Phuang phdp chung ta phai chuyen khoang each giua hai duang thang cheo nhau ve khoang each tu mot diem den mot du&ng thang hoac khoang each tif mot diem den mot mat phang. Thudmg xay ra nhung truang hap sau day:
1. Neu duong thang a thuoc mpt mat phang (P), va duong thang b song song voi mat phang (P). Thi khoang each giira a va b bang khoang each tu duong thang b den mat phang (P). CHQN mot diem M thich hop thupc b sao c6 c6 the tinh khoang each de dang den mat phSng (P). Khoang each tir M den (P) la khoang each giOa hai duong a va b.
-Chu y: Neu khong tim dupe mpt mat phing ehua duong thing nay va song song voi duong thang kia, thi ta phai dung m^t phang (P) ehua duong thang nay va song song voi duong thang kia.
Neu duong thang a thupe mat phang (P), duong thang b thuoc mat phang (Q). Ma hai mat phang (P) va (Q) song song vol nhau, thi khoang each giiia a va b bing khoang each (P) va (Q).
Cu the tinh khoang each giira hai duong thang cheo nhau a va b, voi a la canh ben con b la mpt canh ciia day. Caeh lam nhu sau:
Gpi I la giao diem cua duong thing a voi mat day. Tu I dung duong thang
^ song song voi b. Luc do b song song voi mat phang (P) ehua a va A . Chpn mpt diem M tren b sao cho c6 the tinh khoang each den mat phang (P). Khoang each tu M den mat phing (P) bing khoang each giCia a va b.
Thong qua cae vi du sau cae ban se hieu ro hon:
D E T H I T U Y E N S I N H D A I H Q C K H O I A N A M 2011
Cho hinh chop S.ABC c6 day ABC la tarn giac v u o n g can tai AB = BC = 2a, hai mat p h i n g (SAB) va (SAC) cung v u o n g goc v o i m^, p h i n g (ABC). Gpi M la trung diem cua A B ; mat p h i n g qua S M va son^
song v o i BC cat A C tai N . Biet goc giua hai mat phSng (SBC) va (ABC) ban^
60". T i n h khoang each giij-a hai d u o n g th^ng A B va SN theo a.
"•J
,vnv;- L O I G I A I Ta c6: m p ( S A B ) r i m p ( S A C ) = SA', va
hai mat phSng (SAB) va (SAC) ciing v u o n g goc v o i mat phang (ABC).
Suy ra S A l m p ( A B C ) .
Mat phang qua SM va song song v o i BC cat A C tai N , suy ra M N // BC va N t r u n g diem cua A C .
Trong tam giac ABC c6: M N = ^ B C = a , B M = ^ A B = a .
Ngoai ra:
V i :
B C I A B B C I S A ( S B C ) n ( A B C ) = B C S B I B C A B I B C S B C ( S B C ) , A B C ( S A B )
B C l m p ( S A B ) = ^ B C l S B . A
•((SBC),(ABC) = SBA = 60' ,0
Xet ASAB v u o n g tai A : SA = AB.tan60° = 2aS.
T i n h khoang each giua A B va SN, day Id bdi loan tinh khodng each giua cgnh ben SN vd eqnh day AB. Do chua cd mat phang ndo ehiea mot trong hai ditang trc"
nen ta phdi dung mot mat phdng (P) chiea duang thdng nay vd song song i"'' duang thang kia. Cach d u n g theo p h u o n g phap 3 6 tren:
N la giao diem cua canh ben SN v o i mat day (ABC). ' ' Tir N , ke N x // A B , suy ra A B // m p ( S N x ) (vi N x c ( S N x ) ) .
N e n d ( A B , S N ) = d ( A B , m p ( S N x ) ) = d ( A , m p ( S N x ) ) .
C h u y: A Id hinh chieu cua dinh, eon mat phang (SNx) Id mat phang ben. Cdc xem Iqi phuang phdp tinh khodng edeh tit hinh chieu cua dinh len mot mat ph('"^
ben 6 bdi khodng cdch tif mot diem den mot mat phang
D y n g A I 1 Nx tai I . Goi H la hinh chieu vuong goc cua A tren SI.
V i Nx 1 A I
Nx 1 SA • Nx ± (SAI) =^ m p { S N x ) ± m p ( S A l ) .
Hai mat phSng (SNx) va (SAI) vuong goc v o i nhau c6 giao tuyen SI, ma A H 1 SI ^ A H 1 (SNx) . Vay d ( A , m p ( S N x ) ) = A H .
Taco: A I = ^ B C = a .
Trong ASAI v u o n g tai A c6: ''i^^'ifevij, ^ằif,^k .Kvk.. ằ?w-;„.ijj,i4^ t
1 1 I
i2- 1 2 a 2 ^ a 2
13a^
A H ' A S ' A I
K e t l u a n d ( A , m p ( S N x ) ) = A H = 12 2aN/39
13
A H = 2 a V 3 9 13
D E T H I T U Y E N S I N H D A I H Q C K H O I A N A M 2012.
