mat nao,
mat di qua 3 dinh nao ciia khoi da dien.
V i du 8: Chung minh khong ton tai khoi da dien loi co 7 canh.
Giai
Gia sir ton tai khoi da dien loi co C = 7.
Ta co dac s6 O-le: D - C + M = 2. nen D + M = 9. V i D > 4, M > 4 nen
hoac D = 4, M = 5 hoac D = 5, M = 4.
V o i D = 4 thi khoi da dien loi la ru dien: loai
V o i M = 4 thi khoi da dien loi la tu dien: loai
Vay khong ton tai khoi da dien loi co 7 canh.
V i du 9: Cho khoi da dien. Chung minh tong so do cac goc ciia cac mat la
T = 2(C-M)7t.
Giai
Goi Cj la so canh ciia mat thu i , i = 1, 2,...,M
M
f M
\
T a c 6 T = £ ( C , - 2 ) T T = £ C , - 2 M n = (2C - 2 J v l > = 2(C - M ) T T .
V i du 10: Hay phan chia mot khoi hop thanh nam khoi tir dien.
Giai
Co the phan chia khoi hop
ABCD.A'B'C'D' thanh nam kh6i
tii dien sau day: A B D A ' , C B D C ,
B'A'C'B, D'A'C'D, B D A ' C .
V i du 11: Hay phan chia mot khoi tii dien thanh bon khoi tir dien bdi hai mat phang.
Giai
Cho khoi t i i dien ABCD. Lay diem M
nam giua A va B, diem N nam giua C
va D. Bang hai mat phang (MCD) va
(NAB), ta chia khoi t i i dien da cho
thanh bon khoi t i i dien: A M C N ,
AMND, BMCN, BMND.
D A N G 2: PHEP D O I HINH
- Mot phep bien hinh F trong khong gian duoc goi la phep ddi hinh neu
no bao toan khoang each giua hai diem bat k i : neu F bien hai diem bat ki
M , N lan luot thanh hai diem M ' , N ' thi M'N' = M N .
Phep ddi hinh bien dudng thang thanh dudng thang, mat phang thanh
mat phang...
8
Hop thanh cua nhirng phep ddi hinh la phep ddi hinh.
'- Phep tinh t i i n : Phep tinh t i i n theo vecto v la phep bien hinh bien moi
diem M thanh d i i m M ' sao cho M M ' = v
Phep doi xiing qua dudng thang (phep doi xung true): Cho dudng thang
d, phep doi xung qua dudng thang d la phep bien hinh bien moi diem thuoc
d thanh chinh no va bien moi d i i m M khong thuoc d thanh diem M ' sao cho
trong mat phang ( M , d), d la dudng trung true cua doan thang M M ' .
Phep doi xung qua mot diem (phep doi xung tam): Cho diem O, phep
ddi xung qua diem O la phep bien hinh bien moi diem M thanh diem M '
sao cho OM + O M ' = 0 , hay O la trung d i i m cua M M ' .
Phep ddi ximg qua mat phang (P) la phep b i l n hinh bien moi diem
thudc (P) thanh chinh nd va bien mdi diem M khdng thudc (P) thanh
diem M ' sao cho (P) la mat phang trung true cua doan thang M M ' .
Hai hinh H va H' goi la bang nhau neu cd mdt phep ddi hinh bien hinh
nay thanh hinh kia.
Ddi vdi cac khdi da dien ldi: Neu phep ddi hinh F bien tap cac dinh ciia
khdi da dien ldi H thanh tap cac dinh ciia khdi da dien ldi H' thi F bien H
thanh H'.
Dinh ly: Hai hinh t i i dien A B C D va A'B'C'D' bang nhau neu chiing cd
cac canh tuong ung bang nhau, nghia la A B = A'B', BC = B'C, CD = CD',
D A = D'A', AC = A ' C , BD = B'D'.
V i du 1: Cho t i i dien ABCD. Chiing td rang phep ddi hinh bien mdi diem A,
B, C, D thanh chinh nd phai la phep ddng nhat.
