trinh cua (T), biet tam giac ABC c6 di^n tich bang — va diem A c6 hoanh dp duong.
Vi AABC vuong tai B nen AC la duong kinh ciia (T).
Gpi ASB = (dj, dj) = t ta c6 BAC = ASB = t (goc c6 canh tuong ung vuong goc) Gia su ban kinh (T) la R ta c6 :
_ BC.BA AC sin t. AC cost „ „ 2 • , .
S A A B C " 2 2 " smtcost. V3.N/3 + 1.(-1)
Mat khac cos t = ^ l ^ ^ ^ l
Suy ra S^^g^-. = — tu do c6 R = 1.
Do A e d p C e d j nen A|a;-a\/3 j,c(c;cV3 j them niia vector chi phuong cua d j la U j ( l ; - > ^ ) c6 phuong vuong goc voi AC nen:
A C u j - 0<=>c-a-3(c + a) = 0 o c = -2ạ
Mat khac AC = 2R = 2 « 7(c^a)^T(V3(c + a)7 == 2
o 2 a N / 3 = 2 v i a > 0 nen a = •
Tam duong tron la trung diem ciia AC la:
I 'a + c S, ^
2 2 ^ \
a sVSâ ^/3 3'
'a + c S, ^
2 2 ^ \ I 2 ' ^ , , 6 ' 2 Vay phuong trinh cua (T) la x + - { ' 3^ y + -^ 2 = 1. Vay phuong trinh cua (T) la x + - { ' 3^ y + -^ 2 = 1.
84
CtyTNHH MTV DWH Khang Viet
Q^h 2:Ta c6 di tiep xlic voi (T) c6 duong kinh la AC nen AC 1 d j
Tfr gia thiet ta c6 : AOx = 60°,BOx = 120" =^ AOB=60°; ACB = 30"
Nen S,^Bc = ^AB.BC = 2^AB^ AB^ = ^ ^ AB = 1 v '
Vi A e d j ==> A(X ; - V 3 X ) , X > 0 ; O A = - ^ . A B = - ^ = > A ( - ^ ; - l ) ; OC = 20A =
V3 I Vs ^;-2
Duong tron (T) duong kinh AC c6: I
^2 / 1 1 Phuong trinh (T): x + 1 2^3. 3 I 2.V3' 2) 2 - 1 .
Vidu 1.5.21. Trong mat phang Oxy cho tam giac ABC c6 trong tam G(2;3).
Gpi H la true tam cua tam giac ABC . Biet duong tron di qua ba trung diem cua ba dean thSng HA,HB, HC c6 phuong trinh : (x -1)^ + (y -1)^ = 10. Viet phuong trinh duong tron ngoai tiep tam giac ABC.
Xffi gidị
Gpi ( C ) la duong tron (x -1)^ + (y -1)^ = 10, suy ra (C) c6 tam 1(1; 1), ban kinh R = %/lO .
Ta CO két qua sau day trong hinh hoc phang:
"Trong tam ginc, 9 diem gom trung diem C i i a ba canh, chan ba duong cao va ba trung diem cua cac doan noi true tam voi dinh nam tren mot duong tron c6 tam I , G, H thang hang va I H = 3IG ".
Gpi E la tam duong tron ngoai tiep tam giac A B C va M la trung diem BC. Ta c6:
Phep vj t u V(G,_2): I - > E, M A va M e ( C ) nentaco: E(4;7) va E A = 2 I M = 2N/IO
Vay phuong trinh duong tron ngoai tiep tam giac A B C la: ( x - l) 2+ ( y - 1 0) 2 =40.
Phuong phdp giai Toan Hinh hqc theo chuyen de- Nguyen Phii Khdnh, Nguyen Tat Tliu
m BAI TAP
Bai 1.5.1. Trong mat phSng vai h? tpa dp Oxy, cho tam giac ABC biét
A (5; 2). Phuang trinh duang trung true canh BC, duong trung tuyen C C laVi lupt la x + y - 6 = 0 va 2 x- y + 3 = 0. Tim tpa dp cac dinh ciia tam giac ABC.
