(D 2011)

Phn I

L DO CHN TITrong chng trnh ton hc lp 11, 12, bi ton v khong cch trong khng gian gi mt vai tr quan trng, n xut hin hu ht cc thi tuyn sinh vo i hc, cao ng; thi hc sinh gii, cc thi tt nghip trong nhng nm gn y. Mc d vy y l phn kin thc i hi hc sinh phi c t duy su sc, c tr tng tng hnh khng gian phong ph nn i vi hc sinh i tr, y l mng kin thc kh v thng mt im trong cc k thi ni trn. i vi hc sinh gii, cc em c th lm tt phn ny. Tuy nhin cch gii cn ri rc, lm bi no bit bi y v thng tn kh nhiu thi gian.

Trong sch gio khoa, sch bi tp v cc ti liu tham kho, loi bi tp ny kh nhiu song ch dng vic cung cp bi tp v cch gii, cha c ti liu no phn loi mt cch r nt cc phng php tnh khong cch trong khng gian.

i vi cc gio vin, th do lng thi gian t i v vic tip cn cc phn mm v hnh khng gian cn hn ch nn vic bin son mt chuyn c tnh h thng v phn ny cn gp nhiu kh khn.

Trc cc l do trn, ti quyt nh vit ti sng kin kinh nghim mang tn: Phn loi cc bi ton v tnh khong cch trong khng gian nhm cung cp cho hc sinh mt ci nhn tng qut v c h thng v bi ton tnh khong cch trong khng gian, mt h thng bi tp c phn loi mt cch tng i tt, qua gip hc sinh khng phi e s phn ny v quan trng hn, ng trc mt bi ton hc sinh c th bt ngay ra c cch gii, c nh hng trc khi lm bi qua c cch gii ti u cho mi bi ton.

Mc d vy, v iu kin thi gian cn hn ch nn s phn loi c th cha c trit v ch mang tnh cht tng i, rt mong c cc bn b ng nghip gp kin chnh sa ti ny c hon thin hn.

Ti xin chn thnh cm n!

Phn IINI DUNG CA TI

A. C S L THUYT1. Khong cch t mt im n mt ng thng

Cho im O v ng thng (. Gi H l hnh chiu ca O trn (. Khi khong cch gia hai im O v H c gi l khong cch t im O n ng thng (. K hiu

* Nhn xt

tnh khong cch t im O n ng thng ( ta c th

+ Xc nh hnh chiu H ca O trn ( v tnh OH+ p dng cng thc

2. Khong cch t mt im n mt mt phngCho im O v mt phng ((). Gi H l hnh chiu ca O trn ((). Khi khong cch gia hai im O v H c gi l khong cch t im O n mt phng ((). K hiu

* Nhn xt

tnh khong cch t im O n mt phng (() ta c th s dng mt trong cc cch sau:

Cch 1. Tnh trc tip. Xc nh hnh chiu H ca O trn (() v tnh OH* Phng php chung.

Dng mt phng (P) cha O v vung gc vi (() Tm giao tuyn ( ca (P) v (() K OH ( ( (). Khi . c bit:

+ Trong hnh chp u, th chn ng cao h t nh trng vi tm y

+ Hnh chp c mt mt bn vung gc vi y th chn ng vung gc h t nh s thuc giao tuyn ca mt bn vi y

+ Hnh chp c 2 mt bn vung gc vi y th ng cao chnh l giao tuyn ca hai mt bn ny

+ Hnh chp c cc cnh bn bng nhau (hoc to vi y nhng gc bng nhau) th chn ng cao l tm ng trn ngoi tip y

+ Hnh chp c cc mt bn to vi y nhng gc bng nhau th chn ng cao l tm ng trn ni tip y

Cch 2. S dng cng thc th tch

Th tch ca khi chp . Theo cch ny, tnh khong cch t nh ca hnh chp n mt y, ta i tnh V v S

Cch 3. S dng php trt nh

tng ca phng php ny l: bng cch trt nh O trn mt ng thng n mt v tr thun li , ta quy vic tnh v vic tnh . Ta thng s dng nhng kt qu sau:

Kt qu 1. Nu ng thng ( song song vi mt phng (() v M, N ( ( th

Kt qu 2. Nu ng thng ( ct mt phng (() ti im I v M, N ( ( (M, N khng trng vi I) th

c bit, nu M l trung im ca NI th

nu I l trung im ca MN th

Cch 4. S dng tnh cht ca t din vungC s ca phng php ny l tnh cht sau: Gi s OABC l t din vung ti O () v H l hnh chiu ca O trn mt phng (ABC). Khi ng cao OH c tnh bng cng thc

