Gii bi ton hnh phng bng vic s dng Tch V Hng.

Li ni u:

Nh chng ta bit,tch v hng l vic lm kh ph bin trong qu trnh gii ton hnh phng Oxy.Tuy n ch n gin l vic tnh hai vecto ra ri nhn vi nhau bng 0 nhng linh hot c n trong cc bi ton th vic lm ny khng h n gin.i khi y l s kh khn khin cho cc bn hc sinh khng lm tt c cu 7 trong thi i hc,quc gia. Bi vit

ny mun chia s i cht v k thut gii hnh phng Oxy bng vic s dng tch v hng!

PHN 1: L THUYT C S

Cho tam gic ABC vung ti A,ta c:

1 1 2 2

1 2 1 2

( , ), (x , ).

. x 0.

AB x y AC y

AB AC AB AC x y y

i khi khng th tham tm c trn mi ng thng 2 im tm vtcp ,v d tnh hung sau:Cho 1 phng trnh

ng thng,ta tnh c vtcp ca 2 im thuc ng thng vung gc vi ng thng cho.V d d c vtcp l u,ng

thng d vung gc

: ( , ),d' : ( , ).

' . 0

d

d

d u x y x a b

d d u x ax by

Chng ta hiu nhanh l nu c c gc vung hoc cc ng thng vung gc th ta c tnh cc vecto ch phng ra ( 2

vecto nm trn 2 ng thng vung gc) ri nhn vi nhau bng 0.L thuyt ch ngn gn nh vy.Sau y chng ta s

n vi cc bi ton c th hiu su phng php cng nh thy r nhng k thut gii!

PHN 2: CC V D C TH

Phn tch:

c thy chi tit : ng thng AC v BD vung gc vi nhau ,ta nh hnh trong u l d dng tch v

hng.Nhng nu ch c 1 phng trnh l tch v hng,chng ta phi tham s 3 im A,C,D theo mt n th bi ton mi

c gii quyt,vic ny kh n gin v ta d dng tham s A,M l trung im AD nn D cng theo A v M. Cn C? ta da

vo chi tit AD=2BC s tham s c C theo cc im kia.

V d 1: Cho hnh thang ABCD c im B(0,6).Hai y AD v BC vi AD=2BC.Hai ng cho AC v BD vung gc vi

nhau,ng thng AC v BD vung gc vi nhau.ng thng AB c phng trnh:3x-y+6=0. Bit M(2,-3) l trung im

cnh AD.Tm ta cc nh cn li ca hnh thang.

Li gii:

2

( ,3 6) D(4 a, 12 3a).C(x, y)

( , 6), (4 2 , 18 6 ).

2 4 2 22 (2 , 3 3 ).

2( 6) 18 6 3 3

(2 2 , 9 6 ), (4 , 18 3 ).

. 0 20

A a a

BC x y AD a a

x a x aBC AD C a a

y a y a

AC a a BD a a

AC BD AC BD a

( 2,0), (4,3),D(6, 6)2

125 170 0 17 27 25 39 33 317, , , , ,

4 4 4 4 4 44

A Ca

aA C Da

KL: Ta cc nh cn tm ( 2,0), (4,3),D(6, 6)A C .

Nhn xt: c nh hnh s dng tch v hng,chng ta cn c mt cht k nng v vic s dng h thc vecto tm C

theo cc im bit. Lm c vic ny coi nh bi ton c gii quyt.

Phn tch:

nh hng gii:Trc tm l giao im 3 ng cao,ta d dng s dng h tch v hng gii quyt bi ton. bi ny,ch

cc im P,Q cho thuc Ab,AC nn ta tham s B hoc C theo 2 n,thng qua nhng im cho d dng lp c h

tch v hng.

V d 2: Cho hnh tam gic ABC c trc tm 5 9,

2 2H

, 3 5,

2 2M

l trung im ca BC.Cc im 1 11,

2 2P

,

(6, 1)Q ln lt thuc AB,AC. Tm ta cc nh ca tam gic ABC.

Li gii:

Tham s B(a,b),do M l trung im BC nn C(3-a,5-b).

Ta c:1 11 5 9 1 1

( , ), ( 3 ,6 ), ( ,b ), ( , )2 2 2 2 2 2

PB a b QC a b HB a HC a b

Do H l trc tm tam gic ABC nn:

2 2

2 2

2

1 1 11 1( )( ) (b )( ) 0

. 0 2 2 2 2

5 9. 0 ( 3 )( ) (6 )( ) 02 2

6 3(1)

21 39(2)

2 2 2

(1) (2) 3 11 3 11.

3 2 ( 2,3),

(1) 10 75 135 0

a a bPB HC

QC HB a a b b

a b a b

a ba b

a b a b

b a B

b b

(5,2) (3,8)

9 5 5 9( , ) ( )

2 2 2 2

C A

b a B H Loai

KL: (3,8)A , ( 2,3), (5,2)B C .

Nhn xt: Bi trn khng kh nhn ra vic s dng h tch v hng.Tuy nhin vic nhn tch v hng,gii h phng

trnh chng ta cng cn phi lu ti n nng v con s,lm khng cn thn s nhm dn ti sai kt qu.

p dng: Cho tam gic ABC c chn ng cao h t B,C xung cnh i din ln lt l K(-2,2),E(2,2).im

16 2,

5 5P

l hnh chiu vung gc ca E xung BC.tm ta cc nh ca tam gic ABC.

Phn tch: y l bi ton kh,chng ta cn vn dng tt k nng v s dng din tch.

