Khoi a dien
Tran S TungTran S Tung
Khoi a dien
1. Hai ng thang song song
a) nh ngha:
b) Tnh chat
(
(
(
2. ng thang va mat phang song song
a) nh ngha:
d // (P) ( d ( (P) = (
b) Tnh chat
(
(
(
3. Hai mat phang song song
a) nh ngha:
(P) // (Q) ( (P) ( (Q) = (
b) Tnh chat
(
(
(
4. Chng minh quan he song song
a) Chng minh hai ng thang song song
Co the s dung 1 trong cac cach sau:
( Chng minh 2 ng thang o ong phang, roi ap dung phng phap chng
minh song song trong hnh hoc phang (nh tnh chat ng trung bnh, nh l
Talet ao, )
( Chng minh 2 ng thang o cung song song vi ng thang th ba.
( Ap dung cac nh l ve giao tuyen song song.
b) Chng minh ng thang song song vi mat phang
e chng minh , ta chng minh d khong nam trong (P) va song song vi
mot ng thang d( nao o nam trong (P).
c) Chng minh hai mat phang song song
Chng minh mat phang nay cha hai ng thang cat nhau lan lt song
song vi hai ng thang trong mat phang kia.
1. Hai ng thang vuong goc
a) nh ngha:
a ( b (
b) Tnh chat
( Gia s la VTCP cua a, la VTCP cua b. Khi o .
(
2. ng thang va mat phang vuong goc
a) nh ngha:
d ( (P) ( d ( a, (a ( (P)
b) Tnh chat
( ieu kien e ng thang ( mat phang:
(
(
(
(
(
(
( Mat phang trung trc cua mot oan thang la mat phang vuong goc
vi oan thang tai trung iem cua no.
Mat phang trung trc cua oan thang la tap hp cac iem cach eu hai
au mut cua oan thang o.
( nh l ba ng vuong goc
Cho , a( la hnh chieu cua a tren (P). Khi o b ( a ( b ( a(
3. Hai mat phang vuong goc
a) nh ngha:
(P) ( (Q) (
b) Tnh chat
( ieu kien e hai mat phang vuong goc vi nhau:
(
(
(
4. Chng minh quan he vuong goc
a) Chng minh hai ng thang vuong goc
e chng minh , ta co the s dung 1 trong cac cach sau:
( Chng minh goc gia a va d bang 900.
( Chng minh 2 vect ch phng cua a va d vuong goc vi nhau.
( Chng minh ma .
( Chng minh d vuong goc vi (P) va (P) cha a.
( S dung nh l ba ng vuong goc.
( S dung cac tnh chat cua hnh hoc phang (nh nh l Pitago, ).
b) Chng minh ng thang vuong goc vi mat phang
e chng minh d ( (P), ta co the chng minh bi mot trong cac cach
sau:
( Chng minh d vuong goc vi hai ng thang a, b cat nhau nam trong
(P).
( Chng minh d vuong goc vi (Q) va (Q) // (P).
( Chng minh d // a va a ( (P).
( Chng minh d ( (Q) vi (Q) ( (P) va d vuong goc vi giao tuyen c
cua (P) va (Q).
( Chng minh d = (Q) ( (R) vi (Q) ( (P) va (R) ( (P).
c) Chng minh hai mat phang vuong goc
e chng minh (P) ( (Q), ta co the chng minh bi mot trong cac cach
sau:
( Chng minh trong (P) co mot ng thang a ma a ( (Q).
( Chng minh
1. Goc
a) Goc gia hai ng thang:
a//a', b//b' (
Chu y: 00 ( ( 900
b) Goc gia ng thang vi mat phang:
( Neu d ( (P) th = 900.
( Neu th = vi d( la hnh chieu cua d tren (P).
Chu y: 00 ( ( 900
c) Goc gia hai mat phang
( Gia s (P) ( (Q) = c. T I ( c, dng (
Chu y:
d) Dien tch hnh chieu cua mot a giac
Goi S la dien tch cua a giac (H) trong (P), S( la dien tch cua
hnh chieu (H() cua (H) tren (Q), ( = . Khi o:S( = S.cos(2. Khoang
cach
a) Khoang cach t mot iem en ng thang (mat phang) bang o dai oan
vuong goc ve t iem o en ng thang (mat phang).
b) Khoang cach gia ng thang va mat phang song song bang khoang
cach t mot iem bat k tren ng thang en mat phang.
c) Khoang cach gia hai mat phang song song bang khoang cach t
mot iem bat k tren mat phang nay en mat phang kia.
d) Khoang cach gia hai ng thang cheo nhau bang:
( o dai oan vuong goc chung cua hai ng thang o.
( Khoang cach gia mot trong hai ng thang vi mat phang cha ng
thang kia va song song vi ng thang th nhat.
( Khoang cach gia hai mat phang, ma moi mat phang cha ng thang
nay va song song vi ng thang kia.
1. He thc lng trong tam giac
a) Cho (ABC vuong tai A, co ng cao AH.
( (
(
(
b) Cho (ABC co o dai ba canh la: a, b, c; o dai cac trung tuyen
la ma, mb, mc; ban knh ng tron ngoai tiep R; ban knh ng tron noi
tiep r; na chu vi p.
( nh l ham so cosin:
( nh l ham so sin:
( Cong thc o dai trung tuyen:
2. Cac cong thc tnh dien tch a) Tam giac:
(
(
(
(
(
( (ABC vuong tai A:
( (ABC eu, canh a:
b) Hnh vuong:
S = a2
(a: canh hnh vuong)
c) Hnh ch nhat:S = a.b
(a, b: hai kch thc)
d) Hnh bnh hanh:S = ay ( cao =
e) Hnh thoi:
f) Hnh thang:
(a, b: hai ay, h: chieu cao)
g) T giac co hai ng cheo vuong goc:
1. The tch cua khoi hop ch nhat:
vi a, b, c la ba kch thc cua khoi hop ch nhat.
2. The tch cua khoi chop:
vi Say la dien tch ay, h la chieu cao cua khoi chop
3. The tch cua khoi lang tru:
vi Say la dien tch ay, h la chieu cao cua khoi lang tru
4. Mot so phng phap tnh the tch khoi a dien
a) Tnh the tch bang cong thc
( Tnh cac yeu to can thiet: o dai canh, dien tch ay, chieu
cao,
( S dung cong thc e tnh the tch.
b) Tnh the tch bang cach chia nho
Ta chia khoi a dien thanh nhieu khoi a dien nho ma co the de
dang tnh c the tch cua chung. Sau o, cong cac ket qua ta c the tch
cua khoi a dien can tnh.
c) Tnh the tch bang cach bo sung
Ta co the ghep them vao khoi a dien mot khoi a dien khac sao cho
khoi a dien them vao va khoi a dien mi tao thanh co the de tnh c
the tch.
d) Tnh the tch bang cong thc t so the tch
Ta co the van dung tnh chat sau:
Cho ba tia Ox, Oy, Oz khong ong phang. Vi bat k cac iem A, A
tren Ox; B, B' tren Oy; C, C' tren Oz, ta eu co:
* Bo sung
( Dien tch xung quanh cua hnh lang tru (hnh chop) bang tong dien
tch cac mat ben
( Dien tch toan phan cua hnh lang tru (hnh chop) bang tong dien
tch xung quanh vi dien tch cac ay.Bai 1. Cho hnh chop t giac eu
S.ABCD co ay ABCD la hnh vuong canh a. Goc gia mat ben va mat ay
bang ( (450 < ( < 900). Tnh the tch hnh chop.
HD: Tnh h =
(
Bai 2. Cho hnh chop t giac eu S.ABCD co ay ABCD la hnh vuong
canh 2a, canh ben SA = a. Mot mat phang (P) i qua AB va vuong goc
vi mp(SCD) lan lt cat SC va SD tai C( va D(. Tnh the tch cua khoi a
dien ADD(.BCC(.
HD: Ghep them khoi S.ABC'D' vao khoi ADD'.BCC' th c khoi
SABCD
(
Bai 3. Cho hnh chop tam giac S.ABC co SA = x, BC = y, cac canh
con lai eu bang 1. Tnh the tch hnh chop theo x va y.
HD: Chia khoi SABC thanh hai khoi SIBC va AIBC (I la trung iem
SA)
(
Bai 4. Cho t dien ABCD co cac canh AD = BC = a, AC = BD = b, AB
= CD = c. Tnh the tch t dien theo a, b, c.
HD: Trong mp(BCD) lay cac iem P, Q, R sao cho B, C, D lan lt la
trung iem cua PQ, QR, RP. Chu y: VAPQR = 4VABCD =
(
Bai 5. Cho hnh chop S.ABC co ay ABC la tam giac eu canh a, SA =
2a va SA ( (ABC).Goi M va N lan lt la hnh chieu cua A tren cac ng
thang SB va SC. Tnh the tch khoi chop A.BCNM.
HD: (
Bai 6. Cho hnh chop S.ABCD co ay ABCD la hnh vuong canh 7cm, SA
( (ABCD), SB = 7cm. Tnh the tch cua khoi chop S.ABCD.
Bai 7. Cho hnh chop S.ABC co ay ABC la tam giac vuong tai A vi
AB = 3 cm, AC = 4cm. Hai mat phang (SAB) va (SAC) cung vuong goc vi
mat phang ay va SA = 5cm. Tnh the tch khoi chop S.ABC.
Bai 8. Cho hnh t dien ABCD co AD ( (ABC). Cho AC = AD = 4cm, AB
= 3cm, BC = 5cm.
a) Tnh khoang cach t A en mp(BCD).
b) Tnh the tch t dien ABCD.
Bai 9. Cho lang tru tam giac eu ABC.A(B(C( co mp(ABC() tao vi ay
mot goc 450 va dien tch (ABC( bang 49cm2. Tnh the tch lang tru.
Bai 10. Cho hnh vuong ABCD canh a, cac na ng thang Bx, Dy vuong
goc vi mp(ABCD) va ve cung mot pha oi vi mat phang ay. Tren Bx va
Dy lan lt lay cac iem M, N va goi BM = x, DN = y. Tnh the tch t
dien ACMN theo a, x, y.
Bai 11. Cho hnh chop S.ABCD co ay ABCD la hnh ch nhat vi AB =a,
AD = a, SA ( (ABCD). Goi M,N lan lt la trung iem cua AD va SC, I la
giao iem cua BM va AC.
a) Chng minh mp(SAC) ( BM.
b) Tnh the tch cua khoi t dien ANIB.
Bai 12. Cho hnh chop S.ABC co ay ABC la tam giac eu canh a, SA =
2a va SA ( (ABC). Goi M va N lan lt la hnh chieu cua A tren cac ng
thang SB, SC. Tnh the tch khoi chop A.BCNM.
Bai 13. (A08) Cho lang tru ABC. ABC co o dai canh ben bang 2a,
ay ABC la tam giac vuong tai A, AB = a, AC = a va hnh chieu vuong
goc cua A tren (ABC) la trung iem cua BC. Tnh theo a the tch cua
khoi chop A.ABC va cosin cua goc gia 2 ng thang AA va BC.HD:
Bai 14. (B08): Cho hnh chop S.ABCD co ay ABCD la hnh vuong canh
2a, SA = a, SB = a va (SAB) vuong goc mat ay. Goi M, N lan lt la
trung iem AB, BC. Tnh theo a the tch cua khoi chop S.BMDN va cosin
cua goc gia hai ng thang SM va DN.HD:
Bai 15. (D08): Cho lang tru ng ABC. ABC co ay ABC la tam giac
vuong, AB = BC = a, canh ben AA = a. Goi M la trung iem cua BC. Tnh
theo a the tch cua lang tru ABC.ABC va khoang cach gia 2 ng thang
AM, B(C.HD:
Bai 16. (A07): Cho hnh chop S.ABCD co ay ABCD la hnh vuong canh
a, mat ben SAD la tam giac eu va nam trong mat phang vuong goc vi
ay. Goi M, N, P lan lt la trung iem SB, BC, CD. Chng minh AM ( BP
va tnh the tch khoi CMNP.HD:
Bai 17. (B07): Cho hnh chop t giac eu S.ABCD co ay ABCD la hnh
vuong canh a. Goi E la iem oi xng cua D qua trung iem cua SA; M la
trung iem cua AE, N la trung iem cua BC. Chng minh MN ( BD va tnh
khoang cach gia hai ng thang MN va AC.HD:
Bai 18. (D07): Cho hnh chop S.ABCD co ay ABCD la hnh thang vi ,
BC = BA = a, AD = 2a. SA((ABCD), . Goi H la hnh chieu vuong goc cua
A tren SB. Chng minh tam giac SCD vuong va tnh khoang cach t H en
(SCD).HD:
Bai 19. (A06): Cho hnh tru co cac ay la hai hnh tron tam O va
O(, ban knh ay bang chieu cao va bang a. Tren ng tron ay tam O lay
iem A, tren ng tron ay tam O( lay iem B sao cho AB = 2a. Tnh the
tch cua khoi t dien OO(AB.
HD:
Bai 20. (B06): Cho hnh chop S.ABCD co ay ABCD la hnh ch nhat vi
AB = a, , SA = a va SA ( (ABCD). Goi M, N lan lt la trung iem cua
AD, SC; I la giao iem cua BM va AC. Chng minh rang (SAC) ( (SMB).
Tnh the tch cua khoi t dien ANIB.HD:
Bai 21. (D06): Cho hnh chop tam giac S.ABC co ay ABC la tam giac
eu canh a, SA = 2a va SA ( (ABC). Goi M, N lan lt la hnh chieu
vuong goc cua A tren SB, SC. Tnh the tch cua hnh chop
A.BCMN.HD:
Bai 22. (D b 1 A07): Cho lang tru ng ABC.A1B1C1 co AB = a, AC =
2a, AA1 = va . Goi M la trung iem CC1. Chng minh MB ( MA1 va tnh
khoang cach d t A en (A1BM).HD:
Bai 23. (D b 2 A07): Cho hnh chop SABC co goc , ABC va SBC la
cac tam giac eu canh a. Tnh theo a khoang cach t B en (SAC).HD:
Bai 24. (D b 1 B07): Cho hnh chop S.ABCD co ay ABCD la hnh vuong
tam O, SA ( (ABCD). AB = a, . Goi H, K lan lt la hnh chieu vuong
goc cua A tren SB, SD. Chng minh SC((AHK) va tnh the tch cua t dien
OAHK.HD:
Bai 25. (D b 2 B07): Trong mat phang (P), cho na ng tron ng knh
AB = 2R va iem C thuoc na ng tron o sao cho AC = R. Tren ng thang
vuong goc vi (P) tai A lay iem S sao cho . Goi H, K lan lt la hnh
chieu cua A tren SB, SC. Chng minh tam giac AHK vuong va tnh the
tch t dien SABC.
HD:
Bai 26. (D b 1 D07): Cho lang tru ng ABC.A1B1C1 co ay ABC la tam
giac vuong, AB = AC = a, AA1 = . Goi M, N lan lt la trung iem oan
AA1 va BC1. Chng minh MN la ng vuong goc chung cua AA1 va BC1. Tnh
the tch cua t dien MA1BC1.
HD:
Bai 27. (D b 2 D07): Cho lang tru ng ABC.A1B1C1 co tat ca cac
canh eu bang a. M la trung iem cua oan AA1. Chng minh BM ( B1C va
tnh khoang cach gia hai ng thang BM va B1C.HD:
Bai 28. (D b 1 A06): Cho hnh hop ng ABCD.A'B'C'D' co cac canh AB
= AD = a, AA' = va . Goi M, N lan lt la trung iem cua cac canh A'D'
va A'B'. Chng minh AC' ( (BDMN). Tnh the tch khoi chop
A.BDMN.HD:
Bai 29. (D b 2 A06): Cho hnh chop S.ABCD co ay ABCD la hnh ch
nhat vi AB = a, AD = 2a, canh SA vuong goc vi ay, canh SB tao vi
mat phang ay mot goc 600. Tren canh SA lay iem M sao cho AM = . Mat
phang (BCM) cat canh SD tai N. Tnh the tch khoi chop S.BCNM.HD:
Bai 30. (D b 1 B06): Cho hnh chop S.ABCD co ay ABCD la hnh thoi
canh a, , SA ( (ABCD), SA = a. Goi C' la trung iem cua SC. Mat
phang (P) i qua AC' va song song vi BD, cat cac canh SB, SD lan lt
tai B', D'. Tnh the tch khoi chop S.AB'C'D'.HD:
Bai 31. (D b 2 B06): Cho hnh lang tru ABC.A'B'C' co A'ABC la hnh
chop tam giac eu, canh ay AB = a, canh ben AA' = b. Goi ( la goc
gia hai mat phang (ABC) va (A'BC). Tnh tan( va the tch khoi chop
A'.BB'C'C.HD: tan( = ;
Bai 32. (D b 1 D06): Cho hnh chop t giac eu S.ABCD co canh ay
bang a. Goi SH la ng cao cua hnh chop. Khoang cach t trung iem I
cua SH en mat phang (SBC) bang b. Tnh the tch khoi chop S.ABCD.
HD:
Bai 33. (D b 2 D06): Cho hnh lap phng ABCD.A(B(C(D( co canh bang
a va iem K thuoc canh CC( sao cho CK = . Mat phang (() i qua A, K
va song song vi BD, chia khoi lap phng thanh hai khoi a dien. Tnh
the tch cua hai khoi a dien o.
HD:
Bai 34. (D b 04): Cho hnh chop S.ABC co SA = 3a va SB ( (ABC).
Tam giac ABC co BA = BC = a, goc ABC bang 1200. Tnh khoang cach t
iem A en mat phang (SBC).Bai 35. (D b 03): Cho hnh chop S.ABC co ay
ABC la tam giac vuong tai B, AB = a, BC = 2a, canh SA vuong goc vi
ay va SA = 2a. Goi M la trung iem cua SC. Chng minh rang tam giac
AMB can tai M va tnh dien tch tam giac AMB theo a.
HD:
Bai 1. Cho hnh chop t giac eu SABCD, co canh ay bang a va .
a) Tnh dien tch xung quanh hnh chop.
b) Chng minh chieu cao cua hnh chop bang
c) Tnh the tch khoi chop.
HD:a) Sxq =
c) V =
Bai 2. Cho hnh chop SABC co 2 mat ben (SAB) va (SAC) vuong goc
vi ay. ay ABC la tam giac can nh A. Trung tuyen AD = a. Canh ben SB
tao vi ay goc ( va tao vi mp(SAD) goc (.
a) Xac nh cac goc (, (.
b) Chng minh: SB2 = SA2 + AD2 + BD2.
c) Tnh dien tch toan phan va the tch khoi chop.
HD:a)
c) Stp =
V =
Bai 3. Cho hnh chop SABCD co ay ABCD la hnh vuong canh a. Mat
ben SAB la tam giac eu va vuong goc vi ay. Goi H la trung iem cua
AB va M la mot iem di ong tren ng thang BC.
a) Chng minh rang SH ( (ABCD). Tnh the tch khoi chop SABCD.
b) Tm tap hp cac hnh chieu cua S len DM.
c) Tm khoang cach t S en DM theo a va x = CM.
HD: b) K thuoc ng tron ng knh HD c) SK =
Bai 4. Tren ng thang vuong goc tai A vi mat phang cua hnh vuong
ABCD canh a ta lay iem S vi SA = 2a. Goi B(, D( la hnh chieu cua A
len SB va SD. Mat phang (AB(D() cat SC tai C(. Tnh the tch khoi
chop SAB(C(D(.
HD:
( VSAB(C(D( =
Bai 5. Cho hnh chop SABCD co ay ABCD la hnh bnh hanh. Mot mat
phang (P) cat SA, SB, SC, SD lan lt tai A(, B(, C(, D(. Chng
minh:
HD: S dung tnh chat t so the tch hnh chopBai 6. Cho t dien eu
SABC co canh la a. Dng ng cao SH.
a) Chng minh SA ( BC.
b) Tnh the tch va dien tch toan phan cua hnh chop SABC.
c) Goi O la trung iem cua SH. Chng minh rang OA, OB, OC oi mot
vuong goc vi nhau.
HD:b) V = ;Stp = .Bai 7. Cho hnh chop t giac eu SABCD co canh
ben tao vi ay mot goc 600 va canh ay bang a.
a) Tnh the tch khoi chop.
b) Qua A dng mat phang (P) vuong goc vi SC. Tnh dien tch thiet
dien tao bi (P) va hnh chop.
HD:a) V =
b) S =
Bai 8. Cho hnh chop t giac eu SABCD co chieu cao SH = h va goc
ay cua mat ben la (.
a) Tnh dien tch xung quanh va the tch khoi chop theo ( va h.
b) Cho iem M di ong tren canh SC. Tm tap hp hnh chieu cua S
xuong mp(MAB).
HD:a) Sxq = ;V =
Bai 9. Tren canh AD cua hnh vuong ABCD canh a, ngi ta lay iem M
vi AM = x (0 ( x ( a) va tren na ng thang Ax vuong goc tai A vi mat
phang cua hnh vuong, ngi ta lay iem S vi SA = y (y > 0).
a) Chng minh hai mat phang (SBA) va (SBC) vuong goc.
b) Tnh khoang cach t iem M en mp(SAC).
c) Tnh the tch khoi chop SABCM.
d) Vi gia thiet x2 + y2 = a2. Tm gia tr ln nhat cua the tch vi
SABCM.
e) I la trung iem cua SC. Tm qu tch hnh chieu cua I xuong MC khi
M di ong tren oan AD.
HD:b) d =
c) V =
d) Vmax =
Bai 10. Cho hnh chop SABCD co ay ABCD la hnh ch nhat co canh AB
= a, canh ben SA vuong goc vi ay, canh ben SC hp vi ay goc ( va hp
vi mat ben SAB mot goc (.
a) Chng minh: SC2 = .
b) Tnh the tch khoi chop.
HD:b) V =
Bai 11. Cho hnh chop S.ABCD co ay ABCD la hnh vuong canh a .Canh
ben SA =2a va vuong goc vi mat phang ay.a) Tnh dien tch toan phan
cua hnh chop.
b) Ha AE ( SB, AF ( SD. Chng minh SC ( (AEF).
Bai 12. Cho hnh chop t giac eu S.ABCD, co ay ABCD la hnh vuong
canh bang a va SA = SB = SC = SD = a. Tnh dien tch toan phan va the
tch khoi chop S.ABCD.Bai 13. Cho hnh chop t giac S.ABCD co ay la
ABCD hnh thang vuong tai A va D, AB = AD = a, CD = 2a. Canh ben SD
( (ABCD) va SD = a .a) Chng minh (SBC vuong. Tnh dien tch (SBC.
b) Tnh khoang cach t A en mat phang (SBC).
Bai 14. Cho hnh chop t giac S.ABCD co ay ABCD la hnh thang vuong
tai A va D, AB = AD = a, CD = 2a. Canh ben SD ( (ABCD), SD . T
trung iem E cua DC dng EK ( SC (K ( SC). Tnh the tch khoi chop
S.ABCD theo a va chng minh SC ( (EBK).
