Chuyn :Luyn tp Hnh Hc Khng Gian
GV: L Minh Tin
CHUYN :
PHNG PHP LUYN TP TH TCH KHI A DIN
N TP 3 KIN THC C BN HNH HC LP 12 A. TH TCH KHI A DIN I/ Cc cng thc th tch ca khi a din: 1. TH TCH KHI LNG
TR:V= B.h vi B:d t h ie c a n y h a u h: c ie c oc ba
h B
Th tch khi hp ch nht: V = a.b.c vi a,b,c l ba kch thc Th tch khi lp phng: V = a3 vi a l di cnh
aa
a
2. TH TCH KHI CHP:V= Bh1 3h
B : die tch a n y vi u h: chie cao 3. T S TH TCH T DIN: Cho khi t din SABC v A, B, C l cc im ty ln lt thuc SA, SB, SC ta c:VSABC VSA'B'C' S S S A B C = S S A' B'S C'A '
B
S C' A'
A
B' C B
B ' C '
4. TH TCH KHI CHP CT:V= h B+ B' + 3
A
B
(
BB'
)
C
B, B': die tch hai a n y vi u h: chie cao II/ Bi tp:
Chuyn :Luyn tp Hnh Hc Khng Gian
GV: L Minh Tin
LOI 1:
TH TCH LNG TR
1) Dng 1:
Khi lng tr ng c chiu cao hay cnh y
V d 1: y ca lng tr ng tam gic ABC.ABC l tam gic ABC vung cn ti A c cnh BC = a 2 v bit A'B = 3a. Tnh th tch khi lng tr.A' B' 3a A a B C C'
a 2
Li gii: Ta c VABC vung cn ti A nn AB = AC = a ABC A'B'C' l lng tr ng AA ' AB VAA 'B AA '2 = A 'B2 AB2 = 8a2 AA ' = 2a 2 Vy V = B.h = SABC .AA' = a3 2
V d 2: Cho lng tr t gic u ABCD.ABCD' c cnh bn bng 4a v ng cho 5a. Tnh th tch khi lng tr ny. C' Li gii: D' ABCD A'B'C'D' l lng tr ng nn A' BD2 = BD'2 - DD'2 = 9a2 BD = 3a B' 3a 4a 5a ABCD l hnh vung AB = 2 C D 2 9a Suy ra B = SABCD = A B 4 Vy V = B.h = SABCD.AA' = 9a3 V d 3: y ca lng tr ng tam gic ABC.ABC l tam gic u cnh a = 4 v bit din tch tam gic ABC bng 8. Tnh th tch khi lng tr.A' B' C'
Li gii: Gi I l trung im BC .Ta c V ABC u nn AI = A 'I BC(dl3 ) 2S 1 SA'BC = BC.A 'I A 'I = A'BC = 4 2 BC AA ' (ABC) AA ' AI . VA 'AI AA ' = A 'I2 AI2 = 2 Vy : VABC.ABC = SABC .AA'= 8 3AB 3 = 2 3 & AI BC 2
A I B
C
V d 4: Mt tm ba hnh vung c cnh 44 cm, ngi ta ct b i mi gc
Chuyn :Luyn tp Hnh Hc Khng Gian
GV: L Minh Tin
tm ba mt hnh vung cnh 12 cm ri gp li thnh mt ci hp ch nht khng c np. Tnh th tch ci hp ny.D' C'
D'A' D
C' C
D'B' C
D
C'
A'A B
AA'
BB'
B'
Gii Theo bi, ta c AA' = BB' = CC' = DD' = 12 cm nn ABCD l hnh vung c AB = 44 cm - 24 cm = 20 cm v chiu cao hp h = 12 cm Vy th tch hp l V = SABCD.h = 4800cm3
V d 5: Cho hnh hp ng c y l hnh thoi cnh a v c gc nhn bng 600 ng cho ln ca y bng ng cho nh ca lng tr. Tnh th tch hnh hp .C' B' D B C
D'
Li gii: Ta c tam gic ABD u nn : BD = aa2 3 v SABCD = 2SABD = 2
A'
Theo bi BD' = AC = 2
A
60
a 3 =a 3 2 VDD'B DD' = BD'2 BD2 = a 2 a3 6 Vy V = SABCD.DD' = 2
2)Dng 2:
Lng tr ng c gc gia ng thng v mt phng.
