C

CHUYN PHNG PHP TA TRONG KHNG GIANI. PHNG PHP GII TON

gii c cc bi ton hnh khng gian bng phng php ta ta cn phi chn h trc ta thch hp. Lp ta cc nh, im lin quan da vo h trc ta chn v di cnh ca hnh.

PHNG PHP

Bc 1: Chn h trc ta Oxyz thch hp. (Quyt nh s thnh cng ca bi ton)

Bc 2: Xc nh ta cc im c lin quan.Bc 3: S dng cc kin thc v ta gii quyt bi ton.Cc dng ton thng gp: nh tnh: Chng minh cc quan h vung gc, song song,

nh lng: di on thng,, gc, khong cch, tnh din tch, th tch, din tch thit din, Bi ton cc tr, qu tch.

Ta thng gp cc dng sau

1. Hnh chp tam gic

a. Dng tam din vungV d : Cho t din OABC c y OBC l tam gic vung ti O, OB=a, OC=, (a>0) v ng cao OA=. Gi M l trung im ca cnh BC. Tnh khong cch gia hai ng thng AB v OM.

Cch 1:

Chn h trc ta nh hnh v. Khi O(0;0;0),

, gi N l trung im ca AC (.

MN l ng trung bnh ca tam gic ABC ( AB // MN

( AB //(OMN) ( d(AB;OM) = d(AB;(OMN)) = d(B;(OMN)).

, vi .Phng trnh mt phng (OMN) qua O vi vect php tuyn

Ta c: . Vy,

Cch 2:

Gi N l im i xng ca C qua O.

Ta c: OM // BN (tnh cht ng trung bnh).

( OM // (ABN)

( d(OM;AB) = d(OM;(ABN)) = d(O;(ABN)).

Dng

Ta c:

T cc tam gic vung OAK; ONB c:

. Vy,

b. Dng khc

V d 1: T din S.ABC c cnh SA vung gc vi y v vung ti C. di ca cc cnh l SA =4, AC = 3, BC = 1. Gi M l trung im ca cnh AB, H l im i xng ca C qua M.

Tnh cosin gc hp bi hai mt phng (SHB) v (SBC).Hng dn giiChn h trc ta nh hnh v, ta c:

A(0;0;0), B(1;3;0), C(0;3;0), S(0;0;4) v H(1;0;0).

mp(P) qua H vung gc vi SB ti I ct ng thng SC ti K, d thy

(1).

, suy ra:

ptts SB: , SC: v (P): x + 3y 4z 1 = 0.

EMBED Equation.DSMT4 = Ch : Nu C v H i xng qua AB th C thuc (P), khi ta khng cn phi tm K.V d 2: Cho hnh chp SABC c y l tam gic ABC vung cn ti A, AB = AC = a (a > 0), hnh chiu ca S trn y trng vi trng tm G ca (ABC. t SG = x (x > 0). Xc nh gi tr ca x gc phng nh din (B, SA, C) bng 60o.

Cch 1:

Gi M l trung im ca BC .Gi E, F ln lt l hnh chiu ca G ln AB, AC. T gic AEGF l hnh vung

Dng h trc ta Axyz, vi Ax, Ay, Az i mt vung gc, A(0;0;0), B(a;0;0), C(0; a; 0), .

, vi

vi .

Mt phng (SAB) c cp vect ch phng nn c vect php tuyn .Mt phng (SAC) c cp vect ch phng nn c vect php tuyn .Gc phng nh din (B; SA; C) bng 60o.

EMBED Equation.DSMT4 Vy,

Cch 2:

Gi M l trung im ca BC ((ABC vung cn)

Ta c: . Suy ra:

Dng v

EMBED Equation.DSMT4 l gc phng nh din (B; SA; C).

EMBED Equation.DSMT4 cn ti I.

.

EMBED Equation.DSMT4 .

Ta c:

EMBED Equation.DSMT4 .

Vy,

V d 3: (Trch thi i hc khi A 2002). Cho hnh chp tam gic u S.ABC c di cnh y l a. Gi M, N l trung im SB, SC. Tnh theo a din tch (AMN, bit (AMN) vung gc vi (SBC).

Hng dn giiGi O l hnh chiu ca S trn (ABC), ta suy ra O l trng tm . Gi I l trung im ca BC, ta c:

EMBED Equation.DSMT4 Trong mt phng (ABC), ta v tia Oy vung gc vi OA. t SO = h, chn h trc ta nh hnh v ta c:

O(0; 0; 0), S(0; 0; h),

EMBED Equation.DSMT4 , ,

, v .

,

.2. Hnh chp t gic

a) Hnh chp S.ABCD c SA vung gc vi y v y l hnh vung (hoc hnh ch nht). Ta chn h trc ta nh dng tam din vung.

b) Hnh chp S.ABCD c y l hnh vung (hoc hnh thoi) tm O ng cao SO vung gc vi y. Ta chn h trc ta tia OA, OB, OS ln lt l Ox, Oy, Oz. Gi s SO = h, OA = a, OB = b ta c

O(0; 0; 0), A(a; 0; 0), B(0; b; 0), C(a; 0; 0), D(0;b; 0), S(0; 0; h).

c) Hnh chp S.ABCD c y hnh ch nht ABCD v AB = b. u cnh a v vung gc vi y. Gi H l trung im AD, trong (ABCD) ta v tia Hy vung gc vi AD. Chn h trc ta Hxyz ta c: H(0; 0; 0),

EMBED Equation.DSMT4 3. Hnh lng tr ng

Ty theo hnh dng ca y ta chn h trc nh cc dng trn.V d: 1. Cho hnh lp phng ABCD A'B'C'D' cnh a. Chng minh rng AC' vung gc vi mt phng (A'BD).Li gii:Chn h trc ta Oxyz sao cho O ( A; B ( Ox; D ( Oy v A' ( Oz . ( A(0;0;0), B(a;0;0), D(0;a;0), A'(0;0;a), C'(1;1;1)( Phng trnh on chn ca mt phng(A'BD): x + y + z = a hay x + y + z a = 0

(Php tuyn ca mt phng (A'BC): v .

Vy AC' vung gc vi (A'BC)

2. Cho lng tr ABC.A'B'C' cc cc mt bn u l hnh vung cnh a. Gi D, F ln lt l trung im ca cc cnh BC, C'B'. Tnh khong cch gia hai ng thng A'B v B'C'.

Gii

Cch 1:

V cc cc mt bn ca lng tr u l hnh vung nn

( cc tam gic ABC, ABC l cc tam gic u.

Chn h trc Axyz, vi Ax, Ay, Az i mt vung gc, A(0;0;0),

Ta c:

, vi

Phng trnh mt phng (ABC) qua A vi vect php tuyn :

Vy,

Cch 2:

V cc cc mt bn ca lng tr u l hnh vung nn

( cc tam gic ABC, ABC l cc tam gic u.

Ta c: .

.Ta c:

Dng

V

(AFD vung c:

Vy,

3. T din ABCD c AB, AC, AD i mt vung gc vi nhau, AB = 3, AC=AD=4. Tnh khong cch t A ti mt phng (BCD)

Li gii+ Chn h trc ta Oxyz sao cho A ( O.D (Ox; C ( Oy v B ( Oz( A(0;0;0); B(0;0;3); C(0;4;0); D(4;0;0)

( Phng trnh mt phng (BCD) l:

( 3x + 3y + 4z - 12 = 0.Suy ra khongr cch t A ti mt phng (BCD).II. Lyuyn tpBi 1: Cho hnh chp SABC c di cc cnh bng 1, O l trng tm ca tam gic (ABC. I l trung im ca SO.1. Mt phng (BIC) ct SA ti M. Tm t l th tch ca t din SBCM v t din SABC.2. H l chn ng vung gc h t I xung cnh SB. Chng minh rng IH qua trng tm G ca (SAC. Li gii1. Chn h trc ta Oxyz sao cho O l gc ta . A(Ox, S(Oz, BC//Oy(;;;;

Ta c: ;;

( Phng trnh mt phng (IBC) l:

Hay: m ta li c: .Phng trnh ng thng SA: .+ Ta im M l nghim ca h: .Thay (1), (2), (3) v (4):

;

( M nm trn on SA v

EMBED Equation.DSMT4 .2. Do G l trng tm ca tam gic (ASC ( SG i qua trung im N ca AC( GI ( (SNB) ( GI v SB ng phng (1)

Ta li c

EMBED Equation.DSMT4

EMBED Equation.DSMT4 T (1) v (2) .Bi 2: Cho hnh chp O.ABC c OA = a, OB = b, OC = c i mt vung gc. im M c nh thuc tam gic ABC c khong cch ln lt n cc mt phng (OBC), (OCA), (OAB) l 1, 2, 3. Tnh a, b, c th tch O.ABC nh nht.

Hng dn giiChn h trc ta nh hnh v, ta c:

O(0; 0; 0), A(a; 0; 0), B(0; b; 0), C(0; 0; c).

d(M, (OAB)) = 3 ( zM = 3.

Tng t ( M(1; 2; 3).

( (ABC):

(1). (2).

.

(2).Bi 3: Cho t din ABCD c AD vung gc vi mt phng (ABC) v tam gic ABC vung ti A, AD=a, AC=b, B=c. Tnh din tch ca tam gic BCD theo a, b, c v chng minh rng .

Gii

Chn h trc ta nh hnh v, ta c: A(0;0;0), B(c;0;0), C(0;b;0), D(0;0;a).

Theo bt ng thc Cachy ta c:

Bi 4: Cho hnh lng tr ABC. A1B1C1 c y l tam gic cnh a. AA1 = 2a v vung gc vi mt phng (ABC). Gi D l trung im ca BB1; M di ng trn cnh AA1. Tm gi tr ln nht, gi tr nh nht ca din tch tam gic MC1D.Li gii+ Chn h trc ta Oxyz sao cho A(O; B(Oy; A1(Oz. Khi : A(0;0;0), B(0;a;0); A1 (0;0;2a)

v D(0;a;a)

Do M di ng trn AA1, ta M(0;0;t) vi t ( [0;2a]Ta c :

Ta c:

EMBED Equation.DSMT4

EMBED Equation.DSMT4

Gi tr ln nht caty thuc vo gi tr ca tham s t.Xt f(t) = 4t2 ( 12at + 15a2f(t) = 4t2 ( 12at + 15a2 (t ([0;2a]) f '(t) = 8t (12a

Lp bng bin thin ta c gi tr ln nht cakhi t =0 hay M ( A.Ch

+ Hnh chp tam gic u c y l tam gic u v cc cnh bn bng nhau, nhng khng nht thit phi bng y. Chn ng cao l trng tm ca y.

+ T din u l hnh chp tam gic u c cnh bn bng y.

+ Hnh hp c y l hnh bnh hnh nhng khng nht thit phi l hnh ch nht.

III. CC DNG BI TP

1. CC BI TON V HNH CHP TAM GIC

Bi 1 (Trch thi i hc khi D 2002). Cho t din ABCD c cnh AD vung gc (ABC), AC = AD = 4cm, AB = 3cm, BC = 5cm. Tnh khong cch t nh A n (BCD).

Bi 2. Cho (ABC vung ti A c ng cao AD v AB = 2, AC = 4. Trn ng thng vung gc vi (ABC) ti A ly im S sao cho SA = 6. Gi E, F l trung im ca SB, SC v H l hnh chiu ca A trn EF.

1. Chng minh H l trung im ca SD.

2. Tnh cosin ca gc gia hai mt phng (ABC) v (ACE).

3. Tnh th tch hnh chp A.BCFE.

Bi 3. Cho hnh chp O.ABC c cc cnh OA = OB = OC = 3cm v vung gc vi nhau tng i mt. Gi H l hnh chiu ca im O ln (ABC) v cc im A, B, C ln lt l hnh chiu ca H ln (OBC), (OCA), (OAB).

1. Tnh th tch t din HABC.

2. Gi S l im i xng ca H qua O. Chng t S.ABC l t din u.

Bi 4. Cho hnh chp O.ABC c OA, OB, OC i mt vung gc. Gi ln lt l gc nh din cnh AB, BC, CA. Gi H l hnh chiu ca nh O trn (ABC).

1. Chng minh H l trc tm ca (ABC.

2. Chng minh

3. Chng minh

4. Chng minh

Bi 5. Cho hnh chp O.ABC c OA = a, OB = b, OC = c vung gc vi nhau tng i mt. Gi M, N, P ln lt l trung im BC, CA, AB.

1. Tnh gc ( gia (OMN) v (OAB).

2. Tm iu kin a, b, c hnh chiu ca O trn (ABC) l trng tm .

3. Chng minh rng gc phng nh din [N, OM, P] vung khi v ch khi

Bi 6. Cho hnh chp S.ABC c (ABC vung cn ti A, SA vung gc vi y. Bit AB = 2, .

1. Tnh di SA.

2. Tnh khong cch t nh A n (SBC).

3. Tnh gc hp bi hai mt phng (SAB) v (SBC).Bi 7. Cho hnh chp O.ABC c OA = a, OB = b, OC = c vung gc vi nhau tng i mt.

1. Tnh bn knh r ca mt cu ni tip hnh chp.

2. Tnh bn knh R ca mt cu ngoi tip hnh chp.

Bi 8 (trch thi i hc khi D 2003). Cho hai mt phng (P) v (Q) vung gc vi nhau, giao tuyn l ng thng (d). Trn (d) ly hai im A v B vi AB = a. Trong (P) ly im C, trong (Q) ly im D sao cho AC, BD cng vung gc vi (d) v AC = BD = AB. Tnh bn knh mt cu ngoi tip t din ABCD v khong cch t nh A n (BCD) theo a.

Bi 9. Cho hnh chp S.ABC c y l tam gic vung ti B, AB = a, BC = 2a. Cnh SA vung gc vi y v SA = 2a. Gi M l trung im ca SC.

1. Tnh din tch theo a.

2. Tnh khong cch gia MB v AC theo a.

3. Tnh gc hp bi hai mt phng (SAC) v (SBC).Bi 10. Cho t din S.ABC c (ABC vung cn ti B, AB = SA = 6. Cnh SA vung gc vi y. V AH vung gc vi SB ti H, AK vung gc vi SC ti K.

1. Chng minh HK vung gc vi CS.

2. Gi I l giao im ca HK v BC. Chng minh B l trung im ca CI.

3. Tnh sin ca gc gia SB v (AHK).

4. Xc nh tm J v bn knh R ca mt cu ngoi tip S.ABC.

Bi 11. Cho hnh chp S.ABC c (ABC vung ti C, AC = 2, BC = 4. Cnh bn SA = 5 v vung gc vi y. Gi D l trung im cnh AB.

1. Tnh cosin gc gia hai ng thng AC v SD.

2. Tnh khong cch gia BC v SD.

3. Tnh cosin ca gc hp bi hai mt phng (SBD) v (SCD).Bi 12. Cho hnh chp S.ABC c y l tam gic u cnh a. SA vung gc vi y v .

1. Tnh khong cch t nh A n (SBC).

2. Tnh khong cch gia hai ng thng AB v SC.

Bi 13. Cho hnh chp tam gic u S.ABC c di cnh y l a, ng cao SH = h. Mt phng (() i qua AB v vung gc vi SC.

1. Tm iu kin ca h theo a (() ct cnh SC ti K.

2. Tnh din tch (ABK.

3. Tnh h theo a (() chia hnh chp thnh hai phn c th tch bng nhau. Chng t rng khi tm mt cu ni tip v ngoi tip trng nhau.

2. CC BI TON V HNH CHP T GIC

Bi 14. Cho hnh chp S.ABCD c y hnh vung cnh a, SA = a v vung gc vi y. Gi E l trung im CD.

1. Tnh din tch (SBE.

2. Tnh khong cch t nh C n (SBE).

3. (SBE) chia hnh chp thnh hai phn, tnh t s th tch hai phn .

Bi 15. Cho hnh chp S.ABCD c y hnh vung cnh a. Cnh bn SA vung gc vi y v .

1. Tnh khong cch t nh C n (SBD).

2. Tnh khong cch gia hai ng thng SD v AC.

3. Tnh gc hp bi hai mt phng (SBC) v (SCD).Bi 16. Cho hnh chp S.ABCD c y hnh vung cnh 3cm. Cnh bn SA vung gc vi y v cm. Mt phng (() i qua A v vung gc vi SC ct cc cnh SB, SC, SD ln lt ti H, M, K.

1. Chng minh AH vung gc vi SB, AK vung gc vi SD.

2. Chng minh BD song song vi (().

3. Chng minh HK i qua trng tm G ca .

4. Tnh th tch hnh khi ABCDKMH.

Bi 17. Cho hnh chp S.ABCD c y l hnh ch nht, AB = a, AD = b. Cnh bn SA vung gc vi y v SA = 2a. Gi M, N l trung im cnh SA, SD.