Cch 5. S dng phng php ta

C s ca phng php ny l ta cn chn h ta thch hp sau s dng cc cng thc sau:

vi ,

vi ( l ng thng i qua A v c vect ch phng

vi l ng thng i qua v c vtcp

Cch 6. S dng phng php vect

3. Khong cch t mt ng thng n mt mt phng song song vi nCho im ng thng ( song song vi mt phng ((). Khong cch gia ng thng ( v mt phng (() l khong cch t mt im bt k ca ( n mt phng ((). K hiu

* Nhn xt

Vic tnh khong cch t ng thng ( n mt phng (() c quy v vic tnh khong cch t mt im n mt mt phng.

4. Khong cch gia hai mt phng song songKhong cch gia hai mt phng song song l khong cch t mt im bt k ca mt phng ny n mt phng kia. K hiu

* Nhn xt

Vic tnh khong cch gia hai mt phng song song c quy v vic tnh khong cch t mt im n mt mt phng.

5. Khong cch gia hai ng thng cho nhauCho hai ng thng cho nhau a v b. ng thng ( ct c a v b ng thi vung gc vi c a v b c gi l ng vung gc chung ca a v b. ng vung gc chung ( ct a ti H v ct b ti K th di on thng MN gi l khong cch gia hai ng thng cho nhau a v b. K hiu .

* Nhn xt

tnh khong cch hai ng thng cho nhau a v b ta lm nh sau:

+ Tm H v K t suy ra

+ Tm mt mt phng (P) cha a v song song vi b. Khi

+ Tm cp mt phng song song (P), (Q) ln lt cha a v b. Khi

+ S dng phng php ta

* c bit

Nu th ta tm mt phng (P) cha a v vung gc vi b, tip theo ta tm giao im I ca (P) vi b. Trong mp(P), h ng cao IH. Khi

Nu t din ABCD c AC = BD, AD = BC th on thng ni hai trung im ca AB v CD l on vung gc chung ca AB v CD.B. CC V D MINH HOI) Phng php tnh trc tipV d 1.

Cho hnh chp SABCD c y ABCD l hnh thoi tm O, cnh a, gc , c SO vung gc mt phng (ABCD) v SO = a.

a) Tnh khong cch t O n mt phng (SBC).

b) Tnh khong cch t ng thng AD n mt phng (SBC).

Li gii.

a) H

Trong (SOK) k

. Ta c u ;

Trong tam gic vung OBC c:

Trong tam gic vung SOK c:

Vy

b) Ta c

K . Do

V d 2. ( thi i hc khi A nm 2010). Cho hnh chp S.ABCD c y ABCD l hnh vung cnh a. Gi M v N ln lt l trung im ca cc cnh AB v AD; H l giao im ca CN vi DM. Bit SH vung gc vi mt phng (ABCD) v . Tnh khong cch gia hai ng thng DM v SC theo a.Li gii.

Ta c:

Do

K

Suy ra HK l on vung gc chung ca DM v SC nn

Ta c:

Vy

II) Phng php s dng cng thc tnh th tch.

V d 3.

Cho hnh chp t gic u S.ABCD c AB = a, SA = . Gi M, N, P ln lt l trung im ca cc cnh SA, SB, CD. Tnh khong cch t P n mt phng (AMN).Phn tch. Theo gi thit, vic tnh th tch cc khi chp S.ABCD hay S.ABC hay AMNP l d dng. Vy ta c th ngh n vic quy vic tnh khong cch t P n mt phng (AMN) v vic tnh th tch ca cc khi chp ni trn, khong cch t P n (AMN) c th thay bng khong cch t C n (SAB).Li gii.

Gi O l tm ca hnh vung ABCD, khi SO ( (ABCD).

M, N ln lt l trung im ca SA v SB nn

. Vy:

. .

Vy

V d 4. Cho hnh chp S.ABCD c y ABCD l hnh vung tm O, SA vung gc vi y hnh chp. Cho AB = a, SA = . Gi H, K ln lt l hnh chiu ca A trn SB, SD. Tnh khong cch t im O n mt phng (AHK).Phn tch. Khi chp AOHK v ASBD c chung nh, y cng nm trn mt mt phng nn ta c th tnh c th tch khi chp OAHK, hn na tam gic AHK cn nn ta tnh c din tch ca n.Li gii. Cch 1:

Trong :

;

Ta c HK v BD ng phng v cng vung gc vi SC nn HK // BD.