Cho hinh chop S.ABC c6 day la tam giac deu canh a. H i n h chieu vuong goc cua S tren mat phang (ABC) la diem H thuoc canh A B sao cho H A = 2HB. Goc giua d u o n g thSng SC va mat ph^ng (ABC) bang 60". Tinh khoang each giua hai d u o n g thgng SA va BC theo a.
L O I G I A I Ap d u n g d i n h ly cosin eho tam giac A H C c6:
CH^ = AC^ + A H ^ -2AC.CH.COSCAH
= a2 + 2a „ 2a ^ n 7a
-2.a. — .eos60 = —
^ C H = av/7
Ta c6: H C la hinh chieu vuong goc cua SC tren mat phang (ABC).
Nen goc giua SC va mat phSng (ABC)
•a goc SCH - 60" .
Trong tam giac SCH vuong tai H eo:
SH = C H . t a n 6 0 " = ^ . y 3 = " ^ • '}>Vf 3 3
Ti'nh khoang each giua SA va BC, day Id bdi todn tinh khodng cdch gii'ea canh SA vd Ci^nh ddy BC. Do chua cd mat phdng ndo chita mot trong hai duang tren
ta phdi dung mot mat phdng (P) chiea duang thdng ndy vd song song vai
^udng thdng kia . Cach d u n g theo p h u o n g phap 3 6 tren :
^ 'a giao Jiem cua eanh ben SA voi mat day (ABC).
"^"^ A ke A x // B C . Suy ra BC // m p ( S A x ) (vi A x c ( S A x ) ) nen:
d(BCSA) = d ( B C , mp{SAx)) = d(B,mp(SAx)) . >, BC song song v&i mat phang (SAx) thi khoang each mot diem tren duang thang BQ
den mat phdng (SAx) deu bang nhau.
Vi sao Thay lai chgn diem B ma khdng chon diem khdc chang hqn IdC ,ld vi diem fi nam tren dit&ng thang c6 chita diem H Id hinh chie'u cua dinh, viec tinh khodn^
each tit H den mat phang ben (SAx) la rat de dang. Thong qua cong thitc tinh ti SQ- khoang each tht ta tinh dugc khoang each tit B. ;f. j f^nhiiy fô
Nen tren ly thuyei Thay c6 noi chgn diem M thich hffp c6 the tinh khoang each deh mp(P), cu the a bdi nay la diem B.
Dung H I 1 Ax tai I . Gpi K la hinh chie'u vuong goc ciia H tren SI.
Vi I ^ Ax 1 (SHI) ^ mp(SAx) 1 mp(SHl). _ ^ Ax _L SH
Hal mat phSng (SAx) va (SHI) vuong goc voi nhau c6 giao tuyen SI.
Ma HK 1 SI => HK 1 (SAx). Vay d(H,mp(SAx)) = HK .
r r
Trong AAIH vuong tai I c6 HI = HA.sin60" = - ^ . ^ = ^ . iBi"}:'
Trong ASIH vuong tai H c6: ' h ' ^ i K i i
1 1 1 9 9 24 „ ^ aV7 , .
= + = + — - = — - => HK = - j = . .'.
HK^ HS^ H r 21a^ 3a^ 7a^ V24
Duong thing di qua hai diem B va H c6 giao diem voi mat phang (SAx) Ki , d(B,mp{SAx)) BA 3 ' i i V v -n
A n e n : ^ ^ = —— = - • - U i i
d(H,mp(SAx)) HA 2 -^^=
^ d ( B , m H ^ A x ) ) = |d(H,mp(SAx)) = | . ^ = ^ " ' ' ' Cho hinh chop S.ABCD c6 day la hinh vuong canh a, SA = SB = SC = SD =
asll . Goi I , K la trung diem ciia AD, BC.
a) Chungminh(SIK)l(SBC). . - . . ^ . u . > , v . r.w;..
b) Tinh khoang each giiJa hai duong thang SB va AD.
LOIGIAI
Goi O la giao diem cua AC va BD. ' J-^''^* -
Ta c6: OA = OB = OC = OD (tinh cha't hinh vuong)
Theo de bai SA = SB = SC = SD = aVz . ' Suy ra: O la hinh chie'u vuong goc ciia S tren mat phang (AB^r*
S O l ( A B C D ) . , ' fBC ± IK
^ ^ ^ B C I S O ^ ^ ^ ^ ^ ^ ^ ^ )
=> (SBC) 1 (SIK) ( vi BC c (SBC)).
Co AD // BC AD / /mp(SBC), nen: ^ ' d ( A a S B ) = d(AD,(SBC))
= d(l,(SBC))(vi l e A D ) .
Goi H la hinh chie'u vuong goc cua O tren SK c6 f O H l S K , ,
' OH 1 BC ° " nip(SBC) => d(0,mp(SBC)) = OH .