1. Tnh khong cch t A n (BCN).

2. Tnh khong cch gia SB v CN.

3. Tnh gc gia hai mt phng (SCD) v (SBC).

4. Tm iu kin ca a v b . Trong trng hp tnh th tch hnh chp S.BCNM.

Bi 18. Cho hnh chp S.ABCD c y l hnh vung cnh a. u v vung gc vi (ABCD). Gi H l trung im ca AD.

1. Tnh d(D,(SBC)), d(HC,SD).

2. Mt phng (() qua H v vung gc vi SC ti I. Chng t (() ct cc cnh SB, SD.

3. Tnh gc hp bi hai mt phng (SBC) v (SCD).Bi 19. Cho hnh chp S.ABCD c y l hnh thoi tm O. SO vung gc vi y v , AC = 4a, BD = 2a. Mt phng (() qua A vung gc vi SC ct cc cnh SB, SC, SD ti .

1. Chng minh u.

2. Tnh theo a bn knh mt cu ni tip S.ABCD.

Bi 20. Cho hnh chp S.ABCD c y l hnh ch nht vi AB = a, AD = 2a. ng cao SA = 2a. Trn cnh CD ly im M, t MD = m .

1. Tm v tr im M din tch ln nht, nh nht.

2. Cho , gi K l giao im ca BM v AD. Tnh gc hp bi hai mt phng (SAK) v (SBK).3. CC BI TON V HNH HP LNG TR NG

Bi 21. Cho hnh lp phng ABCD.ABCD cnh a. Gi I, K, M, N ln lt l trung im ca AD, BB, CD, BC.

1. Chng minh I, K, M, N ng phng.

2. Tnh khong cch gia IK v AD.

3. Tnh din tch t gic IKNM.

Bi 22 (Trch thi i hc khi A 2003). Cho hnh lp phng ABCD.ABCD. Tnh gc phng nh din [B,A'C,D].

Bi 23. Cho hnh lp phng ABCD.ABCD cnh a. Tm im M trn cnh AA sao cho (BDM) ct hnh lp phng theo thit din c din tch nh nht.

Bi 24. Cho hnh lp phng ABCD.ABCD cnh a.

1. Chng minh AC vung gc vi (ABD).

2. Tnh gc gia (DAC) v (ABBA).

3. Trn cnh AD, DB ly ln lt cc im M, N tha AM = DN = k

a. Chng minh MN song song (ADBC).

b. Tm k MN nh nht. Chng t khi MN l on vung gc chung ca AD v DB.

Bi 25. Cho hnh hp ch nht ABCD.ABCD c AB = 2, AD = 4, AA = 6. Cc im M, N tha Gi I, K l trung im ca AB, CD.

1. Tnh khong cch t im A n (ABD).

2. Chng minh I, K, M, N ng phng.

3. Tnh bn knh ng trn ngoi tip .

4. Tnh m din tch t gic MINK ln nht, nh nht.

Bi 26. Cho hnh lp phng ABCD.ABCD c di cnh l 2cm. Gi M l trung im AB, N l tm hnh vung ADDA.

1. Tnh bn knh R ca mt cu (S) qua C, D, M, N.

2. Tnh bn knh r ca ng trn (C) l giao ca (S) v mt cu (S) qua A, B, C, D.

3. Tnh din tch thit din to bi (CMN) v hnh lp phng.

Bi 27 (trch thi i hc khi B 2003) Cho hnh lng tr ng ABCD.ABCD c y hnh thoi cnh a, Gi M, N l trung im cnh AA, CC.

1. Chng minh B, M, D, N cng thuc mt mt phng.

2. Tnh AA theo a BMDN l hnh vung.

Bi 28. Cho hnh lng tr ng tam gic ABC.ABC c y l tam gic vung ti A. Cho AB = a, AC = b, AA = c. Mt phng (() qua B v vung gc vi BC.

1. Tm iu kin ca a, b, c (() ct cnh CC ti I (I khng trng vi C v C).

2. Cho (() ct CC ti I.

a. Xc nh v tnh din tch ca thit din.

b. Tnh gc phng nh din gia thit din v y.(((((((((((((((((((((((((((((A'

C

B'

B

C'

D'

D

D

C

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x

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EMBED Package

B

C

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x

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EMBED Equation.DSMT4

EMBED Equation.DSMT4

A

x

B

x

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M

O

N

C

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EMBED Equation.DSMT4

EMBED Equation.DSMT4

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H

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C

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PAGE 2Chuyn : PHNG PHP TA TRONG KHNG GIAN

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