V d 1: Cho lng tr ng tam gic ABC A'B'C' c y ABC l tam gic vung cn ti B vi BA = BC = a ,bit A'B hp vi y ABC mt gc 600 . Tnh th tch lng tr.A' C'
B'
A 60o B
C
Li gii: Ta c A 'A (ABC) A 'A AB& AB l hnh chiu ca A'B trn y ABC . ABA Vy gc[A 'B,(ABC)] = ' = 60o VABA ' AA ' = AB.tan 600 = a 3 1 a2 SABC = BA.BC = 2 2 a3 3 Vy V = SABC.AA' = 2
V d 2: Cho lng tr ng tam gic ABC A'B'C' c y ABC l tam gic
Chuyn :Luyn tp Hnh Hc Khng Gian
GV: L Minh Tin
vung ti A vi AC = a , ACB = 60 o bit BC' hp vi (AA'C'C) mt gc 300. Tnh AC' v th tch lng tr. o A' C' Li gii: VABC AB = AC.tan60 = a 3 . Ta c: AB AC;AB AA ' AB (AA 'C'C) nn AC' l hnh chiu ca BC' trn (AA'C'C). B' o 30 Vy gc[BC';(AA"C"C)] = BC'A = 30o AB VAC'B AC' = = 3a tan30o C A a V =B.h = SABC.AA' o 60 VAA 'C' AA ' = AC'2 A 'C'2 = 2a 2 B 2 VABC l na tam gic u nn SABC = a 3 2 3 Vy V = a 6
V d 3: Cho lng tr ng ABCD A'B'C'D' c y ABCD l hnh vung cnh a v ng cho BD' ca lng tr hp vi y ABCD mt gc 300. Tnh th tch v tng din tch ca cc mt bn ca lng tr . Gii: A' D' Ta c ABCD A'B'C'D' l lng tr ng nn ta c: DD' (ABCD) DD' BD v BD l hnh chiu ca BD' trn ABCD . o C B Vy gc [BD';(ABCD)] = DBD' = 300 30 D a 6 A VBDD' DD' = BD.tan 300 = 3 a 3 a 6 4a 2 6 Vy V = SABCD.DD' = S = 4SADD'A' = 3 3 V d 4: Cho hnh hp ng ABCD A'B'C'D' c y ABCD l hnh thoi cnh o o a v BAD = 60 bit AB' hp vi y (ABCD) mt gc 30 . Tnh th tch ca hnh hp.C' B'B' C'
Gii VABD u cnh a SABD =C
A'
D' B
a2 3 4
o 30 A 60 o
a
D
a2 3 2 VABB' vung tiB BB' = ABtan30o = a 3 3a3 V = B.h = SABCD .BB' = Vy 2 SABCD = 2SABD =
3) Dng 3:
Lng tr ng c gc gia 2 mt phng
Chuyn :Luyn tp Hnh Hc Khng Gian
GV: L Minh Tin
A'
A
V d 1: Cho lng tr ng tam gic ABC A'B'C' c y ABC l tam gic vung cn ti B vi BA = BC = a ,bit (A'BC) hp vi y (ABC) mt gc 600 .Tnh th tch lng tr. C' Li gii: Ta c A 'A (ABC)& BC AB BC A 'B B' ABA Vy gc[(A 'BC),(ABC)] = ' = 60o VABA ' AA ' = AB.tan 600 = a 3 C 1 a2 o 60 SABC = BA.BC = 2 2 B a3 3 Vy V = SABC.AA' = 2
V d 2: y ca lng tr ng tam gic ABC.ABC l tam gic u . Mt (ABC) to vi y mt gc 300 v din tch tam gic ABC bng 8. Tnh th tch khi lng tr. Gii: VABC u AI BC m AA' (ABC) C' A' nn A'I BC (l 3 ). Vy gc[(A'BC);)ABC)] = A 'IA = 30oB'
A
30o B xI
2x 3 = x 3 .Ta c 2 2 AI 2 x 3 A' AI : A' I = AI : cos 30 0 = = = 2x 3 3
Gi s BI = x AI =
C
3 =x 3 Vy VABC.ABC = CI.AI.AA = x3 3 M SABC = BI.AI = x.2x = 8 x = 2 Do VABC.ABC = 8 3AA = AI.tan 300 = x 3.