Trong ASAO vuong tai O c6 SO = V s A ^ - A O ^ = Jla^ -— = a^/6 rong ASOK vuong tai O c6:
1 1 4 4
—- + — 14 a^/47
•H' OS' OK^ 6a' a' 3a' ' ' 14
Hai diem I va O ciing n^m tren duong thing c6 giao diem voi mp(SBC) tai K n i n : ^('-'"PW) ^ ^ ^ ^
d(0,mp(SBC)) OK
'HI rj !
iO-y ,v. yrryr, r,f,<^;.j;^ ;^':Uli;1
> d(I, mp(SBC)) = 2d (O,mp(SBC)) = 2 . ^ = ^ . ,, v.vi> ô 14
Ketluan: d(AD,SB) = aV42
Cho lang try tam giac ABC.AiBiC. c6 tat ca cac canh bSng a, goc tao boi canh ben va mat phang day bSng 30". Hinh chie'u H cua diem A tren mat phing (AiBiCi) thuoc duong thing BiCi. Tinh khoang each giua hai duong thing AAi va BiCi theo a.
'!! LOIGIAI Do A H l ( A , B , C j ) nen goc giua AAi
va (AiBiCi) la goc A A ^ = 30°.
Xet tam giac vuong AHAi c6: AAi = a;
A j H = AA,.cos30° = A H = ^ A A i . s i n 3 0 " = - .
^ 2
D o t a m g i a c A i B i C i la A d e u c a n h a; H t h u p c B i C i v a A ^ H = n e n : A i H v u o n g goc v o i B i C i . ::: 3 ^ , ; > - j .
M a t k h a c : A H 1 B j C j n e n : B^C^ 1 ( A A j H )
K e d u o n g cao H K c u a t a m giac A A i H t h i H K c h i n h la k h o a n g each g i i i a A A i v a B i Q > . . • . , , • ;,,\ J , ? , , : " ; ' ,
A i R A H aS ^ J i i b - T a c6: A A i . H K = A i H . A H => H K =
A A ,
mi ^irn 1. HO
D A N G 2: Xac d i n h d u o n g v u o n g goc c h u n g . P h u o n g p h a p g i a i :
T a C O cac t r u o n g h o p sau d a y : iv (.,) A<?/:> .v- a) G i a sir a v a b la h a i d u a n g t h S n g cheo n h a u v a a 1 b .
- T a d u n g m a t p h S n g ( a ) c h u a a v a v u o n g goc v o i b t a i B . 1 ^ * ' ' - T r o n g ( a ) d u n g B A l a t a i A , ta
d u o c d o d a i d o a n A B la k h o a n g each
" g i u a hai d u o n g t h a n g cheo n h a u a v a b . b) G i a sir a v a b la h a i d u o n g t h a n g cheo n h a u n h u n g k h o n g v u o n g goc v o i n h a u . C a c h 1:
- T a d u n g m a t p h S n g ( a ) e h u a v a s o n g s o n g v o i b . ' - L a y m p t d i e m M t u y y t r e n b ,
d u n g M M ' ± ( a ) t a i M ' .
, - T u M ' d u n g b ' / / b cat a t a i A . T i r A d u n g A B / / M M ' cat b t a i B , d p d a i d o a n A B la k h o a n g each g i i i a h a i j d u o n g t h a n g c h e o n h a u a v a b . C a c h 2:
- T a d u n g m a t p h a n g ( a ) 1 a t a i O , ( a ) eSt b t a i I . - D u n g h i n h chie'u v u o n g goc cua b la b ' t r e n ( a ) . - T r o n g m a t p h S n g ( a ) , v e O H l b ' , H e b ' . - T i r H d u n g d u a n g t h i i n g s o n g s o n g ' v o i a eat b t a i B .
- T i r B d u n g d u a n g t h a n g s o n g s o n g v o i O H cat a t a i A .
- D p d a i d o a n t h i n g A B l a k h o a n g each g i i i a h a i d u a n g t h i i n g c h e o n h a u a v a b .
B - J ' s
N b' /
C h o h i n h c h o p S . A B C D c6 d a y la h i n h v u o n g c ^ n h a, S A = h v a J . d a y . D u n g v a t i n h d p d a i d o a n v u o n g goc c h u n g cua: '
a) SB v a C D . b) A D v a SB c ) A B v a S D / d ) SC v a B D . e) SC v a A B . f) SC v a A D
L O I G I A I
a) D u n g v a t i n h d p d a i d o a n v u o n g goc c h u n g cua SB v a C D B C l A B ( A B C D h i n h v u o n g )
B C 1 S A ( V i S A 1 ( A B C D ) ) J ' ' ' '
= > B C l m p ( S A B ) = > B C l S B (1).
M a : B C 1 C D ( A B C D h i n h v u o n g ) (2).
T i r (1) v a (2) ta eo: B C la d u o n g v u o n g goc c h u n g c i i a SB v a C D , v a B C c u n g la khoang each giira hai d u o n g thSng SB va C D . K e t i u a n d ( S B , C D ) = B C = a .
b) D u n g v a t i n h d p d a i d o a n v u o n g goc c h u n g c i i a SB v a A D
T r o n g A S A B k e A K 1 SB t a i K (3) <^*"^'' A D I A B ( A B C D h i n h v u o n g ) ;
A D I S A ( V i S A K A B C D ) , - 1 A K ( A K c ( S A B ) ) , 4 ) T i r (3) v a (4) t h i A K la d u a n g v u o n g goc c h u n g ciia SB v a A D , v a A K c u n g la k h o a n g each g i i r a h a i d u o n g t h i n g SB v a A D .
T r o n g A S A B v u o n g t a i A : ^ v if f; ,
1 1 . + . 1 - 1 . 1 a ^ + h ^ a.h ^"-^ ' " 1 1
+ —
A K ^ A S ^ A B ^ h
K e t l u a n d (SB, A D ) = A K = •
. A K =
4
a.h V a 2 + h 2 •
^) D u n g v a t i n h d p d a i d o a n v u o n g goc c h u n g cua A B v a S D . T r o n g A S A D ke A H 1 S D t a i H (5)
A B I A D ( A B C D h i n h v u o n g ) , ^ , ^ 1 ' A B I S A ( V i S A 1 ( A B C D ) ) ; - . r ^ M • A B l m p ( S A D ) ^ A B l A H ( A H c ( S A D ) ) (6)
Ta eo:
Tu (5) va (6) thi A H la duong vuong goc chung ciia SD va AB, va A H cung la khoang each giCra hai duong thiing SD va AB.
1 '?V,'/
Trong ASAD vuong tai A : a.h
1 1
- + •
1 1 a^+h^ - + •
. A H =
'a^+h^
A H 2 " A S 2 AD^'h' a ^ " a\h'
d)
Ket luan: d (SD, AB) = A H = . .
Va^ + h , .••
Dyng va tinh dp dai doan vuong goc chung cua SC va BD.
Gpi O = AC n BD . Trone ASAC ke OG 1 SC t^i G (7) Ta c6: BD 1 A C (^'"^ ^^^^ ^^^^ vuong)
B D I S A ( V i S A l ( A B C D ) )
=^ BD 1 mp(SAC) ^ BD 1 0 G ( 0 G c (SAC)) (8).
Tu (7) va (8) thi O G la duong vuong goc chung ciia SC va BD, va O G cung la khoang each giiia hai duong th^ng SC va BD.
Trong ASAC c6: AC = ajl; CO = - AC = ^ SC = VSA^ + AC^ = N/ITTZ?
Taco: ACGO - ACAS(g.g) CO OG J . /'.S r/i ' A c CI/
CS SA ,rvTTW>; f i i m.MiQ. >
.OG = ^ . S A = .h = a.h.Vi
= , = . l l - , •
CS V2a2+h2 2V2a2+h2 t:
Ketluan: d(SC,BD) = OG= "
I
e)
Ta c6: • A H 1 mp(SCD) => A H 1 SC Dung va tinh dp dai doan vuong goc chung cua SC va AB.
A H I S D
A H I C D ( C D I ( S A D ) ) T u H k e H I / / C D .
Suy ra H I va AB ciing thupc mpt mat phSng vi cung song song voi CD.
Trong mat phing (AB,HI) ke IJ // A H , IJ cat A B tai J. > ^r,^.u J--. I ,
Ta c6: IJ // A H
A H ± SC, A H ± AB (Da chung minh 6 tren) I J ± S C I J X A B
V|y: IJ la duong vuong goc chung cua SC va AB, va IJ cung la khoang each giua hai duong thiing SC va AB.
Ket lu^n: d (SC, AB) = IJ = A H = . Va^+h^
f) Dyng va tinh dp dai do^n vuong goc chung cua SC va AD. '
Tir K ke KL // BC KL va AD cung thupc mpt m|it phang v i ciing song song voi BC.
Trong mat phing (AD,KL) ke L M // A K , L M cat AD tai M . Ta c6: L M / / A K
AK 1 SC, AK 1 AD (^^ ^hung minh 6 tren)
L M I S C L M I A D
Vay: L M la duong vuong goc chung cua SC va AD, va L M cung la khoang each giiia hai duong thSng SC va AD. ' , • ^ • Ket luan: d (SC, AD) = L M = AK = .
Va^+h^
Cho hinh chop S.ABCD c6 day ABCD la hinh thoi c^nh a, c6 goc BAD = 120'',SA = hva SA vuong goc voi day (ABCD). Dyng va tinh dp dai doan vuong goc chung ciia:
a) SBvaCD.
b) BDvaSC.
c) SC va AB.
' I I
L6 I G I A I a) Dung va h'nh dp dai doan vuong goc
chung ciia SB va CD.
Trong mat phiing (ABCD) ke CL 1 AB t?i L (1) Vi S A 1 (ABCD), ma S A c ( S A B ) nen:
( S A B ) 1 (ABCD) ^ CL 1 ( S A B ) '
ri> CL 1 SB (2)
Trong m^t phang (ABCD), tir B ke:
BN // LC voi N G CD (3)
227
Tu (1), (2), (3) thi B N la duang vuong goc chung ciia hai duong thing SB va CD va khoang each giua chiing la CL.
Its Vi ABCD la hinh thoi c6 BAD = 120° nen: AABC deu. - Ma CL duang cao ciia tarn giac: CL = ——.