AI ct SO ti G l trng tm ca tam gic SAC, G thuc HK nn

. Tam gic AHK cn tai A, G l trung im ca HK nn AG ( HK v

T din ASBD vung ti A nn:

Tam gic OHK cn ti O nn c din tch S bng

Cch 2: Ta chng minh

Ta c:

Cch 3: Gii bng phng php ta nh sau:

Chn h ta Oxyz sao cho O ( A, B(a ; 0 ; 0), D(0 ; a ; 0), S(0 ; 0 ; ).

Tnh SH, SK suy ra ta ca H, K, O

p dng cng thc

Cch 4: SC ( (AHK) nn chn ng vung gc h t O xung (AHK) c th xc nh c theo phng SC.* AH ( SB, AH ( BC (do BC ( (SAB)) ( AH ( SC

Tng t AK ( SC. Vy SC ( (AHK)

* Gi s (AHK) ct SC ti I, gi J l trung im ca AI, khi OJ // SC ( OJ ( (AHK).

SA = AC = ( (SAC cn ti A ( I l trung im ca SC.

Vy

III) Phng php trtV d 5. ( thi i hc khi B nm 2011).

Cho lng tr ABCDA1B1C1D1 c y ABCD l hnh ch nht . Hnh chiu vung gc ca im A1 trn mt phng (ABCD) trng vi giao im ca AC v BD, gc gia hai mt phng (ADD1A1) v (ABCD) bng 600. Tnh th tch ca khi lng tr cho v khong cch t im B1 n mt phng (A1BD) theo a.

Phn tch. Do B1C // (A1BD) nn ta trt nh B1 v v tr thun li C v quy vic tnh thnh tnh

Bi gii.

* Gi O l giao im ca AC v BD

Gi E l trung im AD

* Tnh :

Cch 1:

Do B1C // (A1BD)

H

EMBED Equation.3 Cch 2:

Trong :

V d 6. Cho hnh chp SABCD c y ABCD l hnh vung tm O c cnh bng a, v vung gc vi mt phng (ABCD). a) Tnh khong cch t O n (SBC).

b)Tnh khong cch t trng tm tam gic SAB n (SAC).

Phn tch: Do , nn thay v vic tnh ta i tnh , tng t nh vy ta c th quy vic tnh thng qua vic tnh hay

Li gii.a) Ta c: nn:

Gi H l hnh chiu ca A trn SB ta c:

Trong tam gic vung SAB c:

b) Gi E l trung im AB, G l trng tm tam gic SAB.Do nn

Ta c:

IV) Phng php s dng tnh cht ca t din vung

1. nh ngha. T din vung l t din c mt nh m ba gc phng nh u l gc vung.

2. Tnh cht. Gi s OABC l t din vung ti O () v H l hnh chiu ca O trn mt phng (ABC). Khi ng cao OH c tnh bng cng thc

Chng minh.

Gi s , (1)

(2)

T (1) v (2) suy ra . Trong cc tam gic vung OAD v OBC ta c

V vy

Mc tiu ca phng php ny l s dng cc php trt quy vic tnh khong cch t mt im n mt mt phng v vic tnh khong cch t nh ca tam din vung n mt huyn ca n v v vy p dng c tnh cht trn

V d 7. Cho lng tr u c tt c cc cnh u bng a. Gi M, N ln lt l trung im ca v . Tnh khong cch gia v CN

Phn tch. tnh khong cch gia v CN ta tm mt mt phng cha CN v song song vi , tip theo ta dng cc php trt quy vic tnh khong cch t mt im n mt mt phng v vic tnh khong cch trong t din vung.

Li gii.

Gi O, D ln lt l trung im ca BC v CN th OACD l t din vung ti O. l hnh bnh hnh . Mt phng (ACN) cha CN v song song vi nn

p dng tnh cht ca t din vung ta c

. Vy

V d 8. Cho hnh lp phng c cnh bng a. Gi M l trung im ca . Tnh khong cch gia hai ng thng CM v .

Li gii.

Gi N l trung im ca th l hnh bnh hnh nn . Mt phng () cha v song song vi nn

vi . Gi th G l trng tm ca tam gic . Do .