Giai
Gia sir phep ddi hinh f bien cac diem A, B, C, D thanh chinh cac diem
do, tiic la f(A) = A, f(B) = B, f(C) = C, f(D) = D. Ta chiing minh r i n g f
bien diem M bat ki thanh M . That vay gia sir M ' = f ( M ) va M ' khac vdi
M . Khi do vi phep ddi hinh khdng lam thay ddi khoang each giua hai
d i i m nen A M = A M ' , B M = B M ' , C M = CM', D M = D M ' , suy ra bon
diem A, B, C, D nam tren mat phang trung true ciia doan M M ' , dieu do
trai vdi gia thiet A B C D la hinh tii dien. Vay M ' triing vdi M va do do f la
phep ddng nhat.
V i du 2: Cho hai tii dien ABCD va A'B'C'D' cd cac canh tuong img bang nhau:
AB = A'B', BC = B'C, CD = CD', D A = D'A', DB = D'B', AC = A ' C . Chung
minh rang cd khdng qua mdt phep ddi hinh bien cac diem A, B, C, D lan luot
thanh cac d i i m A', B', C, D'.
Giai
Gia sir cd hai phep ddi hinh f j va f deu bien cac diem A, B, C, D lan
luot thanh cac diem A', B', C , D'. NeU f i va f khac nhau thi cd it nhat
mdt diem M sao cho neu M i = f\(M) va M = f ( M ) thi M | va M la hai
d i i m phan biet. Khi dd vi f, va f deu la phep ddi hinh nen A ' M i = A M
2
2
2
2
2
2
9
ya A ' M = A M , vay A ' M i = A ' M , tuong tu B ' M , = B ' M , C'Mi = C ' M ,
D ' M , = D ' M , do do bon diem A', B', C , D' cung nam tren mat phang
trung true ciia doan thing M ) M , trai voi gia thiet A'B'C'D' la hinh tu
dien. Do do vdi moi diem M ta deu co f i ( M ) = f ( M ) , tiic la hai phep ddi
hinh f i va f trung nhau.
Vay co khong qua mot phep ddi hinh bien cac diem A, B, C, D lan luot
thanh cac diem A', B', C , D'.
du3:
a) Cho hai diem phan biet A, B va phep ddi hinh f bien A thanh A, bien
B thanh B. Chiing minh rang f bien moi diem M nam tren dudng
thang A B thanh diem M .
b) Cho tam giac ABC va phep ddi hinh f bien tam giac A B C thanh chinh
no, tiic la f(A) = A, f(B) = B, f(C) = C. Chiing minh rang f bien moi
diem M ciia mp(ABC) thanh chinh no, tiic la f ( M ) = M .
Giai
Ta co f ( A ) = A, f(B) = B. Gia sir diem M thuoc dudng thang A B va
f ( M ) = M ' . K h i do M ' thuoc dudng thing A B va A M = A M ' , B M = BM'.
Suy ra M ' triing M , tiic la f bien M thanh chinh no.
Vay f bien moi diem M nam tren dudng thang A B thanh chinh diem M .
V i f(A) = A, f(B) = B va f(C) = C nen f bien mp(ABC) thanh mp(ABC).
Bdi vay neu M thuoc mp(ABC) va f ( M ) = M ' thi M ' thuoc mp(ABC) va
A M = A M ' , B M = B M ' , C M = CM'. Neu M ' va M phan biet thi ba diim
A, B, C thuoc dudng thang trung true ciia doan thang M M ' tren
mp(ABC), trai vdi gia thiet ABC la tam giac. Vay f ( M ) = M .
du 4:
a) Cho hai tam giac bing nhau ABC va A ' B ' C (AB = A'B', BC = B'C,
AC = A ' C ) . Chiing minh rang co diing hai phep ddi hinh, moi phep
bien tam giac ABC thanh tam giac A'B'C.
b) Cho tam giac ABC. Co nhung phep ddi hinh nao bien tam giac ABC
thanh chinh no?
Giai
Tren dudng thang a vuong goc vdi mp(ABC) tai
A lay diem D khac A, tren dudng thang a' vuong
goc vdi mp(A'B'C) tai A' co hai diem phan biet
Di va D sao cho A'Di = A ' D = A D .