Ket luan: d (SB, CD) = CI = — . , * .
b) . Dung va tinh do dai doan vuong goc chung cua BD va SC. A. ] Goi O = AC n BD . Trong ASAC ke OM 1 SC tai M (4) , ! ' '
^ ' TaCO- \^^'"^^^"^^'""^^^^r'jyri^'On w jiu>i
• | B DI S A ( V i S A l (ABCD)) '^"-•^^^•',1
r:> BD 1 mp(SAC) => BD 1 0 M ( 0 M a (SAC)) ^^5 si^^ffSjI^m sno Tu (4) va (5) thi O M la duong vuong goc chung cua SC va BD, va O M cung la khoang each giiia hai duong thang SC va BD.
^eimtm'ii: . , , - Trong ASAC c6: AC = a X O = - A C = - , S C = VSA^ + AC^ =yjh^+a^ .
2 2
Taco: ACMO~ A C A S ( g . g ) ^ = ^ • - ( C I A ; n $ u l ; f f v ^
^^^^ CS SA
i l i l ^ O M = ^ . S A = , ^ .h = i^-^- . , '\^ md- Mt-=€1(1
Ket luan: d(SC,BD) = O G = , ^"^^ . / iJ^iviff? (R c) Dung va tinh do dai doan vuong goc chung cua SC va AB. . ij / ')f! (o
Ke AE 1 CD tai E. " "
^ ° ^ ^ ^ ^ C D 1 ( S A E ) = ^ ( S C D ) I ( S A E ) ( C D C ( S C D ) ) ! ' ' ^ " - f
C D I S A ^ ^ ^ ^ ^ ^ " loiifwrlvi
( S A E) n( S C D ) = SE
Ke A H I S E t a i H v a c o ) [ ) [ => A H 1 ( S C D ) A H 1 SC .
( S A E ) I ( S C D ) ^ '
Tu H ke HI // CD.
Suy ra H I va AB cung thuoc mot mat phang vi cung song song voi CD.
Trong mat ph^ng (AB,HI), ke IK // A H , IK cat AB tai K.
9 9 8
[AH 1 SC, A H 1 AB(vi A H 1 CD, CD // AB) ^ j l K 1 AB , Vay IK la duong vuong goc chung cua SC va AB, va IK ciing la khoang each giiia hai duong thang SC va AB.
Trong ASAE vuong tai A: = :r + - 1 4 3a^+4h2 + • . A H =
AH^ AS^ AE^ h^ 3a^ 3a^.h^
a.h^/3 ---Timrr:fff-^^-^_
Ket luan: d (SC, AB) = IK = A H = a.hV3 Vsa^ + 4h2
Cho hinh hop dung ABCD.A'B'C'D' c6 day la hinh thoi canh a, goc A bang 60°, goc ciia duong cheo A C va mp day bang 60°.
a) Tinh duong cao cua hinh hop do. • > • "' ' • •' ' ' • ' " *"
b) Tim duong vuong goc chung ciia A'C va BB'. Tinh khoang each giira hai duong thang do.
= 6 0 ° . , L O I G I A I
Vi A B C D . A ' B ' C ' D ' la hinh hpp dung nen:
A A' l ( A B C D ) .
Ta c6: A C la hinh chieu vuong goc cua
A ' C tren mat phang ( A B C D ) nen :
A ' ^ C ( A B C D ) J = ( A ^ C A C ) = A ^ = (
Vi A B C D la hinh thoi eo A = 60° nen:
A B A D deu A C = 2 A O = a S .
Trong A A ' A C vuong t?i A : A A ' = A C . tan A ' C A = a Vs. Vs = 3a .
Gpi O la giao diem ciia A C va B D . I trung diem cua A ' C . Ta c6 O I la duong trungbinhciia A A ' A C .
Vay: O I / / A A ' / / B B '
B O 1 A C (Vi A B C D hinh thoi)
B O I A A (Vi A A ' 1 ( A B C D ) )
"y—-—
D' C
D'
K
" ^ ^
Bt — - — C
a , ' '
<
D
Ta co:
B O I A A ' B O I C A '
= > B 0 1 m p ( A ' A C ) :
Trong mat phSng (BDD'B') ke IK // BO
(2)
(3)
" ' . 1':.':
2 2 9
Tir (1), (2) va (3) ta suy ta IK la duang vuong goc chung cua A'C va BB'.
IK cung la khoang each giua A'C va BB'. i B D a
" 2 • Ket lugn: d ( A ' C B B ' ) = IK = BO = •
Cho chop tam giac deu S.ABC c6 c^nh day bang 3a, canh ben bang 2a. Go!
G la trpng tam tam giac ABC. Dyng va tinh doan vuong goc chung aia hai duong thang SA va BC.
LOIGIAI
Trong tam giac ABC deu, keo dai AG cat BC tai M
=:>AG1BC
Chop S . A B C deu, ma G la tam A A B C nen:
S G 1 ( A B C ) = > S G 1 B C
Vi BC 1 S G v a B C l A M t u d o B C l ( S A M ) . Trong ASAM ke M N 1 SA ( N e S A ) ' ' '' ^
=>MN1BC ( v i M N c i ( S A M ) ) .