Bai 15. Cho hnh chop t giac S.ABCD co ay ABCD la hnh thang vuong
tai A va D. Biet rang AB = 2a, AD = CD = a (a > 0). Canh ben SA
= 3a va vuong goc vi ay.
a) Tnh dien tch tam giac SBD.b) Tnh the tch cua t dien SBCD theo
a.
Bai 16. Cho hnh chop tam giac S.ABC co ay ABC la tam giac vuong
B. Canh SA vuong goc vi ay. T A ke cac oan thang AD SB va AESC.
Biet AB = a, BC = b, SA = c.
a) Tnh the tch cua khoi chop S.ADE.
b) Tnh khoang cach t iem E en mat phang (SAB).
Bai 17. Cho lang tru tam giac eu ABC.A(B(C(, canh ay bang a, ng
cheo cua mat ben BCC(B( hp vi mat ben ABB(A( mot goc (.
a) Xac nh goc (.
b) Chng minh the tch lang tru la: .
HD:a) vi I( la trung iem cua A(B(
Bai 18. Cho lang tru t giac eu ABCD.A(B(C(D(, chieu cao h. Mat
phang (A(BD) hp vi mat ben ABB(A( mot goc (. Tnh the tch va dien
tch xung quanh cua lang tru.
HD:V = ,Sxq =.Bai 19. Cho lang tru ng ABC.A(B(C(, ay ABC vuong
tai A. Khoang cach t AA( en mat ben BCC(B( bang a, mp(ABC() cach C
mot khoang bang b va hp vi ay goc (.
a) Dng AH ( BC, CK ( AC(. Chng minh: AH = a, = (, CK = b.
b) Tnh the tch lang tru.
c) Cho a = b khong oi, con ( thay oi. nh ( e the tch lang tru
nho nhat.
HD:b) V =
c) ( = arctan
Bai 20. Cho lang tru eu ABCD.A(B(C(D( canh ay bang a. Goc gia ng
cheo AC( va ay la 600. Tnh the tch va dien tch xung quanh hnh lang
tru.
HD:V = a3;Sxq = 4a2
Bai 21. Cho lang tru t giac eu, co canh ben la h. T mot nh ve 2
ng cheo cua 2 mat ben ke nhau. Goc gia 2 ng cheo ay la (. Tnh dien
tch xung quanh hnh lang tru.
HD:Sxq = 4h2.Bai 22. Cho lang tru tam giac eu ABc.A(B(C(, canh
ay bang a. Mat phang (ABC() hp vi mp(BCC(B() mot goc (. Goi I, J la
hnh chieu cua A len BC va BC(.
a) Chng minh = (.
b) Tnh the tch va dien tch xung quanh hnh lang tru.
HD:b) V = ; Sxq = 3a2.Bai 23. Cho lang tru xien ABC.A(B(C(, ay
la tam giac eu canh a, AA( = A(B = A(C = b.
a) Xac nh ng cao cua lang tru ve t A(. Chng minh mat ben BCC(B(
la hnh ch nhat.
b) nh b theo a e mat ben ABB(A( hp vi ay goc 600.
c) Tnh the tch va dien tch toan phan theo a vi gia tr b tm
c.
HD:b) b = a
c) Stp =
Bai 24. Cho hnh lang tru xien ABC.A(B(C(, ay ABC la tam giac
vuong can nh A. Mat ben ABB(A( la hnh thoi canh a, nam tren mat
phang vuong goc vi ay. Mat ben ACC(A( hp vi ay goc nh dien co so o
( (0 < ( < 900).
a) Chng minh: = (.
b) Tnh the tch lang tru.
c) Xac nh thiet dien thang qua A. Tnh dien tch xung quanh lang
tru.
d) Goi ( la goc nhon ma mp(BCC(B() hp vi mat phang ay.
Chng minh: tan( = tan(.