Jiimng đn gidi
Gpi C = (c;2c + 3) va I = ( m ; 6 - m ) la trung diem cua BC Suy ra: B = (2m - c;9 - 2m - 2c).
^2m-c + 5 l l- 2 m - 2 c ^
Vi C la trung diem ciia AB nen: C = , 2 m - c + 5. l l - 2 m - 2 c G C C Nen 2(- -)-- 2 2 Phuong trinh BC: 3x - 3y + 23 = 0. ' 2 x - y + 3 = 0 + 3 = 0=>m = - - ^ I = (--;—). 6 ^6 6'
Tpa dp cua C la nghif m ciia he:
Tpa dp B
=>C = 'U 37^ 3 ' 3 3 x - 3 y + 23 = 0
'[ 3 ' 3 /
Bai 1.5.2. Tim tpa dp dinh B, C ciia tam giac deu ABC , biet A( 3 ; - 5 ) va trpng tam C ( l ; l ) .
Jiu&ng dan gidi
Gpi M la trung diem BC suy ra AG = 2GM => M( 0 ; 4 ) Phuong trinh B C : x - 3 y + 12 = 0.
Giasu B(3yQ-12;yQ) va t u GA = GB dan toi phuong trinh y o - 4 + V3
J o= 4 - V 3 '
• Neu y o = 4 + 7 3 = > B ( - 3 V 3 ; 4 + V3)=>c(3V3;4-V3' .
• Neil yo = 4 - N / 3 ^ B ( 3 7 3 ; 4 - V 3 ) = ^ C ( - 3 V 3 ; 4 + N/3).
Bai 1.5.3. Trong mat phang Oxy cho tam giac ABC c6 M(6;4) la trung diem
canh BC. Duong thSng AB va duong phan giac trong goc A Ian lupt c6 phuong trinh x - y + l = 0, x - 2 y + 3 = 0. Tim tpa dp cac dinh ciia tam giac ABC
( 3 y o - 1 3 ) ' + { y o - l f =40.
Cty TNHH MTV DWH Khang Viet
Jliccmg đn gidi Gpi d : x - 2 y + 3 = 0; A B : x - y + l = 0. x - y + l = 0 7 i:' ^ / Ta c6: A = d n AB => A : • x - 2 y + 3 = 0 Ăl;2). B e AB => B(b;b +1) AB = (b - l ; b - 1 ) .
Do C doi xiing voi B q u a M n e n t a c o C ( 1 2 - b ; 7 - b ) => AC = (11 - b ; 5 - b ) .
Gpi u = (2; 1) la VTCP ciia duong thing d. • Taco: cos^u,ABJ = cos^u,ACJ - ụAB _ ỤAC
AB " AC 2 { b - l ) + l ( b - l ) 2(11-b) +1(5-b) 9 - b b - 1 yjlih-lf ^J{n-hf +{5-hf V2b2-32b + 146 ^l2{h-lf l < b < 9 {9-hY 1 o b = 4 =^ B(4; 5), C(8; 3). Vay A ( l ; 2), B(4; 5), C(8; 3) .2b2-32b + 146 2
Bai 1.5.4. Trong mat phang Oxy cho tam giac ABC c6 M(6;4) la trung diem
canh BC. Duong thang AB va duong phan giac ngoai goc A Ian lupt c6 phuong trinh x - y + l = 0, 2x + y - 4 = 0. Tim tpa dp cac dinh ciia tam giac ABC
Jiuong đn gidi
Dat d:2x + y - 4 = 0
x - y + l = 0 2x + y - 4 = : 0
Lay N(2; 3) e A B , gpi N ' dol xung voi N qua d, suy ra N ' e AC Phuang trinh N N ' : x - 2y + 4 = 0
Ta c6: A = AB n d A : , ^ A ( 1 ; 2 ) .