V d 3: Cho lng tr t gic u ABCD A'B'C'D' c cnh y a v mt phng (BDC') hp vi y (ABCD) mt gc 60o.Tnh th tch khi hp ch nht.
Chuyn :Luyn tp Hnh Hc Khng GianC' B' D' A'
GV: L Minh Tin
C
60 0
D O Ba
A
Gi O l tm ca ABCD . Ta c ABCD l hnh vung nn OC BD CC' (ABCD) nn OC' BD (l 3 ). Vy gc[(BDC');(ABCD)] = COC' = 60o Ta c V = B.h = SABCD.CC' ABCD l hnh vung nn SABCD = a2 VOCC' vung nn CC' = OC.tan60o = a 6 2 3 a 6 Vy V = 2
V d 4: Cho hnh hp ch nht ABCD A'B'C'D' c AA' = 2a ; mt phng (A'BC) hp vi y (ABCD) mt gc 60o v A'C hp vi y (ABCD) mt gc 30o .Tnh th tch khi hp ch nht. Ta c AA' (ABCD) AC l hnh chiu ca A'C trn (ABCD) . Vy gc[A'C,(ABCD)] = A 'CA = 30o BC AB BC A'B (l 3 ) . Vy gc[(A'BC),(ABCD)] = A 'BA = 60o VA 'AC AC = AA'.cot30o = 2a 3 2a 3 VA 'AB AB = AA'.cot60o = 3 4a 6 VABC BC = AC2 AB2 = 3 3 16a 2 Vy V = AB.BC.AA' = 3
A' B' 2a A o 30 C C'
D'
D
o 60 B
4) Dng 4:
Khi lng tr xin
V d 1: Cho lng tr xin tam gic ABC A'B'C' c y ABC l tam gic u cnh a , bit cnh bn l a 3 v hp vi y ABC mt gc 60o . Tnh th tch lng tr.A' B' C'
A a B
C
6 0
o
H
Li gii: Ta c C'H (ABC) CH l hnh chiu ca CC' trn (ABC) C'CH = 60o Vy gc[CC',(ABC)] = 3a VCHC' C'H = CC'.sin 600 = 2 2 a 3 3a 3 3 SABC = = .Vy V = SABC.C'H = 4 8
Chuyn :Luyn tp Hnh Hc Khng Gian
GV: L Minh Tin
V d 2: Cho lng tr xin tam gic ABC A'B'C' c y ABC l tam gic u cnh a . Hnh chiu ca A' xung (ABC) l tm O ng trn ngoi tip tam gic ABC bit AA' hp vi y ABC mt gc 60 . 1) Chng minh rng BB'C'C l hnh ch nht. 2) Tnh th tch lng tr . Li gii: 1) Ta c A 'O (ABC) OA l hnh chiu ca AA' trn (ABC) Vy gc[AA ',(ABC)] = OAA ' = 60o B' Ta c BB'CC' l hnh bnh hnh ( v mt bn ca lng tr) AO BC ti trung im H ca BC nn o BC A 'H (l 3 ) 60 A BC (AA 'H) BC AA ' m AA'//BB' C O nn BC BB' .Vy BB'CC' l hnh ch nht. a H 2 2a 3 a 3 2) VABC u nn AO = AH = = 3 3 2 3 B VAOA ' A 'O = AO t an60o = a a3 3 Vy V = SABC.A'O = 4 V d 3: Cho hnh hp ABCD.ABCD c y l hnh ch nht vi AB = 3 AD = 7 .Hai mt bn (ABBA) v (ADDA) ln lt to vi y nhng gc 450 v 600. . Tnh th tch khi hp nu bit cnh bn bng 1.A' C'D' C'
A' B'
Li gii: K AH ( ABCD ) ,HM AB , HN AD A' M AB, A' N AD (l 3 ) A 'MH = 45o,A 'NH = 60o
