. Dovay dÚcJng tron (K) tiep xuc trong vdi dtfdng
b) Gpi Ila trung diem AK, Sla giaodi em ciJa BI vhO M Tac 6I trung diem M N (do A M K N la hinh vuong)
X 6 t A I B N va A I S M c6: I N = I M , B I N = S I M ( d o i dinh), i B N = I S M (so le trong va A B // M S ) .
D o do A I B N = A I S M (g.c.g) B N = S M S M = B N = A B - A N = 3a - 2a = a
N e n OS = O M + S M = — + a = — => S thuoc di/c(ng tron (O) 2 2
Ta c6: OS 1 A C AS = SC !::> A B S = SBC => BS la tia phan giic cua goc A B C .
A A B C c6 B S , A K 1^ hai dÚcJng phan giac c^t nhau tai I . V a y I la t a m dúclng tron n p i tiep tam giac A B C .
N h f i n x e t : B a i toan nay khong kh6 l ^ m , viec l ^ m xuát h i e n d i e m S (giao d i e m cua B I va O M ) r o i chilng m i n h S thuoc diTdng tron (O) de tir 66 c6 BS la tia phan giac cua goc A B C , that dac s^c.
B a i 5:
— , • , „ , . Uiii-'''
a) = = - <=>c.ab = b.ca <:>c(10a+ b) = b(10c + a) ' i f
ca c
« lOac + be = lObc + ab o 10c(a - b) = b(a - c) <•; -^^ *1
Ta c6: b(a - c ) : 5 => b =.5 hoSc a - c = 5 hoSic c - a = 5 ( v l 0 < b < 9 ; - 9 < a - c < 9)
Luy^n g\i\e IMII-C ti'i v V i I'l; '0 h i mien oac, Irung. Nam mon loan _ Nguyen u u c l a n " a . . 9 <=> c = 2a - 9 2a - 9 <=> 2c = 1 + 2a - 9 > 0 ; 2a - 9 la irdc cua 9 ; 2a - 9 ^ 1 T a c 6 : 2 a - 9 € {3;9} c>a e {6 ; 9} + a = 9 thi 2c = 2 c = 1 (a ; b ; c) = (9 ; 5 1) (a ; b ; c) = (6 ; 5 ; 2) : 3 ^ .1 V : . + a = 6 thi 2c = 4 c = 2 Xet a - c = 5 <=> a = c + 5 , , Ta c6: 2c(c + 5 - b) = b o 2c^ + 10c - 2bc = b o b(2c + 1) = 2c^ + 10c « b = 2c^ +10c « 2b = 2c + 9 - 2c + 1 2c + 1
Do do 2c + 1 la iTdc difdng cua 9. Ma c ?i 0 nen 2c + 1 ?t 1
Taco: 2c + 1 e { 3 ; 9 } o c e {l ;4}
+ c = 1 thi b = 4, a = 6 ncn (a ; b ; c) = (6 ; 4 ; 1) + c = 4 Ihi b = 8, a = 9 nen (a ; b ; c) = (9 ; 8 ; 4) • X e t c - a = 5 o c = a + 5
Ta c6: 2(a + 5)(a - b) = - b o 2(a + 5)(b - a) = b. V i b > 0 nen b - a > 0. b = 2(a + 5)(b - a) > 2.5 = 10. V6 lị
Cac bo (a ; b ; c) can tim la: (a ; b ; c) = (9; 5 ; 1), (6 ; 5 ; 2), (6 ; 4 ; 1), (9 ; 8 ; 4) b) So do mol goc b^ng trung binh cua so do hai goc con lai, suy ra ba lan goc
do bang tdng ba goc cua tarn giac va bang 180".
So do goc do la: 180" : 3 = 60". Vay tarn gidc nay c6 mot goc b^ng 60". (1) Mat khac, tir Va + b - c = Va + ^/b - Vc => Va + b - c + Vc = Va + Vb
^1 ]
=> {yx + b - c + Vc) = (Va + Vb)
=>a + b - c + c + 2 Vc(a + b - c) = a + b + 2Vab
=> Vc(a + b - c) = Vab => c(a + b - c) = ab => ca + c(b - c) = ab ăc - b) + c(b - c) = 0 => (c - b)(a - c) = 0 c - b = 0
a - c = 0
c = b a = c (2) a = c (2) Tuf (1) va (2) la c6 tam giac nay la tam giac deụ
Nhanxet:
a) Day la bai loan vc cau tao so, dan den giai phiTdng irinh nghiem nguyen. Lap luan de CO b = 5 hoSc a - c = 5 hoSc c - a = 5 tir gia thuyet ^ - la rat haỵ