N N ' n d = I:-^ 2x + y - 4 = 0 j , 4 12 x - 2 y + 4 = 0 ^ 5' 5
2 9.
I V i I la trung diem N N ' nen N ' ( - - ; - ) => A N ' = 5 5
7 _ n
Phuang trinh A C : x - 7 y + 13 = 0. Tirdosuy ra C(7c-13;c) va B(b;b + 1) Ma M la trung diem BC nen
V$yĂl;2), B(4;5), C(8;3). b + 7c-13 = 12 c + b + l = 8 b = 4 c = 3 • 87
Phutnigphdpgiai Todn Hiiilt tiQc thco chuyen de- Nguyen Phu Khanh, Nguyen Tat Thu
Bai 1.5.5. Trong mat phang voi h^ toa dp Oxy cho tarn giac ABC voi duong cao ke t u d i n h B va d u o n g phan giac trong ciia goc A Ian l u g t c6 p h u o n g trinh la : 3x + 4y + 10 = 0 va x - y+1 = 0, diem M(0 ; 2) thupc d u o n g thSng AB dong
thoi each C mot khoang bang ^2 . T i m toa do cac d i n h ciia tam giac A B C .
^Jdu&ng đn giai
Cho d j :3x + 4y + 10 = 0 ; d 2: x - y + l = 0
Goi A la d u o n g thang d i qua diem M va vuong goc v o i di, A cat d2 tai I va
cat A C tai N => N doi x i i n g voi M qua d j Ta CO p h u o n g trinh A : x + y - 2 = 0.
Goi I = A n d j =i> I : x + y - 2 = 0 x - y + 1 = 0
I la trung diem cua M N => N ( l ; l )
A C d i qua N ( l ; l ) va vuong goc v o i di => A C : 4x - 3y - 1 = 0 . 4 x - 3 y - l = 0 [x = 4
A = A C n d2 => A :
x - y + 1 = 0 y = 5 Ă4;5).
A B la d u o n g thang d i qua diem M ( 0 ; 2 ) nhan M A = (4;3) lam vec to chi p h u o n g A B : - = ^^—^ <=> 3x - 4y + 8 = 0. 4 3 B = A B n d j => B : < 3 x - 4 y + 8 = 0 3x + 4 y + 10 = o ' x = -3 _ l = ^ B ( - 3 ; - - ) .
V i C each M mot khoang bang Vi nen C thuoc d u o n g tron (S) c6 tam M ban k i n h bang V2 (S): x^ + (y - 2)^ = 2.
Tpa dQ ciia C la nghiem cua he p h u o n g trinh : 4 x - l 4 x - 3 y- l = 0 x 2+ ( y - 2) 2= 2 y = 4 x - l <=> 2 5 x ^ - 5 6 x + 31 = 0 x = l = i > y = l 31 33 25 ^ 25
V i d2 la phan giac trong ciia goc A nen B va C nMm khac phia bo la d2
« ( ^ B - YB + l ) ( ' < c - Yc +1) < 0 « - yc +1 > 0. 88
Cty TNim MTV DWII Khang Viet
Ca hai diem C tren deu thoa man 1 ^
Vay Ă4;5), B -3; — 4 , C ( l ; l ) hoac C / 3 1 . 3 3 ' . 2 5 ' 2 5 /
gai 1.5.6. Trong mat phSng Oxy cho tam giac ABC c6 dien tich bang 14. H a i diem M ( 2 ; - l ) , N ( - 2 ; 4 ) Ian luot la trung diem ciia hai canh AB, A C . D u o n g thang A : x + 2y + 5 = 0 d i qua d i n h B. Hay xac d i n h tpa do cac dinh ciia tam giac ABC. Jiizang đn gidi V i B G A = ^ B ( - 2 b - 5 ; b ) r : > Ă2b + 9 ; - 2 - b ) D u o n g thSng M N : 5x + 4y - 6 = 0 , M N = \/41 Ta CO S ^ A M N = ^ S ^ A B C = I ^ M N . d ( A , M N ) = 7 (*) Ú; Ma: d ( A , M N ) = 6 b + 31 N/4T >(*)<» 6b + 31 = 7 c ^ b = -4 3 •J'iv:' * b = - 4 ^ A ( l ; 2 ) , B(3;-4), C(-5;6) ^ ^ 19 ^, 11 13, „ , 2 3 19, ^ , 1 11, s i r b = = > A ( ; — ) , B ( — ; ), C ( — ; — ) . 3 ^ 3 3 ' ^ 3 3^ 3 3 '