D N A C H M
t AH = x . Khi 2x AN = x : sin 600 = 33 4x 2 AN = AA' A' N = = HM 3 M HM = x.cot 450 = x2 2
B
3 4x 2 3 Ngha l x = x= 3 7 Vy VABCD.ABCD = AB.AD.x 3 = 3. 7. =3 7
Chuyn :Luyn tp Hnh Hc Khng Gian
GV: L Minh Tin
LOI 2: 1) Dng 1:
TH TCH KHI CHP
Khi chp c cnh bn vung gc vi y
V d 1: Cho hnh chp SABC c SB = SC = BC = CA = a . Hai mt (ABC) v (ASC) cng vung gc vi (SBC). Tnh th tch hnh chp . Li gii: Ta c (ABC) (SBC) AC (SBC) (ASC) (SBC) 1 1 a2 3 a3 3 Do V = SSBC .AC = a= 3 3 4 12
A
a_ B \ S
C /
/
V d 2: Cho hnh chp SABC c y ABC l tam gic vung cn ti B vi AC = a bit SA vung gc vi y ABC v SB hp vi y mt gc 60o. 1) Chng minh cc mt bn l tam gic vung . 2)Tnh th tch hnh chp . Li gii: 1) SA (ABC) SA AB &SA AC m BC AB BC SB ( l 3 ). Vy cc mt bn chp l tam gic vung. 2) Ta c SA (ABC) AB l hnh chiu ca SB trn (ABC). Vy gc[SB,(ABC)] = = 60o . SAB a VABC vung cn nn BA = BC = 2 2 1 a SABC = BA.BC = 2 4 a 6 VSAB SA = AB.tan60o = 2 2 1 1a a 6 a3 6 Vy V = SABC .SA = = 3 34 2 24
S
A 60o
a
C
B
V d 3: Cho hnh chp SABC c y ABC l tam gic u cnh a bit SA vung gc vi y ABC v (SBC) hp vi y (ABC) mt gc 60o. Tnh th tch hnh chp .
Chuyn :Luyn tp Hnh Hc Khng GianS
GV: L Minh Tin
A 60 o a B M
C
Li gii: Ml trung im ca BC,v tam gic ABC u nn AM BC SA BC (l3 ) . Vy gc[(SBC);(ABC)] = = 60o . SMA 1 1 Ta c V = B.h = SABC .SA 3 3 3a VSAM SA = AM tan60o = 2 1 1 a3 3 Vy V = B.h = SABC .SA = 3 3 8
V d 4: Cho hnh chp SABCD c y ABCD l hnh vung c cnh a v SA vung gc y ABCD v mt bn (SCD) hp vi y mt gc 60o. 1) Tnh th tch hnh chp SABCD. 2) Tnh khong cch t A n mt phng (SCD). Li gii: 1)Ta c SA (ABC) v CD AD CD SD ( l 3 ).(1) Vy gc[(SCD),(ABCD)] = SDA = 60o . VSAD vung nn SA = AD.tan60o = a 3 1 1 a3 3 Vy V = SABCD .SA = a2a 3 = 3 3 3 2) Ta dng AH SD ,v CD (SAD) (do (1) ) nn CD AH AH (SCD) Vy AH l khong cch t A n (SCD). 1 1 1 1 1 4 VSAD = + = 2+ 2= 2 2 2 2 AH SA AD 3a a 3a a 3 Vy AH = 2
S H
A
60o
D
B
a
C
2) Dng 2 :
Khi chp c mt mt bn vung gc vi y
V d 1: Cho hnh chp S.ABCD c y ABCD l hnh vung c cnh a Mt bn SAB l tam gic u nm trong mt phng vung gc vi yABCD, 1) Chng minh rng chn ng cao khi chp trng vi trung im cnh AB.