T din vung ti A nn

.

Vy

V) S dng phng php ta .* Phng php:

Bc 1: Chon h to Oxyz gn vi hnh ang xt.

Bc 2: Chuyn bi ton t ngn ng hnh hc sang ngn ng to - vc t

Bc 3: Gii bi ton bng phng php to , ri chuyn sang ngn ng hnh hc.V d 9.Cho hnh lp phng ABCDABCD cnh bng 1. Mt mt phng bt k i qua ng cho BD.

a) Tnh khong cch gia hai mt phng (ACD) v (ABC) b) Xc nh v tr ca mt phng sao cho din tch ca thit din ct bi mp v hnh lp phng l b nht.Phn tch: Vi mt hnh lp phng ta lun chn c mt h to thch hp, khi to cc nh bit nn vic tnh khong cch gia hai mt phng (ACD) v (ABC) tr nn d dng. Vi phn b, ta quy vic tnh din tch thit din v vic tnh khong cch t M n ng thng DB.Li gii.

Chn h to sao cho gc to

Gi M l im bt k trong on thng CD, tc

a) D dng chng minh c (ACD) // (ABC)

Mt phng (ACD) c phng trnh:

b) Gi s ct (CDDC) theo giao tuyn DM, do hnh lp phng c cc mt i din song song vi nhau nn ct (ABBA) theo giao tuyn BN//DM v DN//MB. Vy thit din l hnh bnh hnh DMBN.Gi H l hnh chiu ca M trn DB. Khi :

.

Ta c:

Du ng thc xy ra khi

Nn din tch nh nht khi , hay M l trung im DCHon ton tng t nu

Vy din tch nh nht khi M l trung im DC hoc M l trung im DA.

V d 10. Cho hnh chp SABCD c y ABCD l hnh vung cnh a. . Gi M l im di ng trn cnh CD. Xc nh v tr ca M khong cch t im S n BM ln nht, nh nht.Li gii.

Chn h to trc chun Oxyz sao cho M l im di ng trn CD nn vi .

Xt hm s trn [0;1]

Ta c bng bin thin:

t 0 1

f(t)-+-

f(t)

2

T bng bin thin ta c , t c khi t = 0

, t c khi t = 1

Do ln nht khi

nh nht khi

VI) S dng phng php vc t vc t.* Phng php:

Bc 1: Chon h vc t gc, a cc gi thit kt lun ca bi ton hnh hc cho ra ngn ng vc t.

Bc 2: Thc hin cc yu cu ca bi ton thng qua vic tin hnh bin i cc h thc vc t theo h vc t gc.Bc 3: Chuyn cc kt lun vc t sang cc kt qu hnh hc tng ng.V d 11. ( thi i hc khi D nm 2007).Cho hnh chp c y l hnh thang. , . Cnh bn vung gc vi y v . Gi l hnh chiu vung gc ca trn . Tnh khong cch t n mt phng .

Li gii.t

Ta c:

Gi l chn ng vung gc h t ln mt phng (SCD)

D dng tnh c

Khi :

Ta c:

Cch 2: Gi ln lt l khong cch t cc im H v B n mp(SCD), ta c:

Trong

Ta c:

Cch 3: S dng tnh cht ca t din vung.Phn tch. Trong bi ton ny, vic tm chn ng vung gc h t H xung mt phng (SCD) l kh khn. V vy, ta s tm giao im K ca AH v (SCD) v quy vic tnh khong cch t H n (SCD) v vic tnh khong cch t A n (SCD)

Gi M l giao im ca AB v CD, K l giao im ca AH vi SM. Ta c:

. Suy ra H l trng tm ca tam gic SAM.T ta c:

Do t din ASDM vung ti A nn:

Vy

* Nhn xt: Vic la chn h vc t gc l rt quan trng khi gii quyt mt bi ton bng phng php vc t. Ni chung vic la chn h vc t gc phi tho mn hai yu cu: + H vc t gc phi l ba vc t khng ng phng. + H vc t gc nn l h vc t m c th chuyn nhng yu cu ca bi ton thnh ngn ng vc t mt cch n gin nht.

V d 12. ( thi H khi B nm 2007)Cho hnh chp t gic u c y l hnh vung cnh . l im i xng ca qua trung im ca . ln lt l trung im ca v . Tnh khong cch gia v .