Ta co cac hinh t i i dien ABCD, A'B'C'D) va
A ' B ' C D co cac canh tuong ung bang nhau. Neu f
la phep ddi hinh bien tam giac ABC thanh tam
giac A ' B ' C thi hoac f bien D thanh Di hoac f bien
D thanh D .
2
2
2
2
2
2
2
Vi
a)
b)
Vi
a)
2
2
2
2
2
Vay co dung hai phep doi hinh bien tam giac ABC thanh tam giac
A ' B ' C . Do la phep ddi hinh f, bien tu dien A B C D thanh tu dien A ' B ' C D ,
va phep ddi hinh f b i l n tu dien ABCD thanh tu dien A'B'CD2.
Day la trudng hop rieng ciia a) khi hai tam giac ABC va A ' B ' C trung
nhau. Vay ta co hai phep ddi hinh bien ABCD thanh chinh no: do la phep
dong nhat va phep doi xiing qua mp(ABC).
du 5: Cho t i i dien deu ABCD va phep ddi hinh f bien A B C D thanh
chinh no, nghTa la bien moi dinh ciia t i i dien thanh mot dinh ciia t i i dien.
T i m tap hop cac diem M trong khong gian sao cho M = f ( M ) trong cac
trudng hop sau day:
a) f ( A ) = B, f(B) = C, f(C) = A
b) f(A) = B, f(B) = A, f(C) = D
c) f ( A ) = B, f(B) = C, f(C) = D
Giai
Theo gia thiet f(A) = B va f(B) = C, f(C) = A. Do do f(M) = M khi va chi khi
M A = M B = MC. Suy ra tap hop cac diem M la true ciia dudng tron ngoai
tiep tam giac ABC.
Theo gia thiet f(A) = B, f(B) = A, f(C) = D. Do do f(M) = M khi va chi khi
M A = M B va MC = M D , tiic la M dong thdi nam tren cac mat phang trung
true ciia AB va CD. Suy ra tap hop cac diem M la dudng thang di qua trung
diem cua AB va CD.
Theo gia thiet f(A) = B, f(B) = C, f(C) = D. Do do f(M) = M khi va chi khi
M A = M B = MC = MD. Suy ra tap hop cac diem M gom mot diem duy nhat
la trong tam tii dien ABCD.
du 6: Chiing minh rang cac phep tinh tien, phep doi xiing tam la cac phep
ddi hinh.
Giai
- Neu phep tinh tien theo vecto v bien hai diem M , N lan luot thanh hai diem
2
b)
Vi
a)
b)
c)
Vi
M', N thi MM*' = NN*' = v*, suy ra M N = WW'
va do do M N = M'N'.
Vay phep tinh tien la mot phep ddi hinh.
Neu phep doi xiing tam O bien hai diem M , N lan luot thanh hai diem
M ' , N ' thi O M ' = - O M
ON'=-ON
Suy ra: M ' N ' = ON' - O M ' = - O N + OM = N M
Do do M ' N ' = M N , suy ra phep doi xung tam O la mot phep ddi hinh.
V i du 7: Chiing minh rang cac phep doi xiing true, doi
M'
ximg qua mat phang la cac phep ddi hinh.
Giai
M
Gia sir phep doi ximg qua dudng thang d
bien hai diem M , N lan luot thanh hai diem
M ' , N ' Goi H va K lan luot la trung diem
ciia M M ' v a N N ' , taco:
M N + M ' N ' = 2HK, M N - WW'
™
d
K
i:
= H N - H M - H N + HM' = N ' N + MM'
V i hai vectc» M M ' va N N ' deu \nong goc voi H K nen:
(MN + M ' N ' ) . ( M N - WW') = 2 H K ( N N + M M ' ) = 0
7
Suy ra M N = WW'
hay M N = M'N' Vay phep d6i xung qua d la phep
ddi hinh.
Gia sir phep doi xung qua mat phang (P) bien M , N thanh M ' , N ' Neu
M N thuoc (P) thi M ' = M , N ' s N nen M'N' = M N .
Neu co it nhat mot trong hai diem M , N khong nam tren (P) thi qua bon
diem M , N , M ' , N ' co mot mat phing (Q) ( M M va N N ' cimg vuong goc
vdi (P) nen song song vdi nhau). Goi A la giao tuyen ciia (P) va (Q) thi
trong mp(Q), phep doi xung qua dudng thang A bien hai diem M . N
thanh hai diem M ' va N ' nen M N = M'N'
V i du 8: Cho hai dudng thang song song a va a , hai mat phang (P) va (P')
cimg vuong goc vdi a. Tim phep tinh tien bien a thanh a' va bien (P)
thanh (P').
Giai
Goi O la giao diem ciia a va (P), O' la giao diem ciia a' va (P). Khi do phep
2
2
;
1
1
tinh tien theo vecto v = OO ' se bien a thanh a' va bien (P) thanh (P').
V i du 9: Cho t i i dien A B C D noi tiep mat cau (S) ban kinh R = A B , mot diem
M thay doi tren mat cau. Goi C , D', M ' la cac diem sao cho: CC' = DD'
= M M = AB . Chung minh rang neu BC'D'M' la hinh tir dien thi tam mat
cau ngoai tiep t i i dien do nam tren (S).
Giai
1
Phep tinh tien T theo vecto v = AB bien A thanh B, C thanh C, D thanh D'
va M thanh M', tiic la bien t i i dien A C D M thanh t i i dien BC'D'M', do do T
bien tarn O ciia mat cau (S) ngoai tiep tii dien A C D M thanh tam O' ciia mat
cau ngoai tiep tti dien BC'D'M', tiic la OO ' = v = AB . V i OO' = AB = R
nen diem O' nam tren mat cau (S).
V i du 10: Chung minh rang hop thanh ciia cac phep tinh tien la mot phep
tinh tien.
Giai
Gia sir Ti va T lan luot la cac phep tinh tien theo vecto
va
2
Neu
Ti b i l n d i i m M thanh diem M i va T bien M , thanh M thi hop thanh T
o T] bien diem M thanh diem M .
2
2
2
2
Vi M M j =
va M M =
:
2
nen M M = M M j + M M
2
X
Vay T o T i la phep tinh tien theo vecto v + v
2
x
2
=
+ V
2
2
Mot each tong quat: hop thanh ciia n phep tinh tien da cho la mot phep tinh
tien co vecto tinh tien bang tong cac vecto ciia cac phep tinh tien da cho.
12
V i du 11: Cho h i dien ABCD. Goi A i . B i . C,. D, lan luot la trong tam cac
tam giac BCD, A C D , A B D , ABC. V o i diem M bat ki trong khong gian
ta goi M , la anh ciia M qua phep tinh t i i n AA^ M la anh ciia M i qua
2
phep tinh tien theo BB^ M3 la anh ciia M qua phep tinh tien theo CC ,
2
M
4
X
la anh ciia M qua phep tinh t i i n theo Dl\
Chimg minh rang M
3
triing voi M4.
Giai
Ta co M la anh ciia M qua 4 phep tinh tien lien tiep. Hop thanh phep
tinh tien do la mot phep tinh tien theo vecto
4
V =
AA7
+
BB7
+ cc\
+
DD7
Goi G la trong tam tii dien, theo tinh chat trong tam thi:
—
4—. 4 —
4—
4 —
v = - — GA - — GB - — GC - — GD
3
3
3
3
= - — (GA + GB + GC + GD) = 0
3
Do do M triing voi M4.
V i du 12: Cho phep doi hinh f thoa man dieu kien phep hop thanh ciia f va f
la phep dong nhat: f 0 f = e, biet rang co mot diem I duy nhat sao cho f
bien I thanh chinh no. Chiing minh rang f la phep doi xiing tam.
Giai
Voi mot diem M bat k i khac I , ta goi M ' la anh ciia M qua f, khi do M va
M ' khong triing nhau. V i f o f = e nen f bien M ' thanh M , vay f bien doan
thang M M ' thanh doan thang M ' M . T i i do suy ra f bien trung diem doan
thang M M ' thanh chinh no va vi vay, theo gia thiet trung diem M M ' phai
la diem I . Vay f la phep doi xiing qua tam I .