2) Tnh th tch khi chp SABCD.S
A B H a C
D
Li gii: 1) Gi H l trung im ca AB. VSAB u SH AB m (SAB) (ABCD) SH (ABCD) Vy H l chn ng cao ca khi chp. 2) Ta c tam gic SAB u nn SA = a 3 2 3 1 a 3 suy ra V = SABCD .SH = 3 6
Chuyn :Luyn tp Hnh Hc Khng Gian
GV: L Minh Tin
C
V d 2: Cho t din ABCD c ABC l tam gic u ,BCD l tam gic vung cn ti D , (ABC) (BCD) v AD hp vi (BCD) mt gc 60o . Tnh th tch t din ABCD. A Li gii: Gi H l trung im ca BC. Ta c tam gic ABC u nn AH (BCD) , m a (ABC) (BCD) AH (BCD) . Ta c AH HD AH = AD.tan60o = a 3 B a 3 o 60 D H & HD = AD.cot60o = 3 VBCD BC = 2HD = 2a 3 suy ra 3 1 11 a3 3 V = SBCD.AH = . BC.HD.AH = 3 32 9
b)
A
V d 3: Cho hnh chp S.ABC c y ABC l tam gic vung cn ti B, c BC = a. Mt bn SAC vung gc vi y, cc mt bn cn li u to vi mt y mt gc 450. a) Chng minh rng chn ng cao khi chp trng vi trung im cnh AC. Tnh th tch khi chp SABC. S Li gii: a) K SH BC v mp(SAC) mp(ABC) nn SH mp(ABC). Gi I, J l hnh chiu ca H trn AB v BC H SI AB, SJ BC, theo gi thit = SJH = 45o SIH 45 C Ta c: SHI = SHJ HI = HJ nn BH l I J ng phn gic ca VABC suy ra H l trung im ca AC. B a 1 a3 b) HI = HJ = SH = VSABC= S ABC .SH = 2 3 12
Chuyn :Luyn tp Hnh Hc Khng Gian
GV: L Minh Tin
3) Dng 3 :
Khi chp u
V d 1: Cho chp tam gic u SABC cnh y bng a v cnh bn bng 2a. Chng minh rng chn ng cao k t S ca hnh chp l tm ca tam gic u ABC.Tnh th tch chp u SABC . Li gii: Dng SO (ABC) Ta c SA = SB = SC suy ra OA = OB = OC Vy O l tm ca tam gic u ABC. Ta c tam gic ABC u nn 2 2a 3 a 3 AO = AH = = 3 3 2 3 11 2 a VSAO SO2 = SA 2 OA 2 = 3 a 11 1 a3 11 SO = .Vy V = SABC .SO = 3 12 3
S 2a
A a O B
C
H
V d 2:Cho khi chp t gic SABCD c tt c cc cnh c di bng a . 1) Chng minh rng SABCD l chp t gic u. 2) Tnh th tch khi chp SABCD. Li gii: Dng SO (ABCD) Ta c SA = SB = SC = SD nn OA = OB = OC = OD ABCD l hnh thoi c ng trn gnoi tip nn ABCD l hnh vung . Ta c SA2 + SB2 = AB2 +BC2 = AC2 nn VASC vung ti S OS =B
S
D O A a
C
a 2 2 3 V = 1 S ABCD .SO = 1 a 2 a 2 = a 2 3 3 2 6
Vy V =
a3 2 6
V d 3: Cho khi t din u ABCD cnh bng a, M l trung im DC. a) Tnh th tch khi t din u ABCD. b)Tnh khong cch t M n mp(ABC).Suy ra th tch hnh chp MABC.