HD:b) V = a3sin(
c) Sxq = a2(1 + sin( + )Bai 25. Cho lang tru xien ABC.A(B(C( ay
la tam giac eu canh a. Hnh chieu cua A( len mp(ABC) trung vi tam ng
tron (ABC). Cho = 450.
a) Tnh the tch lang tru.
b) Tnh dien tch xung quanh lang tru.
HD:a) V =
b) Sxq = a2(1 + ).Bai 26. Cho lang tru xien ABC.A(B(C(, ay ABC
la tam giac eu noi tiep trong ng tron tam O. Hnh chieu cua C( len
mp(ABC) la O. Khoang cach gia AB va CC( la d va so o nh dien canh
CC( la 2(.
a) Tnh the tch lang tru.
b) Goi ( la goc gia 2 mp(ABB(A() va (ABC) (0 < ( <
900).
Tnh ( biet ( + ( = 900.
HD:a) V =
b) tan( = ;( = arctan
Bai 27. Cho lang tru xien ABC.A(B(C( co ay la tam giac vuong tai
A, AB = a, BC = 2a. Mat ben ABBA( la hnh thoi, mat ben BCC(B( nam
trong mat phang vuong goc vi ay, hai mat nay hp vi nhau mot goc
(.
a) Tnh khoang cach t A en mp(BCC(B(). Xac nh goc (.
b) Tnh the tch lang tru.
HD:a) . Goi AK la ng cao cua (ABC; ve KH ( BB(. = (.
b) V = .Bai 28. Cho hnh hop ng ABCD.A(B(C(D(, ay la hnh thoi.
Biet dien tch 2 mat cheo ACC(A(, BDD(B( la S1, S2.
a) Tnh dien tch xung quanh hnh hop.
b) Biet = 1v. Tnh the tch hnh hop.
HD:a) Sxq = 2
b) V =
Bai 29. Cho hnh hop ch nhat ABCD.A(B(C(D(, ng cheo AC( = d hp vi
ay ABCD mot goc ( va hp vi mat ben BCC(B( mot goc (.
a) Chng minh: .
b) Chng minh the tch hnh hop la: V = d3sin(.sin(
c) Tm he thc gia (, ( e A(D(CB la hnh vuong. Cho d khong oi, (
va ( thay oi ma A(D(CB luon la hnh vuong, nh (, ( e V ln nhat.
HD:c) 2(cos2( sin2() = 1; Vmax = khi ( = ( = 300 (dung Cosi).Bai
30. Cho hnh hop ABCD.A(B(C(D co ay la hnh thoi ABCD canh a, = 600.
Chan ng vuong goc ha t B( xuong ay ABCD trung vi giao iem 2 ng cheo
cua ay. Cho BB( = a.
a) Tnh goc gia canh ben va ay.
b) Tnh the tch va dien tch xung quanh hnh hop.
HD:a) 600
b) V = ; Sxq = a2.Bai 31. Cho hnh hop xien ABCD.A(B(C(D(, ay
ABCD la hnh thoi canh a va = 600; A(A = A(B = A(D va canh ben hp vi
ay goc (.
a) Xac nh chan ng cao cua hnh hop ve t A( va goc (. Tnh the tch
hnh hop.
b) Tnh dien tch cac t giac ACC(A(, BDD(B(.
c) at ( = . Tnh ( biet ( + ( = .
HD:a) Chan ng cao la tam cua tam giac eu ABD.
b) SBDD(B( = ; SACC(A( = a2tan(
c) ( = arctan
Chan thanh cam n cac ban ong nghiep va cac em hoc sinh a oc tap
tai lieu nay.
[email protected]
I. QUAN HE SONG SONG
CHNG 0
ON TAP HNH HOC KHONG GIAN 11
ON TAP KHOI A DIEN
IV. Nhac lai mot so cong thc
trong Hnh hoc phang
CHNG I
KHOI A DIEN VA THE TCH CUA CHUNG
III. GOC KHOANG CACH
II. QUAN HE VUONG GOC
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