Do vay: M N la doan vuong goc chung cua BC va SA.
Trong ASAG vuong tai G c6:
SG = VSA^-AG^ 4 a 2 - 3 a 2 =a
Trong ASAM c6: MN.SA = SG.AM
3a>/3 3aS o MN.2a = a . — — => MN = — ^ ,
iji V.J u jf\.'0,jUJ\
. | a ằ , ) , . , . . ' A ;
.'.ifiHq '.f.r'i n'tti D''
Cho hinh chop tam giac S.ABC c6 SA vuong goc voi mp(ABC) va SA = ay/l..
Day ABC la tam giac vuong tai B voi BA=a. Gpi M la trung diem cua AB.
Tim dp dai doan vuong goc chung cua 2 duong thang SM va BC.
Ta c6: S A I B C
„„ LOI GIAI BCl(SAB)taiB. .
A B I B C
Dyng B H 1 S M ( H e S M ) .
Ta thay: B C 1 B H ( B H c ( S A B ) ) .
Vay: B H chinh la doan vuong goc chung cua S M va B C .
Ta tinh B H nhu sau:
H B M B Co A M H B - A M A S :
A S M S
Oft
HB MB
A S V A S ^ + A M ^ ^ 3
2
= 1 ^ H B = ^ = ^
Trong mat phang (P) cho hinh thoi ABCD c6 tam la O, canh a va OB = Tren duong thang vuong goc vai mp(ABCD) tai O, lay diem S
3
sao cho SB = a. Dung va tinh doan vuong goc chung ciia hai duong thang:
a) BP va SC b) ABvaSD a) De dang chung minh dupe: B D l ( S A C ) (vi B D l A C , B D l S O )
Trong mp(SAC), ke OM I S C ( M e S C )
=> OM la doan vuong goc chung cua SC va BD.
Trong ASOB vuong tai O c6:
SO = V s B 2 - B 0 2 = J a 2 - ^ = ^
V 3 3
Trong ABOC vuong tai O c6:
OC = V B C 2 - B 0 2 = J a 2 - ^ =
V 3 3 B
Trong ASOC vuong tai O c6: SC = VsO^ + OC^ = la^Js aVe aje
•OM = Va OM.SC = OS.OC<=>OM.
3 3 3 3 Gpi G, H Ian lupt la trung diem ciia AB va CD.
CD _L GH
Ta c6: C D 1 ( S G H ) ( S C D ) 1 ( S G H ) ( C D C ( S C D ) ) . C D _L SO
Tu O dung Ol 1 S H la giao tuyen cua hai mat phSng vuong goc nhau ( S C D ) va (SGH), suy ra O I 1 ( S C D ) . • . ^
Trong mat phSng (SGH) ke GJ // OI (J € SH) ^ GJ1 ( S C D ) . Tu J dung duong thang song song voi CD cat SD tai K.
Suy ra AB va JK cung thupc mpt mat phSng.
Trong mat (AB,JK) dung KL // GJ (L € AB).
Suy ra KL la doan vuong goc chung cua AB va SD.
That vay: Vi G J 1 ( S C D ) : G J I S D
G J I C D
G J I S D
GJ1AB(AB//CD)
K L I S D K L I A B
231
Trong ASOH : — - = -t - , • , ,
O I ^ O S ^ O H ^ 2a^ 2a^
GJ = 2 0 I = 2a>/22
1 1
1 1
I, Ke't luan: d (AB, S D ) =
Cho t u di?n ABCD vai AB = CD = a, AC = BD = b, BC = A D = c. Goi I va J Ian lugt la trung diem ciia AB va CD. H5y tinh dp dai doan vuong goc chung ciia AB va CD.
^1
L O I G I A I Ta c6: ACAB = ADBA (c.c.c) nen hai duong trung tuyen CI va D I tuong ving , r , j f i
bang nhau , nen tam giac ICD can tai I . s Suy ra: I J l C D . ^ \
Chung minh tuang t u ta c6: IJ 1 A B . ^ Kei luan: IJ la doan vuong goc chung ciia AB va CD.
BJ la duong trung tuyen ciia ABCD c6:
B f = BC^ + BD^ CD ^2 c^+b^ a2
2 4 Trong ABJI vuong tai I c6:
IJ^ = BJ^ - B I ^ = c2+b2 a
T
+ c - a IJ = .
2 2 2 3 + C - a
2
Cho hinh chop S.ABCD c6 day ABCD la hinh vuong canh a; mat phang (SAB) vuong goc voi mat phSng (ABCD); goc giCra mat phang (SAD) va mat ph^ng (ABCD) bang 45". Tinh khoang each t u C toi mat phang (SAD).
LOI G I A I Gpi H la hinh chieu cua S tren AB.
Vi (SAB) 1 (ABCD) theo giao tuyen AB nen:
S H I (ABCD) Vi: A D l AB
A D 1 S H • • A D I S A .