V i du 13: Chung minh rang hop thanh ciia mot so chan cac phep doi ximg
tam la mot phep tinh tien, hop thanh ciia mot so le ciia phep doi ximg
tam la phep doi xiing tam.
Giai
- Gia sir Di va D la cac phep doi xiing tam co tam lan luot la Oi va 0 .
Goi M la mot diem bat k i . M i = D i ( M ) va M ' = D ( M i ) thi phep hop
thanh D o D i bien M thanh M ' .
2
2
2
2
Ta co: M M ' = M M , + M M ' = 2 0 ^ + 2 M 0 = 2 0 0
X
1
2
x
Suy ra D o D i la phep tinh tien theo vecto v = 2 0 0
2
x
2
2
Voi diem M ta lay M i doi xiing voi M qua O, va lay M ' sao cho M M ' = v
X
Khi do hop thanh T- o Do bien M thanh M ' . Neu goi I la trung diem cua
MM' thi 01 = — Vay diem I co dinh. Suy ra T- o D la phep d6i xiing
2
qua I .
0
•
V
Tuong tir Do o T- la phep doi xung qua diem I' ma OI = - —
Vi hop thanh ciia hai phep d6i xiing tam la mot phep tinh tien nen hop
thanh ciia 2n phep ddi ximg tam la hop thanh ciia n phep tinh tien va do
do la mot phep tinh tien.
Hop thanh ciia 2n + 1 phep d6i xiing tam la hop thanh cua mot phep tinh
tien va mot phep doi ximg tam nen la mot phep doi ximg tam.
V i du 14: Chiing minh rang mot hinh tir dien khong the co tam doi xiing,
tong quat mot hinh chop khong co tam doi ximg.
Giai
Trudc het ta thly rang neu mot hinh chop co tam doi xung O. thi so mat
chin. That vay neu M la d i i m bat ki thuoc mot mat nao do ciia hinh
chop, thi diem M ' doi xiing vdi M phai thuoc mot mat hinh chop (vi phep
d6i xiing bien mat thanh mat, canh thanh canh va dinh thanh dinh). Dieu
do chiing to moi cap mat ciia hinh chop ling vdi mot doan thang M M ' . V i
so cac doan nhu vay la nguyen, nen so mat la chan. Vay day ciia hinh
chop co tam doi ximg da giac vdi so le canh nen O khong thuoc mat
phang day va khong thuoc cac mat ben. Goi (T) la thiet dien ciia hinh
chop di qua O va song song vdi day ((T) ton tai v i phep doi xiing qua O
bien dinh hinh chop thanh diem thuoc day chop), khi do (T) la da giac co
tam doi ximg lai co so le canh (vi cac canh ciia (T) chi nam tren cac mat
xung quanh ciia hinh chop). Mau thuan do chiing minh bai toan, va suy
cho tii dien bat ki.
V i du 15: Cho mat phang (P) va tii dien ABCD. Vdi moi diem M thuoc (P) ta xac
dinh diem N theo cong thiic: MA + MB + MC + MD = 2MN
(1)
Tim tap hop N , khi M di dong trong (P).
Giai
Goi G la trong tam ciia tii dien ABCD thi G co dinh.
(1) » 4MG = 2MN <=> MG = GN <=> GM = - G N .
Do do N la anh ciia M qua phep doi ximg tam G.
Vay tap hop N la mat phang doi xung vdi (P) qua G.
V i du 16: Chiing minh rang:
a) Hop thanh ciia hai phep d6i xiing true co cac true doi xiing song song
la mot phep tinh tien
b) Hop thanh ciia mot phep ddi xiing true va mot phep tinh t i i n theo
vecto vuong goc vdi true d6i xiing la mot phep doi xiing true.