Li gii:

Phng trnh IC qua I vung gc vi AB l: 2x + y 10 = 0

Tham s ha ta B(2b ; b), C(c ; 10 2c). Theo bi ra ta c h:

2 2 2 2

9 92 4 2 10 2 3, 2 6;32 4 2 2

2 2 10 2;61 2, 6 2 22 4 2 4 8 2 100. 10

b b b c b c b b c Bb

b c b c Cb cb b c cIB IC

Do I l trung im AB suy ra A( 2 ; 1 ).

Vy ta cc im tha mn l A(2 ; 1), B(6 ; 3), C(2 ; 6).

V d 3: Cho tam gic ABC cn ti C c phng trnh cnh AB l x-2y=0.im I(4,2) l trung im cnh AB,im

thuc cnh BC,din tch tam gic ABC bng 10.tm ta cc nh tam gic bit tung ca im B ln hn hoc bng 3.

V d 4: Cho hnh hnh ch nht ABCD c E l trung im cnh BC v F l im nm trn cnh AD sao cho FA=3FD.

Phng trnh ng thng BF:5x+y-5=0,phng trnh ng thng i qua B v vung gc vi DE l d:y=5.im C thuc

ng thng d:x+2y-6=0 v c tung dng.Tm 4 nh hnh ch nht.

Phn tch: y l bi ton kh hay v vic s dng tch v hng.Da vo nhng gi thit cho,ta hon ton c th thit

lp h phng trnh v tch v hng,t bi ton c gii quyt.Bi ny cn lu 3 vn ln sau:

1, bit v tham s c B,C,F,ta d dng suy ra D theo nhng im trn nh h thc vecto.

2, . 0BC DC BC DC

3, . 0dDE d DEu

Li gii:

Ta B tha mn:

5 5 0 0(0,5)

5 5

F FB F(b,5 5b),C d' C(6 2c,c)(DK : c 0).

x y xB

y y

Do E l trung im BC nn :5

(3 , )2

cE c

Ta c:

2 3 20 154 4 4 ( , ).

2 4

9 3 2 3 20 15 3 2 20 5(6 2 , 5), ( , ), ( , ), (1,0)

2 4 2 4d

b c c bFD AD FD BC FD BC D

c b c b c b c bBC c c DC DE u

Do

9 3 2 3 20 15(6 2 )( ) ( 5)( ) 0

. 0 2 4

3 2 20 5. 0 .1 .0 02 4

d

c b c bc c

BC DCBC DC

DE d c b c bDE u

2 2 3 01( ) 1

(4,1), (2, 1) ( 2,3)33( )

2

c cc tm b

C D Acc Loaib

: ( 2,3), (0,5), (4,1), (2, 1).KL A B C D

Nhn xt: Bi ton cn c nhn mnh ch tm D thng qua h thc vecto biu th n theo cc im bit,cn vic

pht hin ra h tch v hng cng khng qu kh.Tc gi bi ton th thch kin nhn ca ngi gii vic

tham s cc im cng nh tnh ton,h phng trnh s,v vy cn kh nng tnh ton tt ca ngi gii!

p dng:

1. Cho hnh ch nht ABCD, c B(2,0).ng thng i qua B vung gc vi ng cho AC c phng trnh d:7x-y-

14=0.ng thng i qua nh A v trung im ca cnh BC c phng trnh x+2y-7=0.Tm ta nh D ca hnh ch

nht bit im A c honh m (p s :D(3,7)).

2. Cho hnh bnh hnh ABCD c A(-4,-2),phng trnh BD:6x-y+2=0.Gi M l trung im ca AB,ng thng i qua C v

vung gc vi DM c phng trnh :x-4y-3=0.Tm ta cc nh cn li ca hnh bnh hnh ABCD.

Phn tch:

Ch I l trung im ca AC,BD,vy khi tham s c 1 trong 4 im th s suy ra thm 1 im i xng vi n. y

hon ton tham s c C,t suy ra A,nh rng AH vung HI,ta tnh tch v hng l tm c A,C,t bi ton tr nn

d dng hn.

V d 5: Cho hnh ch nht ABCD,c tm I(2,3).Hnh chiu vung gc ca nh A ln BD l im 7 6,

5 5H

.im C

thuc ng thng d:2x-y-6=0.Tm ta cc nh hnh ch nht.

Li gii:

Tham s C(a,2a-6),I l trung im ca AC nn A(4-a,12-2a).

13 54 3 9( ,2 ), ( , )

5 5 5 5

13 3 9 54. 0 ( ).( ) .(2 ) 0 a 5

5 5 5 5

( 1,2), (5,4)

AH a a IH

AH BD AH IH a a

A C

Ta c :

2

: 3 3, ( ,3 3).

( 1,3 5), ( 5,3 7).

. 0.

( 1)( 5) (3 5)(3 7) 0

1 (1,0) (3,6)10 40 30 0

3 (3,6) (1,0)

( 1,2) (1,0), (5,4), (3,6):

( 1,2), (3,6)

DB x y B a a

AB a a CB a a

AB BC ABCB

a a a a

a B Da a

a B D

A B C DKL

A B

, (5, 4), (1,0)C D

Qua nhng v d trn,hy vng bn c phn no nm c k thut gii hnh phng Oxy nh vic s dng tch v

hng.Phng php ny dng kh ph bin ,min c gc vung,ng thng vung gc vi nhaul ta c th p dng tch

v hng,n rt hu hiu khi bi ton ch c 1 n.Cng ty vo ngi gii m bi ton th hin c s linh hot,nhanh

chng.Xin chc cc bn hc tt phn hnh phng Oxy cng nh thnh cng trong k thi i hc !

Nhng ngi thc hin

Ni dung: Nguyn Vn Ph Trnh by: Nguyn nh Huynh