Gii: t :

Ta c :

Gi l on vung gc chung ca v , ta c:

Cch 2:

Ta c: ; nn t gic l hnh bnh hnh

Do hnh chp SABCD u

C. BI TP NGH

Bi 1. ( thi i hc khi D nm 2011). Cho hnh chp S.ABC c y ABC l tam gic vung ti B, BA = 3a, BC = 4a; mt phng (SBC) vung gc vi mt phng (ABC). Bit SB = v . Tnh th tch khi chp S.ABC v khong cch t im B n mt phng (SAC) theo a.

Bi 2.

Cho hnh chp t gic SABCD, y ABCD l hnh thoi cnh a, tm O, gc . Cc cnh bn SA = SC; SB = SD .

a) Tnh khong cch t im O n mt phng (SBC).

b) Tnh khong cch gia cc ng thng SB v AD.

Bi 3. Cho t din OABC c OA, OB, OC i mt vung gc v . Gi M, N theo th t l trung im cc cnh Tnh khong cch gia hai ng thng OM v CN.Bi 4. ( thi i hc khi A nm 2011). Cho hnh chp S.ABC c y ABC l tam gic vung cn ti B, AB = BC = 2a; hai mt phng (SAB) v (SAC) cng vung gc vi mt phng (ABC). Gi M l trung im ca AB; mt phng qua SM v song song vi BC, ct AC ti N. Bit gc gia hai mt phng (SBC) v (ABC) bng 60o. Tnh th tch khi chp S.BCNM v khong cch gia hai ng thng AB v SN theo a.

Bi 5. ( thi i hc khi D nm 2008). Cho lng tr ng ABC.A'B'C' c y ABC l tam gic vung, AB = BC = a, cnh bn AA' a 2. Gi M l trung im ca cnh BC. Tnh theo a th tch ca khi lng tr ABC.A'B'C' v khong cch gia hai ng thng AM, B'C.Bi 6. ( thi i hc khi D nm 2009). Cho hnh lng tr ng ABCABC c y ABC l tam gic vung ti B, AB = a, AA = 2a, AC = 3a. Gi M l trung im ca on thng AC,I l giao im ca AM v AC. Tnh theo a th tch khi t din IABC v khong cch t A im n mt phng (IBC)Phn III

KT LUN V KIN NGH

Sng kin kinh nghim ca ti gii quyt c nhng vn sau:

1. Gip hc sinh c ci nhn tng qut v c h thng v bi ton tnh khong cch, t c k nng gii thnh tho cc bi ton thuc ch ny v hn th c th ng dng chng vo bi ton tnh th tch v mt s bi ton thc t khc. 2. Gii quyt mt cch tng i trit bi ton v tnh khong cch ca cc i tng im, ng thng v mt phng.

3. Thng qua vic v hnh, tnh ton, tm con ng ti u tnh khong cch, to cho cc em kh nng lm vic c lp, sng to, pht huy ti a tnh tch cc ca hc sinh theo ng tinh thn phng php mi ca B gio dc v o to. iu quan trng l to cho cc em nim tin, khc phc c tm l s bi ton v hnh hc khng gian.

Qua thc t p dng ti thy cc em hc sinh khng nhng nm vng c phng php, bit cch vn dng vo nhng bi ton c th m cn rt hng th khi hc tp phn ny. Khi hc trn lp v qua cc ln thi th i hc, s hc sinh lm c bi v tnh khong cch cao hn hn cc nm trc v cc em khng c hc chuyn ny.

Mt s xutMi bi ton thng l c nhiu cch gii, vic hc sinh pht hin ra nhng cch gii khc nhau cn c khuyn khch. Song trong nhng cch gii cn phn tch r u im v hn ch t chn c cch gii ti u. c bit cn ch ti nhng cch gii bi bn, c phng php v c th p dng phng php cho nhiu bi ton khc. Vi tinh thn nh vy v theo hng ny cc thy c gio cng cc em hc sinh c th tm ra c nhiu kinh nghim hay vi nhiu ti khc nhau. Chng hn, cc bi ton v tnh gc gia cc i tng hnh hc hay chng minh ng thc hnh hc; cc bi ton v ng dng ca phng php ta gii cc bi ton hnh hc khng gian,

Cui cng xin chn thnh cm n Ban gim hiu, Ban gim c S, Ban gim kho v cc ng nghip gip v gp cho ti hon thnh ti SKKN ny.

J

G

I

D

B

PAGE 17

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