Bai 1.5.7. Trong mat p h i n g v o i he toa do Oxy, cho tam giac A B C can tai A CO dinh A ( - l ; 4 ) va cac d i n h B, C thuoc d u o n g thang A : x - y - 4 = 0 . Xac d j n h toa do cac diem B va C, biet dien tich tam giac ABC bang 18. v
Jiu&ng đn gidi
Gpi H la h i n h chieu cua A len A => H la trung diem BC . A H = d ( A , B C ) - - ^ ; BC = ^ ^ ^ ^ = 4V2, A B = A C =
V2 A H A H ^ +
BC^ '97
Toa do B,C la nghiem ciia h f :
9 9 97
(x + l) 2+ ( y _ 4) 2 =
X - y- 4 = 0 Giai he ta duoc (x;y) = ^11 3^
Vay B 11 3 2 ' 2 , C 3 _ 5 2 ' 2 2 ' 2 hoac B hoac (x;y) = '3 _ 5 ^ 2 ' 2 ,C 3 _ 5 fll 3 ' U ' 2
Bai 1.5.8. Cho tam giac ABC nhon, viet p h u a n g trinh d u o n g t h i n g A C , ''iet toa do chan cac d u o n g cao ha t u cac d i n h A, B, C Ian l u o t la A j ( - l ; - 2 ) /
B : ( 2 ; 2 ) , q ( - l ; 2 ) .
Phuang phiip gidi Todn Hitih hoc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu
Ta CO CBjX = AB,Ci
Jiicang dan gidi
(1)
Tii giac ABiHCj npi tiep => ABjCi = AHCj (2) va A H C j = A j H C (3) Tii giac AjMBjC noi tiep ==> A ^ = A ^ ^ (4).
Tir (1), (2), (3) va (4) suy ra: CBjX = AiBjC => AC la phan giac ngoai goc Bj ciia tarn giac A j B j C j .
Taco: ÂBj : 4x - 3y - 2 = 0;BiCi : y - 2 = 0
Phuong trinh duong phan giac cua goc tao boi hai dt AjB^BjCj 4 x - 3 y - 2
4^+3^
y - 2 x - 2 y + 2 = 0 2 x + y - 6 = 0 Vay phuong trinh canh AC:2x + y - 6 = 0.
Bai 1 . 5 . 9 . Trong h^ tpa dp Oxy, cho tarn giac ABC can tai B vdi Ă1;-1), C(3;5). Diem B nam tren duong thang (d): 2x - y = 0 . Viet phuong trinh cac duong thang AB va BC .
JJuang đn gidi
Taco Bed=>B(2y;y)
Tarn giac ABC can tai B => AB - BC
« ^(2y -1)^ . (y . if = ^(2y - 3 ) ^ (y - sf « y = | B ( | ; | ) .
Phuong trinh canh A C : 3x - y - 4 = 0 . Phuong trinh canh A B : 13x - l l y - 24 = 0 .
Bai 1 . 5 . 1 0 . Trong he tpa dp Oxy, cho tam giac ABC c6 dinh Ă2;1), duong cao ke t u B c6 phuong trinh x - 3y - 7 = 0 va trung tuyen ke tir C c6 phuong trinh x + y +1 = 0 . Xac dinh tpa dp cac dinh B,C ciia tam giac.