14
Giai
a) Gia sir D va Db la cac phep doi xirng true co
true lan luot la cac duong thang a va b song
song voi nhau. Lay hai diem I va J lan luot
a
nam tren
bat k i , ta
phep hop
gpi H la
diem ciia
a va b sao cho IJ _L a. V o i diem M
gpi M , = D ( M ) va M ' = D ( M , ) thi
thanh D o D bien M thanh M ' . Neu
trung diem cua M M , va K la trung
M | M ' thi:
a
b
b
a
M M ' = MM + M M ' = 2HM7 + 2Mji
l
= 2HK = 2IJ
X
Vay hop thanh Db o D chinh la phep tinh tien theo vecto v = 2IJ
a
b) Gia sir D la phep doi xiing qua duong thang a, T- la phep tinh tien theo
a
vecto v
vuong goc voi a. Gpi b la anh ciia a qua phep tinh tien theo
vecto thi phep tinh tien T- la hpp thanh ciia hai phep doi ximg D va
a
Db qua cac duong thang a va b: T- = Db o D .
a
Bdi vay T- o D = Db o D o D = Db o e = Db.
a
a
a
Gpi b' la anh cua a qua phep tinh tien theo vecto — thi phep tinh tien T- la
hpp thanh cua hai phep doi ximg Db va D qua cac duong thang b' va a:
T- = D o D >.
1
a
a
b
Do do: D o T- = D o D o D > = e o D > = D '.
a
a
a
b
b
b
V i du 17: Cho tii dien deu ABCD. Gpi M , N lan lupt la trung diem cac canh A B
va CD. Gpi O la trung diem ciia doan MN. Chung minh rang vdi moi diem K
nam trong tii dien ta co KA + KB + KC + KD > OA + OB + OC + OD.
Giai
Ta co M N la true doi xiing ciia tii dien deu ABCD.
Gpi K' la diem doi xiing vdi K qua M N , H la giao ciia K K ' va M N . Ta co
KA + KB = A K + AK' > 2AH va KC + KD = CK + C K > 2CH. Ta
chiing minh rang A H + CH > OA + OC.
Xet trong mat phang (MCD), diem A' sao cho tia M A ' vuong goc vdi
M N , ngupc chieu vdi tia NC va dp dai M A ' = M A .
Ta co HA' = H A nen HA + HC = HA' + HC > A'C. V i A'C di qua O nen
A'C - OC + OA' - OC + OA.
Vay KA + KB + KC + KD > OA + OB + OC + OD.
V i du 18: Cho lang tru dirng ABC.A'B'C. co day la tam giac can ABC
(AB = AC). Tren cac canh AC va A'B' ta lay cac diem tuong img M va
M ' sao cho A M = A ' M ' . Tim tap hpp trung diem cua doan M M ' .
1-3
Vi
a)
b)
c)
d)
Vi
a)
Giai
Goi I , J la trung diem canh ben A A ' va giao cac duong cheo hinh chu
nhat BCC'B'. Ta co IJ la true ddi xung cua hai doan AC va A'B', do do M
va M ' doi xung voi nhau qua IJ. Vay tap hop cac trung diem ciia M M '
thuoc doan IJ
du 19: Goi D la phep ddi xung qua mat phang (P) va a la mot duong
thang nao do. Gia sir D bien duong thang a thanh duong thang a'. Trong
truong hop nao thi:
a) a triing voi a'
b) a song song voi a'
c) a cat a'
d) a va a' cheo nhau?
Giai
a triing voi a' khi a nam tren mp(P) hoac a vuong goc voi mp(P).
a song song voi a' khi a song song voi mp(P).
a cat a' khi a cat mp(P) nhung khong vuong goc voi (P).
a va a' khong bao gio cat nhau.
du 20: Chung minh:
a) Hop thanh ciia hai phep doi xiing qua hai mat phang song song (P) va
(Q) la mot phep tinh tien.
b) Hop thanh ciia hai phep doi xiing qua hai mat phang (P) va (Q) vuong
goc voi nhau la mot phep doi xung qua duong thang.
Giai
Lay hai diem A va B lan luot nam tren (P) va
A» .H
(Q) sao cho A B _L (P). V o i mot diem M bat k i ,
ta goi M i la diem doi xiing voi M qua mp(P) va
M ' la diem doi ximg voi M i qua mp(Q).
Goi H va K lan luot la trung diem cua M M ] va
M , M ' thi taco:
1
MM' = MM, + M,M
1
= 2 (HTVL\ + M T K ) = 2HK
= 2AB
Vay phep hop thanh la phep tinh tien theo
vecto 2 A B .
b) Goi d la giao tuyen ciia (P) va (Q).