Chuyn :Luyn tp Hnh Hc Khng Gian
GV: L Minh Tin
D M
A O I B H a
C
Li gii: a) Gi O l tm ca ABC DO ( ABC ) 1 V = S ABC .DO 3 a2 3 2 a 3 , OC = CI = S ABC = 3 3 4 a 6 DOC vung c : DO = DC 2 OC 2 = 3 1 a2 3 a 6 a3 2 V = . = 3 4 3 12 b) K MH// DO, khong cch t M n mp(ABC) l MH 1 a 6 MH = DO = 2 61 1 a2 3 a 6 a 3 2 VMABC = S ABC .MH = . = 3 3 4 6 24
Vy V =
a3 2 24
Bi tp tng t: Bi 1: Cho hnh chp u SABC c cnh bn bng a hp vi y ABC mt gc 3a3 o 60 . Tnh th tch hnh chp. s: V = 16 Bi 2: Cho hnh chp tam gic u SABC c cnh bn a, gc y ca mt bn l 45o. a 1) Tnh di chiu cao SH ca chp SABC . s: SH = 3 a3 2) Tnh th tch hnh chp SABC. s: V = 6 Bi 3: Cho hnh chp tam gic u SABC c cnh y a v mt bn hp vi y a3 3 o mt gc 60 . Tnh th tch hnh chp SABC. s: V = 24 Bi 4 : Cho chp tam gic u c ng cao h hp vi mt mt bn mt gc 30o . h3 3 Tnh th tch hnh chp. s: V = 3 Bi 5 : Cho hnh chp tam gic u c ng cao h v mt bn c gc nh h3 3 bng 60o. Tnh th tch hnh chp. s: V = 8 = 60o . Bi 6 : Cho hnh chp t gic u SABCD c cnh y a v ASB a2 3 1) Tnh tng din tch cc mt bn ca hnh chp u. s: S = 3
Chuyn :Luyn tp Hnh Hc Khng Gian
GV: L Minh Tin
a3 2 6 Bi 7 : Cho hnh chp t gic u SABCD c chiu cao h ,gc nh ca mt bn 2h3 o bng 60 . Tnh th tch hnh chp. s: V = 3 o Bi 8: Cho hnh chp t gic u c mt bn hp vi y mt gc 45 v khong cch t chn ng cao ca chp n mt bn bng a. 8a3 3 Tnh th tch hnh chp . s: V = 3 Bi 9: Cho hnh chp t gic u c cnh bn bng a hp vi y mt gc 60o. a3 3 Tnh th tch hnh chp. s: V = 12 Bi 10: Cho hnh chp SABCD c tt c cc cnh bng nhau. Chng minh rng SABCD l chp t gic u.Tnh cnh ca hnh chp ny khi th tch ca 9a3 2 n bng V = . s: AB = 3a 2 4) Dng 4 : Khi chp & phng php t s th tch 2) Tnh th tch hnh chp. s: V = V d 1: Cho hnh chp S.ABC c tam gic ABC vung cn B, AC = a 2 , SA vung gc vi y ABC , SA = a 1) Tnh th tch ca khi chp S.ABC. 2) Gi G l trng tm tam gic ABC, mt phng ( ) qua AG v song song vi BC ct SC, SB ln lt ti M, N. Tnh th tch ca khi chp S.AMN Li gii:S
N A M I B G C
1 a)Ta c: VS . ABC = S ABC .SA v SA = a 3 + ABC cn c : AC = a 2 AB = a 1 2 1 1 2 a3 S ABC = a Vy: VSABC = . a .a = 2 3 2 6 b) Gi I l trung im BC. SG 2 = G l trng tm,ta c : SI 3 // BC MN// BC SM = SN = SG = 2 SB SC SI 3 VSAMN SM SN 4 = . = VSABC SB SC 9
Vy: VSAMN
4 2a3 = VSABC = 9 27
Chuyn :Luyn tp Hnh Hc Khng Gian
GV: L Minh Tin
D
a
V d 2: Cho tam gic ABC vung cn A v AB = a . Trn ng thng qua C v vung gc vi mt phng (ABC) ly im D sao cho CD = a . Mt phng qua C vung gc vi BD, ct BD ti F v ct AD ti E. a) Tnh th tch khi t din ABCD. b) Chng minh CE ( ABD) c)Tnh th tch khi t din CDEF. Li gii: 1 a3 a)Tnh VABCD : VABCD = SABC .CD = F 3 6 AB AC , AB CD AB ( ACD ) b)Tac:E
AB ECB
Ta c:
DB EC EC ( ABD )VDCEF DE DF = . (*) VDABC DA DB
C a A
c) Tnh VDCEF :Ta c:
M DE.DA = DC 2 , chia cho DA2DE DC 2 a2 1 = = 2= 2 DA DA 2a 2 2 DF DC a2 1 Tng t: = = = 2 2 2 DB DB DC + CB 3
T(*)
3 VDCEF 1 = .Vy VDCEF = 1 V ABCD = a VDABC 6 6 36
V d 3: Cho khi chp t gic u SABCD. Mt mt phng ( ) qua A, B v trung im M ca SC . Tnh t s th tch ca hai phn khi chp b phn chia bi mt phng .S
N M D O
Li gii: K MN // CD (N SD) th hnh thang ABMN l thit din ca khi chp khi ct bi mt phng (ABM).ASAND = = VSANB = VSADB = VSABCD +V SD 2 2 4 SADB
V
SN
1
1
1
C
B
VSBMN SM SN 1 1 1 1 1 = . = . = VSBMN = VSBCD = VSABCD VSBCD SC SD 2 2 4 4 8 3 M VSABMN = VSANB + VSBMN = VSABCD . 8 5 Suy ra VABMN.ABCD = VSABCD 8
Chuyn :Luyn tp Hnh Hc Khng Gian V
GV: L Minh TinSABMN = Do : V 5 ABMN . ABCD
3
V d 4: Cho hnh chp t gic u S.ABCD, y l hnh vung cnh a, cnh bn to vi y gc 60 . Gi M l trung im SC. Mt phng i qua AM v song song vi BD, ct SB ti E v ct SD ti F. a) Hy xc nh mp(AEMF) S b) Tnh th tch khi chp S.ABCD c) Tnh th tch khi chp S.AEMF d) M Li gii: E a) Gi I = SO AM . Ta c (AEMF) //BD I EF // BD B CF O A D
2 b) VS . ABCD = S ABCD .SO vi S ABCD = a
1 3
+ VSOA c : SO = AO.tan 60 = Vy : VS . ABCD
a 6 2
a3 6 = 6
c) Phn chia chp t gic ta c VS . AEMF = VSAMF + VSAME =2VSAMF
VS . ABCD = 2VSACD = 2 VSABCXt khi chp S.AMF v S.ACD SM 1 = Ta c : SC 2 SAC c trng tm I, EF // BD nn:
VSAMF SM SF 1 SI SF 2 = . = = = VSACD SC SD 3 SO SD 3
VSAMF
1 1 a3 6 = VSACD = VSACD = 3 6 36
a 3 6 a3 6 VS . AEMF = 2 = 36 18
V d 5: Cho hnh chp S.ABCD c y ABCD l hnh vung cnh a, SA vung gc y, SA = a 2 . Gi B, D l hnh chiu ca A ln lt ln SB, SD. Mt phng (ABD) ct SC ti C. a) Tnh th tch khi chp S.ABCD. b) Chng minh SC ( AB ' D ') c) Tnh th tch khi chp S.ABCD
Chuyn :Luyn tp Hnh Hc Khng GianC' D' A O D C I B B'
GV: L Minh Tin
Li gii: a) Ta c: VS . ABCD = S ABCD .SA =
b) Ta c BC (SAB ) BC AB ' & SB AB ' Suy ra: AB ' (SBC ) nn AB' SC .Tng t AD' SC. Vy SC (AB'D') c) Tnh VS . A B 'C ' D ' +Tnh VS . AB 'C ' : Ta c:
1 3
a3 2 3
VSAB' C' SB ' SC ' = . (*) VSABC SB SC SC ' 1 = SAC vung cn nn SC 2 2 2 SB ' SA 2a 2 a2 2 Ta c: = = = = SB SB 2 SA2 + AB2 3 a2 3 VSAB C ' 1 ' = T (*) VSABC 31 a 3 2 a3 2 VSAB 'C ' = . = 3 3 9
+ VS . AB 'C ' D ' = 2VS . AB 'C '
2a 3 2 = 9
5) Dng 5 : n tp khi chp v lng tr V d 1: Cho hnh chp S.ABCD c ABCD l hnh vung cnh 2a, SA vung gc y. Gc gia SC v y bng 60 v M l trung im ca SB. 1) Tnh th tch ca khi chp S.ABCD. 2) Tnh th tch ca khi chp MBCD. Li gii:S
a)Ta c V = S ABCD .SA2 2 + S ABCD = (2a ) = 4a + SAC c : SA = AC tan C = 2 a 6
1 3
A 60o D 2a
H
B
C
1 2 a3 8 6 V = 4 a .2 a 6 = 3 3 b) K MH / / SA MH (DBC ) 1 1 Ta c: MH = SA , S BCD = S ABCD 2 2
.
Chuyn :Luyn tp Hnh Hc Khng Gian
GV: L Minh Tin
VMBCD
1 23 6 a = V = 4 3
V d 2: Cho hnh chp tam gic S.ABC c AB = 5a, BC = 6a, CA = 7a. Cc mt bn SAB, SBC, SCA to vi y mt gc 60o .Tnh th tch khi chp. Li gii: H SH ( ABC ) , k HE AB, HF BC, HJ AC suy ra SE AB, SF BC, SJ AC . Ta c = SFH = SJH = 60O SEH SAH = SFH = SJH nn HE =HF = HJ = r J ( r l bn knh ng trn ngai tip ABC ) C 60 Ta c SABC = p ( p a )( p b)( p c) H a+b+c E F = 9a Nn SABC = 9.4.3.2 a 2 vi p = 2 B S 2 6a Mt khc SABC = p.r r = = p 3 Tam gic vung SHE: 2 6a SH = r.tan 600 = . 3=2 2 a 3 1 2 3 Vy VSABC = 6 6 a .2 2 a = 8 3 a . 3 V d 3: Cho hnh hp ch nht ABCD.ABCD c AB = a 3 , AD = a, AA = a, O l giao im ca AC v BD. a) Tnh th tch khi hp ch nht, khi chp OABCD b) Tnh th tch khi OBBC.S
A
c)Tnh di ng cao nh C ca t din OBBC.A O M C B
Li gii: a) Gi th tch khi hp ch nht l V. Ta c : V = AB. AD.AA ' = a 3. a2 =3 a
3
D
A' ' C' ' D' '
B' '
ABD c : DB = AB2 +AD2 =2 a * Khi OABCD c y v ng cao 1 a3 3 ging khi hp nn: VOA' B' C' D' = V = 3 3 ' ') b) M l trung im BC O M ( B B C1 1a 2 a 3 a 3 3 VO BB' C' = S BB' C' OM . = . . = 3 3 2 2 12 c) Gi CH l ng cao nh C ca t 3VOBB 'C ' din OBBC. Ta c : C ' H = SOBB '
Chuyn :Luyn tp Hnh Hc Khng Gian
GV: L Minh Tin
ABD c : DB = AB2 + AD2 =2 a SOBB ' = 1 2 a C ' H = 2a 3 2
V d 4: Cho hnh lp phng ABCD.ABCDc cnh bng a. Tnh th tch khi t din ACBD. Li gii: Hnh lp phng c chia thnh: khi ACBD v bn khi CBDC, BBAC, DACD, ABAD. +Cc khi CBDC, BBAC, DACD, ABAD c din tch y v chiu cao bng nhau nn c cng th tch.2 Khi CBDC c V1 = . a .a =A' B'
A
B
D
C
1 1 3 2
1 3 a 63
+Khi lp phng c th tch: V2 = a VACB ' D ' = a 3 4.
C' D' a
1 3 1 3 a = a 6 3
V d 5: Cho hnh lng tr ng tam gic c cc cnh bng a. a) Tnh th tch khi t din AB BC.
b) E l trung im cnh AC, mp(ABE) ct BC ti F. Tnh th tch khi CABFE.A I C E F B
A' J C'
B'
Li gii: a) Khi AB BC:Gi I l trung im AB, 2 3 1 VA ' B ' BC = S A ' B ' B .CI = 1 a . a 3 = a 3 3 3 2 2 12 b)Khi CABFE: phn ra hai khi CEFA v CFAB. +Khi ACEFc y l CEF, ng cao AA nn VA 'CEF = S CEF .A 'A SCEF 1 a2 3 a3 3 VA 'CEF = = S ABC = 48 4 16
1 3
+Gi J l trung im BC. Ta c khi ABCF cy l CFB, ng cao JA nn
VA ' B 'CF
1 1 a2 = S CFB' .A 'J SCFB' = SCBB ' = 3 2 4
Chuyn :Luyn tp Hnh Hc Khng Gian
GV: L Minh Tin
1 a 2 a 3 a3 3 VA ' B ' CF = = 3 4 2 24
+ Vy : VCA'B'FE
a3 3 = 16