232
Suy ra goc giua mat phSng (SAD) va mat phSng (ABCD) la goc giua 2 duong thang SA va AB va bMng 45".
Vi BC // m p (SAD) => d(C; mp (SAD)) = d(B; mp (SAD)). '
Gpi K la hinh chieu vuong goc ciia B len SA c6: ,
| B K i ^ ^ ^ ^ ^ ^ ^ ^ => d(B,mp(SAD)) = BK . , Trong AABK vuong tai K c6 BAK = 45°, suy ra AABK vuong can t ^ i K nen
BK = A K = ^ = ^ . Ket luan d (C, mp(SAD)) = ^ . '
Cho hinh lap phuang ABCD.AjBjCjDj c6 dp dai canh bang a. Tren cac canh AB va CD lay Ian lupt cac diem M , N sao cho B M = C N := x. Xac dinh vi tri diem M sao cho khoang each giiJa hai duong thJing A j C va M N bang - .
3 L O I G I A I
Ta eo: M N // BC => M N // m p ( A j B C )
= > d ( M N , A i C ) = d ( M N , m p ( A j B C ) ) Gpi H = A j B n A B i va M K / / H A , K € A i B Ta c6: ABMKvuong can tai K , ^ .
B M : •, o\r.:, vr.
• M K = BK = •
Vi A j B l A B j ^ ^ M K l A j B ; vC Va C B 1 ( A B B J A j ) =^ C B 1 M K .
T u do suy ra M K 1 ( A J B C ) M K = d ( M N , ( A i B C ) ) = d ( M N , A^C)
A , /
\ c,
fx/
1 '• ' ' '
c
D ,
D e ' M K = - 3
x72 a a>/2
= - = > x =
2 3 3
'A! J,
Vay M thoa man BM = aN/2
Cho hinh hop ABCD.A'B'C'D' c6 day ABCD la hinh thoi canh a, DAB = 60*^, BB' = asfl . H i n h chieu vuong goc ciia diem D tren BB' la diem K nam tren BB' va BK = —BB', hinh chieu vuong goc ciia diem B' tren mat phang (ABCD) la diem H nam tren doan thang BD. Tinh khoang each giiJa hai duong thSng B'C va D C .
Taco: B K = 1 B B ' = ^ 4 4
LOI GlAl
Trong tam giac vuong B K D c6: '
D K = V B D 2 - B K 2 = ^ ('-•-^'^i
4 Taco: B ' K = ^ B B - ^
4 4 ' Trong tam giac vuong B ' K D :
DB' = V K B - ^ K D 2 . , ^ + l ^ = a V i . -"^'^ ^5^''
Suy ra tam giac B'BD can tai B' do do H chinh la giao diem ciia AC va BD.
D C B ' C ) ~'^(DC',mp(AB'C)) ~ °(B,mp(A'AC))
D C // A B' 3: > d. n r R r ^ =d/^^._--/.o.^x\„ = B H = - Cho hinh lang tru diing ABC.A'B'C c6 A C = a, BC = 2a, ACB = 120° va duong thSng A'C tao voi mat phling (ABB'A') goc 30". Gpi M trung diem cua BB'.
Tinh khoang each gii>a hai duong thSng A M , C C theo a.
L O I G I A I Ke C H 1 AB . Vi A A' 1 (ABC) nen:
A A ' 1 CH ^ CH 1 mp(ABB'A').
Vay (A^C(ABB'X')) = C A T I = 30°. '* ? Sir dung djnh ly cosin va cong thiic
tinh di^n tich cho tam giac ABC, ta c6:
AB = a^/7,CH = ^ g M B C . a . 2 a . s i n l 2 0 " a V 2 T
AB aV7 7
Mat phang (ABB'A') chua A M va song song voi CC nen:
d(AM,CC') = d(C,(ABB'A')) = C H = - ^ y i t
B'
, 1 2 0 ° ^ / 2 a
Cho hinh chop S.ABCD c6 day ABCD la hinh thoi canh a, goc BAD = 60° . 0 la giao diem cua AC va BD, H la trung diem cua BO, S H I (ABCD)
. Tim khoang each giua AB va SC. <•
SH=-
LOI GIAI
'^1 A
h = SH = 'AHCD = - A C . D H . Xa c6: AABD deu nen BD=a =^ D H = - a ICe H N song song AB N € AD;
Ke HK vuong goc voi HN, K e CD H I vuong goc voi SK, I thupc SK
=>HI1(SCD).
Khoang each tu H toi (SCD) la HI:
1 1 1
- + •
HI^ HK^ HS^ 'U '
C6:HS = ^ ; H K = HD.sinHDK = . ^ ^ ^ 2
4 64 100
HI^ 3a^ 27a^ 27a^ •HI =
d(AB,SC) = d(AB,(SCD)) = ^d(HN,(SCD)) = -d(H,(SCD)) = ^ H I =
3 3 3 5 Cho hinh chop S.ABCD c6 day ABCD la hinh vuong canh a, SA vuong goc voi day ABCD va SA = a. Tinh :
a) Khoang each tu diem S den mat phang MCD voi M la trung diem ciia SA.
b) Khoang each giira AC va SD.
a) Ta c6: • CD I A D
LOI GIAI CD 1 mp(SAD) CD 1 SA
Ma CD c (MCD). 1 , Vay: mp (SAD) 1 mp (MCD). A • i - i , , ,
K e S H l M D t a i H . ! /
(SAD) ± (MCD) ' • ^'^
(SAD)n(MCD) = M D ' '
= > S H l m p ( M C D ) ^ d ( 3 ^ ^ ^ ^ ^ j p S H Vi M trung diem ciia SA nen: S
Ta c6:
ASMD - 2 ^ASAD <ằSH.MD--SA.AD 2
<^SH. >V5 •SH =
Co: M D = V M A ^ + A D ^ = v2;
2 a
+ a = Ket luan: d ( S , m p ( M C D ) ) - ^ " - 5
D ; . Khoang each giua AC va SD. ' :^ ; A . Trong mat phSng (ABCD), t i r D k e D x / / AC. •'•' J / v<>;-. . - Ggi (a) la mat phing chiia duong thang SD va Dx. 0- .o:^ is J
A C / / D X ^ . ^ , '
^ ' | D X e mp(a) ^ ""^ ^""^ (^^'^°) ^ '^(AC,-p(a)) = ^(A,mp(a)) • Tu A ke AK 1 Dx t^i K. Ke AI 1 SK tai I. Thi A I la khoang each tir A den matphSng ( a ) . * '
Dx 1 AK ^ , , ^ " • "
Th|t vay ta co: <^ Dx 1 (SAKI.
Dx _L SA
Ma Dx c (a) ^ mp (SAK) 1 mp (a).
mp(SAK)nmp(a) = SK
• mp(SAK) 1 mp(a) ^ AI 1 mp(a) d^A,„p(a)) = AI A I ± S K •• -n.^ • .-1 .am^W:.
AD a '^.iJ^^A / n i Ta c6: AADK vuong can tai K nen KA = KD = = ^,, .,
Trong tam giac ASAK vuong tai K : _ " - ô : J—
a a Vs !
1 1 1
— I —
AV- AS^ AK
1 2 3 . - - = — + — = — =>AI =
2 a^ a^ a^
DE THI TUYEN SINH DAI HQC KHOI B NAM 2007
Cho hinh chop tu giac deu S.ABCD c6 day la hlnh vuong canh a. Goi E la diem doi xung cua D qua trung diem SA, M la trung diem ciia AE, N la trung diem cua BC. Chung minh M N vuong goc vol BD va tinh (theo a) khoang each giua hai duong thSng M N va AC.
L O I G I A I
GQI O giao diem eua A C va B D . P trung diem cua SA.
Trong tam giac E A D c6 M P la duong trung binh cua tam giac nen:
M P / / A D , M P = - A D (l).,^'.: •-: ' ''•> 'i'''-^'
Vi N trung diem ciia B C nen: N C // A D , N C = ^ A D (2) !f' j ' j p
fix (1) va (2) suy ra: tu giac CPMN la ^J* v;
hinh binh hanh nen MN // PC. ^ • Ta eo: BD ± (SAC) =^ BD 1 CP(CP c (SAC)) . ICe't luan: BD 1 M N
Vi MN // mp(SAC) nen:
d(MN,AC) = d(MN,mp(SAC)) = d(N,mp(SAC)).
Hai diem B va N nam tren duong thSng CO giao diem vai mat phSng
(SAC) tai diem C nen: • J ' ' ^
d(N,mp(SAC)) _ NC ^ !_ , r^:^^,:,^,^.
d(B,mp(SAC)) BC 2
Ket luan: d(MN,AC) =
d(N,mp(SAC)) = i-d(B,mp(SAC)) = l.BO = l .
Cho hinh lang try ABC.A'B'C c6 cae mat ben la hinh vuong canh bang a. Go E, F Ian luot la trung diem cua cae canh BC, A C , B C . Tinh khoang each giua va AT.
A'
LOI GIAI
Vi cae mat ben ABB'A', ACC'A', BCC'B' la cae hinh vuong eo:
A A ' I A B va A A ' I A C r ^ A A ' l ( A B C ) .
Vay ABC.A'B'C la hinh lang try dung va day la tam giac deu . Trong mp(A'B'C) dung E I / / A ' F ( l € B'C)
=>A'F// mp ( A D I E ) .
Ta CO DE thuoc mat phMng (ADIE).
Vay: d(A'F,DE) = d ( A ' F , ( A D I E ) ) = d(F,(ADIE)).
Taco: AD 1 (BCC'B') => ( A D I E ) 1 (BCC'B') : Hai mat phang nay vuong goc voi ) '^hau theo giao tuyeh D I .
Dvng B ' H l D l ( H e D l ) =^ B'H 1 ( A D I E ) . '
Suy ra: khoang each tu B' den mat ph^ng (ADIE) la B'H . Hinh vuong BCC'B' du^xc ve lai 6 hinh 2.
'^""--^^
/ B'
\
B