Jiucrng đn gidi
Phuong trinh duong cao B H : x - 3y - 7 = 0 va A C : 3x + y - 7 = 0. Vi BeBH=>B(3y + 7;y).
'3y+_9 y + l
2 ' 2
Tpa dp trung diem cua AB la I , v i I thupc trung tuyen ke tit dinhC = = > ^ ^ + ^ + l = 0 o y = - 3 : ^ B ( - 2 ; - 3 ) .
2 2 ^ ' Tu do tim dupe C(4;-5). Tu do tim dupe C(4;-5).
90
Cty TNHH MTVDWH Khang Viet
pai 1 . 5 . 1 1 . Cho tam giac ABC c6 phan giac trong A D : x - y = 0, duong cao CH:2x + y + 3 = 0, AC di qua M ( 0 ; - 1 ) va AB = 2 A M . Tim tpa dp cac dinh
ciia tam giac ABC. _
Jimang đn gidi
Gpi N la diem doi xiing ciia M qua duong phan giac A D khi do N e AB. Taco N{-1;0).
C?inh AB di qua N va vuong goc voi CH => AB: x - 2y +1 = 0 Tpa dp diem A la nghiem ciia hf : x - y = 0 | x = l
x- 2 y + l = 0 [y = l • Vay A ( l ; l ) .
X V +1
Phuong trinh canh AC : - = ^^-^<;=>2x-y-l = 0 Tpa dp diem C la nghiem ciia h^ phuong trinh:
V l y C ( - - ; - 2 ) 2 x + y + 3 = 0 2 x - y - l = 0 ' ^ 2 • y = -2 Taco A M- V( 1 + 1)2+1 =V5;AB = 2 A M = 2 7 5 . G p i B( 2 y o- l ; y o ) e A B . ' ' ^ ' AB ^ ^(2yo - 2f + (yo -1)^ = 2Vs « ( y ^ -1)^ = 4 o yo = - 1 , yo = 3 .
Voi Yo = 3 thi B(5; 3) khong thoa man vi hai diem B, C 6 cimg phia so voi duong phan giac A D .
Voi y o = - l thi B(-3;-l) thoa man.
Vay tpa dp cac dinh ciia tam giac la: A ( l ; l ) , B ( - 3 ; - l ) , C( - ^ ; - 2 ) .
Bai 1 . 5 . 1 2 . Trong mat phang tpa dp Oxy cho diem A( 2 ; 1 ) , lay diemBeOx CO hoanh dp khong am va diem C € Oy c6 tung dp kliong am sao cho tam giac ABC vuong tai Ạ Tim B, C sao cho di^n tich tam giac ABC Ion rvhat. : i
Jiudng đn gidi
Gpi B(x;0)eOx;C(0;y)eOy,x>0,y>0
I
Ta CO AB = (x - 2; -1), AC = (-2; y -1) Tam giac ABC vuong tai A suy ra
AB. AC = - 2 x - y + 5 = 0r:>y = 5 - 2 x > 0 = > x < -
Phuang phlip gilii Toan Hinh hpc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu
r
Dien tích tam giac ABC :
S = ^ AB.AC = ^^{x-2f +1.yl4 + {y-lf
5 T
V a i O < x < - = > S = x^-4x + 5<5 2
maxS = 5 k h i x = 0. VayB(0;0),C(0;5).
B a i 1 . 5 . 1 3 . Trong mat phSng Oxy cho tam giac ABC c6 M(l;0),N(4;-3) Ian
lugt la trung diem ciia AB, AC; D(2;6) la chan duong cao ha tix A len BC. Tim
tpa dp cac dinh cua tam giac ABC.