V o i mot diem M bat ki, ta goi M i la
diem doi ximg voi M qua mp(P) va
M ' la diem doi xung ciia M i qua
mp(Q).
Neu M nam tren (P) hoac tren (Q)
thi thay M ' la diem doi xiing ciia M
qua d.
Neu M khong nam tren ca (P) va (Q) thi ba diem M , M i va M ' xac dinh
mat phang (R) vuong goc voi (P) va (Q), do do vuong goc voi d.
16
Goi giao tuyen cua (R) voi (P) va (Q) lan luot la p. q, con O la giao diem
cua p va q.
Xet trong mat phing (R) thi diem M ' la anh cua diem M qua hop thanh cua
phep doi xung qua duong thang p va phep doi xung qua duong thang q. Suy
ra O la trung diem ciia MM'.
Mat khac M M ' _L d nen phep hop thanh la phep doi xung qua dudng
thang d.
V i du 21: Cho mat phang (P) va cho phep ddi hinh f co tinh chat: f bien diem
M thanh diem M khi va chi khi M n i m tren (P). Chung to rang f la phep
doi xung qua mat phang (P).
Vi
a)
b)
c)
Giai
Phep ddi hinh f bien moi diem M nam tren (P)
thanh M . V d i diem A khong nam tren (P) ta
goi a la dudng thang di qua A va vuong goc
vdi (P). Neu H la giao diem cua a va (P), vi
f(H) = H nen f bien a thanh dudng thang di
qua H va vuong goc vdi (P), vay f(a) = a.
Tir do suy ra diem A bien thanh diem A' nam tren a,
A' khac vdi A va H A = HA'.
Vay (P) la mat phang trung true ciia doan thang AA'. Suy ra f la phep doi
xung qua mp(P).
du 22: Tim cac mat phang doi ximg ciia cac hinh sau day:
a) Hinh chop tii giac deu.
b) Hinh chop cut tam giac deu.
c) Hinh hop chu nhat ma khong co mat nao la hinh vuong.
Giai
Hinh chop t i i giac deu S.ABCD co 4 mat phang doi ximg: mp(SAC),
mp(SBD), mat phang trung true ciia A B (dong thdi ciia CD) va mat
phang trung true ciia A D (dong thdi ciia BC).
Hinh chop cut tam giac deu ABC.A'B'C co ba mat phang doi xiing, do la ba
mat phang trung true ciia ba canh AB, BC. CA.
Hinh hop chir nhat ABCD.A'B'C'D' (ma khong co mat nao la hinh
vuong) co ba mat phang doi xiing, do la ba mat phang trung true ciia ba
canh A B . A D , A A ' .
s
D
A'
c
/
/
C
A'
V
17
V i d u 23:
a) Tim cac true ddi xung cua hinh tu dien deu A B C D .
b) Tim tat ca cac mat phang ddi xung cua hinh tu dien deu ABCD.
i
^
A;1—-\.
D
1
B'
C
Giai
a) Gia sir d la true doi xung cua tir dien deu ABCD, tire la phep doi ximg D
qua duong thang d bien cac dinh ciia tir dien thanh cac dinh ciia tir dien.
Trudc het ta nhan thay rang true doi ximg d khong the la dudng thang di
qua hai dinh nao do cua hinh tir dien, vi hien nhien phep doi ximg qua
dudng thang d nhu the khong bien hinh tir dien thanh chinh no.
Bay gid ta chung to rang true doi ximg d cung khong di qua mot dinh
nao ciia t i i dien. That vay, neu d di qua A thi vi B khong the nam tren d
nen B bien thanh C hoac D. Neu B bien thanh C thi C bien thanh B nen
D bien thanh D va do do d di qua A va D, vo li. Neu B bien thanh D thi
D bien thanh B va do do C bien thanh C va d di qua A va C, vo li.
Vay phep doi xung D qua dudng thang d bien diem A thanh mot trong ba
diem B, C hoac D.Do do tir dien deu co 3 true doi xiing la 3 dudng thang
di qua trung diem 2 canh doi dien (dudng trung binh).
b) Gia sir a la mat phang doi ximg ciia tir dien deu ABCD, tiic la phep doi
xiing D„ bien tap hop {A, B, C, D} thanh chinh no. V i D khong the bien
moi dinh thanh chinh no (vi khi do D„ la phep dong nhat) nen phai co
mot dinh, A chang han, bien thanh mot dinh khac, B chang han. Khi do a
la mat phang trung true ciia doan thang A B (hien nhien a di qua C va D).
Vay t i i dien ABCD co 6 mat phang doi xiing, do la cac mat phang trung
true ciia cac canh.
V i d u 2 4 : Cho hinh lap phuong ABCD.A'B'C'D' Tim
a) Tam doi ximg
b) Mat doi ximg
c) True doi ximg.
Giai
a) Tam ddi xung O la giao diem ciia 4 dudng cheo A C , BD', CA' va DB'.
b) Gpi a la mat doi xiing ciia hinh lap phuong thi phep doi xiing qua a bien
hinh vuong ABCD thanh chinh no, hoac thanh hinh vuong chung canh
hoac thanh hinh vuong A'B'C'D'.
Tir do thi hinh lap phuong co 9 mat phang doi ximg la 3 mat phang trung
true ciia cac canh va 6 mat phang chua hai canh doi.
c) 9 true ddi ximg gom 3 true ciia cac mat va 6 dudng thang di qua trung
diem cua hai canh doi.
a
18
V i du 25: Cho hinh lap phuong ABCD.A'B'C'D'. Chung minh rang:
a) Cac hinh chop A.A'B'C'D' va C'.ABCD bang nhau.
b) Cac hinh lang tru ABC.A'B'C va AA'D'.BB'C bang nhau.
Giai
a) Goi O la tam ciia hinh lap phuong. V i phep doi xiing tam O bien cac dinh
ciia hinh chop A.A'B'C'D' thanh cac dinh ciia hinh chop C'ABCD. Vay hai
hinh chop do bang nhau.
b) Phep doi xung qua mp(ADC'B') bien cac dinh ciia hinh lang try ABC. A'B'C
thanh cac dinh cua hinh lang tru AA'D'.BB'C nen hai hinh lang tru do
bang nhau.
V i du 26: Chung minh 2 hinh lap phuong co canh bang nhau thi bang nhau.
Giai
Gia sir ABCD.A'B'C'D' va MNPQ.M'N'P'Q' la hai hinh lap phuong co
canh deu bang a. Hai t i i dien ABDA va M N Q M ' co cac canh tuong ling
bang nhau nen bang nhau, tuc la co phep ddi hinh F bien cac diem A, B,
D, A' lan luot thanh M , N , Q, M ' . V i F la phep ddi hinh nen F bien hinh
vuong thanh hinh vuong, do do F bien diem C thanh diem P, bien diem
B' thanh N ' , bien diem D' thanh Q' va bien diem C thanh P' Vay hai hinh
lap phuong da cho bang nhau.
1
D A N G 3: KHOI D A DIEN DEU V A PHEP VI TU
• Cho so k khong doi khac 0 va mot diem O co dinh. Phep bien hinh
trong khong gian bien moi diem M thanh diem M ' sao cho O M ' = k OM
goi la phep v i tu. Diem O goi la tam vi tu, so k goi la t i so v i tu.
Neu phep v i tu ti so k bien hai diem M , N thanh hai diem M ' , N ' thi
WW' = k M N va do do M'N' - | k j M N .
Phep vi tu bien ba diem thang hang thanh ba diem thang hang, bon diem
dong phang thanh bon diem dong phang.
- Hinh H duoc goi la dong dang vdi hinh H' neu co mot phep vi tu bien
hinh H thanh hinh H i ma hinh Hi bang hinh H'.
Khoi da dien deu ma moi mat la da giac deu n canh va moi dinh la dinli
chung ciia p canh duoc goi la khoi da dien deu loai {n, p } .
Khoi tii dien deu la loai {3; 3}; Khoi bat dien deu la loai {3; 4}
Kh6i lap phuong la loai {4; 3}; Khoi 20 mat deu la loai {3; 5}
Kh6i 12 mat deu la loai {5; 3 } .
Khoi tii dien deu
Khoi bat dien deu
Khoi lap phuong
19