J-Iuang đn gidi
Ta CO M N = (3;-3) va M N / / B C nen phuong trinh BC:x + y - 8 = 0 Suy ra B(b;8 - b). Do M la trung diem AB nen Ă2 - b;b - 8) Matkhac: AD 1 M N =>AD.MN = 0 <=>3b-3(14-b)==0c:>b = 7 Vay Ă-5;-l),B(7;l),C(13;-5)
B a i 1 . 5 . 1 4 . Trong mat phang Oxy, Cho tam giac ABC can tai A npi tiep duong tron (C): x^ + y^ + 2x - 4y +1 = 0 . Tim toa do cac diem A,B,C biet
M(0;l)la trung diem AB, diem A c6 hoanh do duong.
Jiuang đn gidi
Duong tron (C), tam I(-l;2), R = 2
Goi diem Ăa, b), khi do ta c6: (a + if + (b - 2)^ = 4 (1)
Taco: M I - ^/2, suy ra: IA = >/2 hay: â + (a-1)^ = 2 (2).
Giai he (1), (2), ta suy ra diem Ă2;l). Tu day suy ra diem B(-2;l).
Tu d\x kien tam giac can tai A, ket hop voi: M I = lA = 72 , nen suy ra tam giac ABC vuong can tai A, hay I chinh la trung diem BC.
Toa do diem C(0;3).
B a i 1 . 5 . 1 5 . Trong mat phang toa do Oxy, cho tam giac ABC vuong tai A biet phuong trinh duong th§ng A chiia canh BC la: sfix - y - >/3 = 0; diem A, B
thuoc true hoanh. Xac djnh toa do trong tam G cua tam giac ABC biet ban kinh duong tron ngi tiep tam giac ABC bang 2.
Jiuang đn gidi
De tháy B(1;0). Vi Ce A =^ c(a;Vs(a-l))
Cty TNllll MTV DVVIl Khang Viet
p^^ B thuoc true hoanh va tam giac A B C vuong nen ĂCI;0)
AB = ( a - l ; 0 ) , A C = ( 0 ; > / 3 ( a - l ) ) , A B C la tam g i c k khi va chi khi A B , A C
laiong Cling phuong hay a 5^ 1
Theo cong thuc tinh dien tich tam giac ta c6 S^g(~ = pr = A B . A C suy ra
2 ( A B + B C + C A ) = A B . A C , mat khae A B = |a - l|,BC = 2|a - l | , C A = v's|a - 1 nen ta c6 2^3 + N/3J|a-l =43{a-lf suy ra a = l(loai), a = 3 + 2 / 3 hoac
V^y CO hai truong hop xay ra ta tim dugc toa do trong tam trong hai truong hgp do la Gj 7 + 4V3 2V3 + 6 -I- 4 V 3 - 2 ^ / 3 - 6
Bai 1 . 5 . 6 . Trong mat phang voi h? tga dp Oxy, cho tam giac ABC c6 goc A = 120°, dinh A c6 tung dp duong va A thuoc duong thang x + y - 1 = 0 . Biet cac canh AB, AC cung tiep xuc voi duong tron + ( y - i r = 4 .
hay lap phuang trinh cua cac canh AB, AC cua tam giac ABC.
Jiuang đn gidi
4 .
Gia su duong tron c6 tam I(-p^,l) tiep xiie voi AB, AC tai D, Ẹ
73
Khi do do A = 120° nen de dang suy ra : lAE = 60°. Xet AIAE vuong, c6 lAE = 60° nen tinh dugc : lA = R
sin 60° 73'
Giasu: A ( a , l - a ) , a < l taco: l = a -I
73 j
2 16
+ a = — =>a = 0. Vay taco: Ă0,1).
Phuong trinh duong thang A di qua A c6 dang: ax + by - b = 0 voi â + b^ > 0 A la tiep tuyen cua duong tron (I) khi va chi khi d(I, A) = R
4
= 2 2 a = 73(â + b^) o â = 3b^ <=> a = ±73b
Suy ra phuong trinh AB, AC la: ±73x + y - 1 = 0.
Phuamg phapgiai Todn Hinh hoc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu