Bai Tap Chuyen de Toan 8

8/8/2019 Bai Tap Chuyen de Toan 8

1/16

Nguyn Ngoc ng

Chuyn : Phan tch a thc thanh nhn t

Mt s phng php phn tch a thc thanh nhn t-Nu a thc c hai nhn t ta them bt sau dng hnh ng thc 3,6,7-Nu a thc c ba hng t ta tch them bt sau dng hng ng thc 1,2-

Nu a thc c bn hng t ta nhn sau dng hng dng thc 4, 5-Nu a thc c 5 hng t tr ln thng nhm tch sau s dng hp l cc hngng thc-Nu a thc mt bin th c th nhm nghim-Nu a thc phc tp th c th i binBi 1 Rt gon :

a))2)(3(

62

+

+

xx

xb)

96

92

2

+xx

xc)

xx

x

43

1692

2

d)42

442

+++

x

xx

e)4

22

2

x

xxg)

8

12633

2

++

x

xxh*)

4 2

4

1x x

x x

+ ++

k*)5 4

3 2

1

2 2 1

x x

x x x

+ ++ + +

Bi 2 Thc hin cc php tnh saua)

62

1

+

+

x

x+

xx

x

3

322 ++

b)62

3

+x xx

x

62

62 +

c) + + + 2 22x 2x x

x 3x x 4x 3 x 1d)

1

3 5x 21 3 15

3 5 25 9

x

x x

+

Bi 3 Phn tch a thc thnh nhn ta) x2 - y2 - 2x + 2yc) 3a2 - 6ab + 3b2 - 12c2e) a2 + 2ab + b2 - ac bc

g) x2

y - x3

- 9y + 9xk) 81x2 - 6yz - 9y2 - z2m) 9x2 + 6x - 575

p) 81x4 + 4s*) (x2 + 8x + 7)(x2 + 8x +15) + 15u*) (4x + 1)(12x - 1)(3x + 2)(x + 1) - 4i*) 3 2 10 8x x x+ +

b)2x + 2y - x2 - xyd)x2 - 25 + y2 + 2xyf)x2 - 2x - 4y2 - 4y

h)x2

(x-1) + 16(1- x)l) 36(x-2)2 -49(2x+3)2n) x2 - x - 12r*) (x2 + x)2- 2(x2+ x) 15t*) (x + 2)(x + 3)(x + 4)(x + 5) 120v*) (x2 -7x + 12 )(x2 -11x +30) + 1q*) 3 24 5 6x x x+ + +

Mt s ng dng ca hng ng thc

Thng thngB i1 : Tm gi tr ln nht nh nht (nu c) caA= 162 + xx E =(x+1) 22 )2( ++ x D= 232 ++ xx

B=4x 9122 + x G =2(x-3) 22 )4( x K =-1

4x2+2x-9

C=-25x 1102 ++ x H =(2x-3) )3)(18(2 + xx

Bi 2 Cho x+y = 5 ; xy=1 Tnh


2/16

Nguyn Ngoc ng

a) xyyx 522 + g)yx

yx1133 +++ n)x yyx +

b)x(x+3y)-y(5x-y) h)

+

+

yy

xx

11 22 p)x xyy +

c)(x+7y)(y+7x) k) 3 4 3 4x x y y+ + + q) 66 yx +

d)(2x-3y)(2y-3x) l) 22 yx r) 5 5x y+

e)xy

yx 1313 +++ m) 11 +++ yx s) 7 7x y+

Bi 3 Cho x 121 +=+ mx ; x 221 =mxa)Tm min ca A=x )1()1( 21

2

2

2

2

2

1 +++ xxx b)Tm max ca B=1-x 22

2

1 x

c)Tm s p ln nht sao cho C=(x )2)(2 1221 xxx ++ pd)Tm s q nho nht sao cho D=(x qxxx )3)(3 1221B i 4 : Rt gn biu thc sau

a) (3x-1)2 + 2(3x-1)( 7-2x) +(2x-7)2

b) (8x-5)

2

-(16x-10)( 4x+3) +(4x+3)

2

c) 3.5(24+1) (28+1) (216+1) (232+1) (264+1) (24+1)d) 1002-992+982-972+962-952+...+22-12

Bi 5: Cho :1

3xx

+ = Tnh

a) 2 21

xx

+ b) 3 31

xx

+ c) 5 51

xx

+ d) 2 21

xx

Bi 6: Cho :1

4xx

= tnh

a) 2 21

xx

+ b) 3 31

xx

+ c) 7 71

xx

+ d) 3 31

xx

Chuyn


3/16

Nguyn Ngoc ng

Biu thc hu t

I-Mt s ch khi gii ton v biu thc1) Tm KX ch : Mu 0 , biu thc chia 02)Rt gn biu thc

-Nu biu thc cha cc phn thc cha rt gn th ta nn rt gn phn thc trc-Nu biu thc c mu i nhau ta nn i du trc khi

-Ngoi ra cn thc hin ng th t cc php tnh ,ch dng ngoc ,du - ,- Mt s bi ton nh : Chng minh ng thc , chng minh biu thc khng ph

thuc vo bin cng quy v Rt gn biu thc3) Tnh gi tr ca biu thc

-Cn rt gn biu thc trc.-Nu gi tr ca bin cn phc tp th ngh n vic rt gn trc khi thay vo

tnh

4) Tm bin biu thc tho mn 1 iu kin no -Cn rt gn biu thc trc-Sau khi tm c gi tr ca bin phi i chiu vi KX

II bi tp (Sau khi rt gn cc em c th t cho thm yu cu khc)

bi Kt qu

1.2 2

2x 2x xA

x 3x x 4x 3 x 1= + +

+

2.2

x 2 4xB

x 2 x 2 4 x= +

+

3.2 2 2 2

2 2 2 2

x x 2 x x 12 y 2y 15 y 4F . . .x x 6 x 3x 4 y 3y 10 y y

+ + = + +

4.4

2

2

x 1G x 1

1 x

+= + +

5.2 3

4x 3 12xH

x 2x 2 x x 4x= + +

+

6.2

3 2 2

4x 3x 17 2x 1 6xI

x 1 x x 1 x x

+ = + +

+ +

7.2

3 2 2

4x 3x 5 1 2x 6xI

x 1 x x 1 x x

+ = +

+ +

8. 2 35 10 15

Kx 1 x (x 1) x 1

= + + +

9.( ) ( ) ( ) ( ) ( ) ( )

2 2 2 2x y x yN

x y 1 y x y 1 x 1 x 1 y=

+ + + +

1.x 2

x 3

+

2.x 2

x 2

+

3. 1

4.2

2

1 x

5.1

x 2+

6.2

12

x x 1

+ +

7.3

12x

x 1

8.2

5x

x x 1 +

9. x-y+xy


4/16

Nguyn Ngoc ng

10.2

2 3

4 3 5x 2 x 2x 4T

x 2 x 2 4 x x 8

+ += + +

+ +

11.A=2

2 2 2

6x 1 6x 1 x 36

x 6x x 6x x 1

+ + + +

12.

2 2x 1 2x x 1 10 x

B . .x 10 x 2 x 10 x 2

= ++ + + +

13.Cx x 1 x x 1

:x 1 x x 1 x

+ = +

14.2

2 2

y y 3y y 3 yD

3 y 2y 3 y 3y y 9

+ += + +

15. 2 2 2x x 6 2x 6 x

E :x 36 x 6x x 6x 6 x

= + + +

16.F=

2 2

3 2

x 6 x 10 x

: x 2x 4x 6 3x x 2x x 2

+ + + + +

17.2x 2 2 2 4x x 3x 1

G 3 :3x x 1 x 1 3x

+ = + + + +

18.2

2

1 2x x 2x 24 12xH .

4 2x 3x 6 3x 12 6 13x

+ = + +

19.3 3

2 2

x x x x 1 x 1 xI :

1 x 1 x 1 x 1 x

+ + = + +

20. 3 2 25x 1 1 2x 2 2x

M :x 1 x x 1 1 x x 1

+ = + +

21. ( )2

2 2a b 4ab a b b aN

a b ab

+ =

+

22.3 2

3 2

x x 8 x 2x 4 4P . :

x 2 x 8 x 4 2 x

+ = + + +

23.2

3 2

x 2 x 1 x 1Q :

x 1 x x 1 1 x 2

+ = + +

+ +

24.2

3 2 2 3 2

x x 1 1 2xR :

x x x 1 x 1 x 1 x x x 1

+ = + + + + + +

25.( ) ( )

3 2

2

x 3x 9 x x 3 x 2S 1 :

x 9 x 3 x 2 x 2 x 3

+ = + + +

10.1

x 2+

11.12

x

12.2

x 1x 2+

13.x 1

x 1

+

14.-1

15.-1

16.1

2 x

17.x 1

3

18.2

x 2+

19.2

2

x

1 x+

20.2x 2

x

+

21.2b

22.1

x 2

+23.

2

2

x x 1+ +

24.x 1

x 1

+


5/16

Nguyn Ngoc ng

26.2

3 2

2x x 1 x 2 T : 1

x 1 x 1 x x 1

+ + = + +

27.( ) ( )22 3 2

2 3

1 x x x2x x 1 2x x xU 1 :

1 x 1 x 2x 1

+ + = + +

28.

2 2

22 34x x 2 2 3x x 4V x .x 4 2x 4 x 4x x 2+ = + +

29.2

2 2 2

2x 1 32x 1 2xY

2x x 1 4x 2x x

+ = + +

+

30.3

3 2

10 x 5x x 1A 1

x 8 x 2x 4

+ = + +

+ +

31.2

3 2

x 5x x 2 1B

x 1 x x 1 x 1

+= + +

+ + +

32. 21 2x 3x 2 3x 2

C 2x 2x 1 2x 4x

= + +

33. 2 2 2 22x y 8y 2x y

D2x xy y 4x 2x xy

+ = + +

+

34.2 2

2 3

x 1 x 3x 1 1 x 1E :

x x 1 x 1 x 1 1 x

+ + = + +

35. 2 2 2x x 4 2x 4 x

F :x 16 x 4x x 4x 4 x

= + + +

36.

2

2

x 2 2x 3x 3 4x x 7G .x 1 x 1 x x x

+ + + + = + +

37.( ) ( )

2

3 2 2

1 3 3 3x 3x 3 2x 2H :

x 1 x 1 x x 1 x 1 x 2 x 2x

+ = + + + + + + +

38. 2 22xy x y x y y

I :x y 2x 2y 2x y x

+= + + +

39.2 2

1 2 2

3x x 9 x x 3A 1 :

9 x x x 6 2 x

= +

40. =2A

+

+

+1

3

22:

9

33

33

22

2

x

x

x

x

x

x

x

x

41.2 2

3 2 2 3

x 2 x 1 1 x 2A :

x 1 x x 1 x 1 x 1

+ = + + + +

42.

++

+

++

=1

2

1

3.

1112

3

2

3

4x

x

x

x

xx

xx

x

xx

xA

25.3

x 2+

26.2

1

x 1

27.2

1

x x 1 +

28.( )

3x

2 x 2+

29.-8

30.1

x 2

31. 3x 1+

32.1

2x

33.( )

4x 2y

x 2x y

+

34.2

1

x x 1+ +

35.-1

36.x 1

x

37.1

x

38.1

39.3

x 2+

40.3

x 3

+

41.2x x 1

x 1

+


6/16

Nguyn Ngoc ng

43.x

x

xxx

xA

+

=

2

1:

2

2

6

93

2

2

5

44.xxx

x

x

xA

+

+++

++

=1

1

1

1

1

223

2

6

45.3

32

1

23

32

111527 +

+

++

=

x

x

x

x

xx

xA

46.

+

+=

2

3

3

2

6

4:

2

11

2

8x

x

x

x

x

x

xA

47.K =x

200x

1x

1x4x

1x

1x

1x

1x2

2 +

+

+

+

48.S =2 3 2 3

2 2 2 2

x x x x:

x y x y x y x y 2xy

+ +

49.T=2

1 1 2 (1 )

3 3 9

x x x x

x x x

+

+

50.A= 2 23 1 1 3( 1) 1 1x x

x x x + + + +

51.2 2

2 2

2 4 2:

2 4 2 2

+ = +

x x x xA

x x x x

52.

= + + + +

2 2 2

2 4 2 2 4 2 2

2:

( ) ( ) 2 ( )( )

y x y x y x A

xy x y x y x x y y y x x y

53.+

= + + + 2 399 1 1 20 4

:5 5 5 5 1

xB x x x x y xy

54. ( )32y 1 3 y 11 y

N y 1y 3 y 1 y 2y 3

= + + +

55. ( )32x 2 x 1 x 5

P x 1x 1 x 2 x x 2

+ + + = + + +

56.2

2

x 2 1 2x 4 2 x 2x 6Q :

x 2 2 x 4 x 2 4x 8

+ + + + = +

57.2

223x 1 x 2x 1 x 2D 1 x x 12x 1 1 4x 2+ + = + +

58.( )23 3

2 2 2

2 x 2x 1x 1 x 1E :

x x x x x 1

+ + = +

59.2

x 1 x 1 x 1F

2 2x x 1 x 1

+ = +

42.22x 4x 2

x

+ +

43.2

x 2

44.2

x

x x 1+ +

45.

46.

47.x 2003

x+

48.x y

x y

+

49.2

x 3

50.( )

2

x 3

x 1

+

51.24x

x 3

52.( )

2

x y

xy

53.-5xy

54. 27y 7y 7+ +

55. 25x 5x 5 +

56.1

2


7/16

Nguyn Ngoc ng

60.2

2 2

4x 8x x 1 2G :

2 x 4 x x 2x x

= + +

61. ( )2 22 2b a

H a b b aa ab ba b

=

62. 3 2 21 2x 2x

I : 1x 1 x x x 1 x 1

= + +

63.( ) ( )

2 2

2

a 3a 2 a a 1 1K :

a 2 a 1 a 1 a 1 a 1

+ + + = + + +

64.2

2

2x 3 3x 2 1 6 26x 4xM

2x 3 2x 3 2 9 4x

+ = +

+

65.2

2 2

32x 1 2xN 8

1 4x 2x x

=

+

66. 1 1 x 1 x 2P :x 1 x x 2 x 1+ + =

57.3x

2

58.x 1

x 1

+

59.2

1 xx

60.24x

x 3

61. 2 2b a

62.1

x 163.

2a 1

2a

+

64.( )

1

2 2x 3

65.2

2x 1

2x x

+

66.x 2

3x

67.2

2 2

2x 2x x x 2xA :

x 3x x 4x 3 x 1 x 3

+ = + + +

68. 2x 2 4x x 2

B .x 2 x 2 4 x x 3

= + + +

69.2

2

1 x 1 2x x x xC :

3 x 3 x 9 x x 3

+ = + +

70.2

2 2 2

5 4 3x 51x 15D 3 :

2x 6x x 9 x 9

= +

71.2

2 2 2 2

3x 2 6 3x 2 x 2x 1E .

x 2x 1 x 1 x 2x 1 5x 5

+ + + = + + + +

72.4

2

2 2

x 1 x 1G x 1 :

1 x x 1

+ + = + +

67.1

x

68.x 2

x 3

++

69.10

3 x

70.1

2x

71. ( )22

x 1

72.2

x 1

+


8/16

Nguyn Ngoc ng

73. 2 3 24x 3 12x x 2

H :x 2x 2 x x 4x x 2x

= + + + +

74.2

3 2 2 2

4x 3x 17 2x 1 6x x 3I :

x 1 x x 1 x x x x 1

+ = + + + + + +

75.K = 2 2 2x 9y 3y x 3y

.x 9y x 3xy x 3y

+ + +

76.2

3 2 2 2

4x 3x 5 1 2x 6x xI :

x 1 x x 1 x x x x 1

+ = + + + + +

77.2

2 3 2

5 10 15 x 2xK :

x 1 x (x 1) x 1 x x 1

= + + + +

78.2

2

6x 5x x x 2xM :

x 9 3 x x 3 x 3

+ = + +

79. 26x x 5x 20P .5x 20 x 8x 16 6x 29 = +

80.2

2 2 2

7 1 1 x 4x 3Q :

8x 18 2x 3x 4x 6 9 4x

+ = + +

81. 2 25x 2 2x 33 4x 8

R :2x 3x 2x 3 9 4x 6x 9

= + + +

82. 21 2x 2x 1 x 3

U :2x 2x 1 2x 4x x

= + +

83.2 2

2 2

x 2006x 2009 x 2008 x 3xA :x 1 x 1 x 1 x x

+ + = + +

73.x

x 2

74.12

x 3

75.1

x

76.12

1 x

77.5

x 2

78.6

x 2+

79.x

x 4

80.1

x

81.6

x 2

82.1

x 383.

1

x 3+


9/16

Nguyn Ngoc ngChuyn : Bt ng thc

Dng 1: Chng minh bt ng thc

BT1: CMR vi mi a; b dng, ta c: 2+a

b

b

a. Khi no xy ra ng thc?

BT2: CMR vi mi a; b; c; d dng , ta c: abcddcba

+++4

4. Khi no xy ra ng thc?

BT3 CMR vi mi a; b; c; d dng , ta c: 161111

)(

++++++

dcbadcba

BT4 CMR vi mi a; b; c dng , ta c: 9111

)(

++++

cbacba

BT5 CMR vi mi a; b; c dng , ta c:2

3

++

++

+ bac

ca

b

cb

a

BT6 CMR nu a, b, c l i di 3 cnh ca mt tam gic, ta c:a) ab + bc + ca cabcabcba 222222 +++++BT7 Cho hai s x ; y tho mn ng thc: 3x + y = 1

CMR:10

122 + yx

BT8 Cho hai s x ; y tho mn ng thc: 4x + 3y 1

CMR:5

194

22 + yx

BT9 Cho hai s a ; b tho mn ng thc: a + b = 1 . CMR:

a)2

122 +ba b)8

144 +ba c)2

25111222

++

++

+

cc

bb

aa

BT10 Cho 0> dcba . CMR: dcbaadbccbda ++++++2222

Dng 2: S dng hng BT chng minh BT

BT1 : CMR: vi mi a > 0, b > 0, ta c: abba

+2

(BT C-si)BT2: CMR: vi mi a > 0, b > 0, ta c:

a) 2+a

b

b

ab) 4

11)(

++

baba

BT3 CMR: vi mi a > 0, b > 0, c > 0 ta c:

9111

)(

++++

cbacba

BT4 CMR: vi mi a > 0, b > 0, c > 0 ta c:

2

3

++

++

+ acb

cb

a

ba

c

( BT Nes bit)HD: p dng BT BT3, ta c:


10/16

Nguyn Ngoc ng

[ ]

2

3

2

9111

9111

)(29111

)()()(

+

++

++

+

+++

+++

+

++

++

+++

++

++

++++++

ac

b

cb

a

ba

c

ac

b

cb

a

ba

c

accbbacba

accbbaaccbba

BT5 CMR: vi mi a, b, c, d ta c:))((

2222 dcbabdac +++

(BT Bu-nhi-a-cp-ski )BT6 CMR: vi a, b, c, d R v c > 0, d > 0 ta c:

dc

ba

d

b

c

a

++

+222 )(

BT7 Chng minh rngVi mi s thc a + b 0 v m, n nguyn dng, ta c:

22

.2

nmnmnnmmbababa

++ +

++

HD: )(2))(( nmnmnnmm bababa ++ +++ 0+ ++ nmnmnmnm bababa 0)()( + nnmnnm abbbaa 0))( nnmm baba

Do a, b c vai tr nh nhau, khng mt tnh tng qut, gi s ba (1)Theo bi: a + b 0 a - b (2)T (1) v (2): 0ba

Ta suy ra:

0

0

nn

mm

nn

mm

nn

mm

ba

ba

ba

ba

ba

ba

0))(( nnmm baba , BT c chng minh.BT8 Cho a + b 0 . Chng minh rng:

(a + b)(a3 + b3)(a5 + b5) 4(a9 + b9)HD:Theo bi: a + b 0 , p dng BT BT7:

Ta c:

+

++

+++

22.

2

22.

2

995544

4433

bababa

bababa

2

.22

.2

.2

99445533bababababa ++

+++

22.

2.

2

995533babababa +

+++

(a + b)(a3 + b3)(a5 + b5) 4(a9 + b9)


11/16

Nguyn Ngoc ngBT9: Cho a, b, c, d, e l cc s thc. Chng minh rng: )(22222 edcbaedcba +++++++BT10: Cho a, b, c l cc s thc. Chng minh rng: )(444 cbaabccba ++++BT11: Chng minh rng: nu ad bc = 1 th 32222 +++++ bdacdcba

BT12: Cho a > 1, b > 1. Chng minh rng 811

22

+

a

b

b

a

BT13: Cho 1,1 0, b > 0. Chng minh rng: 2)(41

baab +

BT15: Cho a, b, c, d l cc s dng. Chng minh rng: 222

)(.3

dcba

cbcada

ca

c

cb

a

++++++

>+

++

BT16: Cho a, b, c, d l cc s dng. Chng minh rng:

+++

++++

+++

+++

+

+

+

+

+

+

+2

22

2

22

)()(

.4

dcba

dcdabb

dcba

cbcada

ba

d

ad

c

dc

b

cb

a

BT17: Vi mi a, b. Chng minh rng:a) )(4)( 333 baba ++ b) )(3)()(3 2222 cbacbacabcab ++++++

BT18: Vi mi a, b, c, d. Chng minh rng:a) cabcabcba ++++ 222 b) abcddcba 44444 +++

BT19: CMR: a)Vi a, b, c l cc s dng, ta c: a) (a + b)(b + c)(c + a) 8abcb) Vi mi a, b, c ta c: )(4)()( 22 cbaabccbba ++++

BT20: Cho a + b = 1. Chng minh rng:

a)2

122 + ba b)8

144 + ba c)128

188 + ba

BT21: a) Cho a + b + c + d = 2. Chng minh rng: 12222 +++ dcbab) Cho a, b, c > 0 v a + b + c = 1. Chng minh rng: b + c 16abc

c) Cho a, b > 0 v a + b = 1. Chmg minh rng: 5,1211

22

++

+

bb

aa

Dng 3. S dng BT tm GTLN, GTNN ca biu thc i s.* Phng php:- Ta a cc biu thc i s cn tm GTLN, GTNN v mt trong 2 trng hp:+ TH1: A2 + k k, (gi tr nh nht l k).+ TH2: - A2 + k k, (gi tr ln nht l k).- Tm gi tr ca bin (nu c) ng thc xy ra.BT1: Tm gi tr nh nht ca biu thc

P = (x + 1)2 + (x - 3)2HD: Ta c: P = (x2 + 2x + 1) + (x2 6x + 9)

= 2x2 4x + 10= 2(x2 2x + 1) + 8= 2(x 1)2 + 8

V (x 1)2 0 vi mi gi tr ca x . P = 2(x 1)2 + 8 8. ng thc xy ra x 1 = 0 x = 1.Vy gi tr nh nht ca P = 8 x = 1.BT2: Tm gi tr nh nht ca biu thc


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Nguyn Ngoc ngP = x2 + 6y2 + 14z2 8yz + 6zx 4xy

HD: P 222 3)(2)32( zzyzyx ++++= P 0 vi mi gi tr ca x, y, zng thc xy ra x = y = z = 0. Vy GTNN ca P = 0 x = y = z = 0BT3: Tm gi tr nh nht ca biu thc

Q = x2 + 2y2 + 3z2 2xy + 2zx 2x 2y 8z + 2007HD: 2001)1()2()1( 222 +++++= zzyzyxTa c: (x y + z 1)2 0, (y + z 2)2 0, (z 1)2 0 vi mi x, y, z. Q 2001. ng thc xy ra x = y = 1.Vy GTNN ca Q = 2001 x = y = 1.

BT4: Tm gi tr ln nht ca biu thcE = xy + yz + zx, bit x + y + z = 3

BT5: Tm gi tr nh nht ca biu thcP = (x + 2005)2 + (y + 2006)2 + (z + 2007)2

BT6: Tm gi tr nh nht ca biu thcQ = (x + a)2 + (y + b)2 + (z + c)2, vi a, b, c l cc hng s.

BT7: Cho x, y l hai s thay i lun tho mn iu kin x + y = 1.Tm GTNN ca biu thc A = x3 + y3 + x2 + y2

BT8: Cho x, y l hai s thay i lun tho mn iu kin x + 2y = 3.Tm GTNN ca biu thc B = x2 + 2y2

BT9: Cho a, b, c, d l cc s thc tho mn a + b = c + d .Tm GTNN ca biu thc C = (x + a)(x + b)(x + c)(x + d)

BT10: Tm GTNN ca biu thc D = 2200720072

22

2

11 )(...)()( axaxax ++++++

bit 200721 ... xxx +++ = 2007 v 200721 ,...,, aaa l cc hng s.BT11: Tm GTNN ca biu thc E = (x + a)2007 + (y + b)2007 + (z + c)2007

bit x + y + z = 6021 v a, b, c l cc hng s.

BT12: Tm GTLN ca biu thcG = 2)2007( +x

x, vi x > 0.

BT13: Tm GTLN ca biu thcH = 2222

yxyx

yxyx

+++ vi x > 0, y > 0.

BT14:Cho x, y > 0 v x + y = 5. Tm GTNN ca biu thc: A =yx

11+

Dng 4. S dng bt ng thc tm cc tr trong hnh hc

BT1: Cho ABC. Qua mt im M bt k thuc cnh AC, k cc ng thng song song vi haicnh kia, chng to thnh vi hai cnh y mt hnh bnh hnh. Tm v tr ca M hnh bnh

hnh y c din tch ln nht.HD:


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Nguyn Ngoc ng

S '

x

y

S2

S1H

B

C

a

M

FK

Gi hbh to thnh l BEMF, din tch (BEMF) = S , din tch (ABC) = S. Ta cn tmGTLN ca S. Ta k AK BC, AK ct EM H. Ta c:

S = EM . HK, S =2

1BC . AK, nn:

AK

KH

BC

EM

S

S..2

'=

t MA = x, MC = y. Mt khc ta c:yx

y

AK

HK

yx

x

BC

EM

+=

+= ; (nh l Talet)

2')(

2

yx

xy

S

S

+= . p dng BT ab

ba

+2

2hay (a + b)2 4ab

2

1

)(

22'

+

=yx

xy

S

S. Vy GTLN ca S =

2

1S. ng thc xy ra x = y hay khi M l trung

im ca AC.

BT2: Cho hbh BEMF. Dng ng thng i qua M ct cc cnh ca gc B to thnh mt tam gicc din tch nh nht.

HD: Xt 22

)(

'

2

+

=xy

yx

S

S

BT3: Trong hnh thang ABCD (BC // AD) c din tch S khng i, E l giao im ca cc -ng cho, hnh thang c thm iu kin g th ABE c din tch ln nht.

HD:

S ' S '

S 1

x

xy

S 2

E

A D

B C

K

Ta c: dt(ABE) = dt(CDE) = S. t dt(CEB) = S1, dt(AED) = S2. Trc ht ta CM: 212

.' SSS = .

Tht vy: 212

2

1

2

1 .''

'

';

'SSS

S

S

S

S

EA

EC

S

S

EA

EC

S

S==== (1)

t BC = x; AD = y, ta biu th cc t sS

S

S

S

S

S ';; 21 theo x v y.

Qua C k ng thng song song vi BD, ct AD K. Ta c DK = BC = x, dt(ACK) = S.


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Nguyn Ngoc ngb.CH.CE = CK.CB

Bi 3. Cho hnh thang cn MNPQ (MN //PQ, MN < PQ), NP = 15cm, ng cao NI = 12cm, QI =16 cm.

a) Tnh IP.

b) Chng minh: QN NP.

c) Tnh din tch hnh thang MNPQ.d) Gi E l trung im ca PQ. ng thng vung gc vi EN ti N ct ng thng PQ ti K.Chng minh: KN2 = KP . KQ

Bi4. Cho tam gic ABC vung to A; AB = 15cm, AC = 20cm, ng cao AH.

a) Chng minh: HBA ng dng vi ABC.b) Tnh BC, AH.

c) Gi D l im i xng vi B qua H. V hnh bnh hnh ADCE. T gic ABCE l hnh g? Tisao?

d) Tnh AE.

e) Tnh din tch t gic ABCE.Bi5.Cho tam gic ABC vung ti A (AB < AC), ng cao AH. T B k tia Bx AB, tia Bx ct tiaAH ti K.

a) T gic ABKC l hnh g ? Ti sao?

b) Chng minh: ABK ng dng vi CHA. T suy ra: AB . AC = AK . CHc) Chng minh: AH2 = HB . HC

d) Gi s BH = 9cm, HC = 16cm. Tnh AB, AH.

Bi6.Cho tam gic ABC c ba gc nhn. ng cao AF, BE ct nhau ti H. T A k tia Ax vung gcvi AC, t B k tia By vung gc vi BC. Tia Ax v By ct nhau ti K.a) T gic AHBK l hnh g? Ti sao?

b) Chng minh: HAE ng dng vi HBF.c) Chng minh: CE . CA = CF . CB

d) ABC cn thm iu kin g t gic AHBK l hnh thoi.Bi 7Cho tam gic ABC, AB = 4cm, AC = 5cm. T trung im M ca AB v mt tia Mx ct AC tiN sao cho gcAMN = gcACB.

a) Chng minh: ABC ng dng vi ANM.b) Tnh NC.

c) T C k mt ng thng song song vi AB ct MN ti K. Tnh t sMK

MN.

Bi8.Cho ABC c AB = 4cm, AC = 5cm, BC = 6cm. Trn tia i ca tia AB ly im D sao choAD = 5cm.

a) Chng minh: ABC ng dng vi CBD.b) Tnh CD.

c) Chng minh: gcBAC = 2.gcACD

Bi9.Cho tam gic vung ABC (gcA = 90o), ng cao AH.

Bit BH = 4cm, CH = 9cm.

a) Chng minh: AB2 = BH . BC


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Nguyn Ngoc ngb) Tnh AB, AC.

c) ng phn gic BD ct AH ti E (D AC). TnhDBA

EBH

S

Sv chng minh:

DA

DC

EH

EA= .

Bi 10.Cho hnh bnh hnh ABCD. Trn cnh BC ly im F. Tia AF ct BD v DC ln lt E vG. Chng minh:

a) BEF ng dng vi DEA. DGE ng dng vi BAE.b) AE2 = EF . EGc) BF . DG khng i khi F thay i trn cnh BC.

Bi11.Cho ABC, v ng thng song song vi BC ct AB D v ct AC E. Qua C ktia Cx song song vi AB ct DE G.

a) Chng minh: ABC ng dng vi CEG.b) Chng minh: DA . EG = DB . DEc) Gi H l giao im ca AC v BG. Chng minh: HC2 = HE . HA

Bi12.Cho ABC cn ti A (gc A < 90o

). Cc ng cao AD v CE ct nhau ti H.a) Chng minh: BEC ng dng vi BDA.

b) Chng minh: DHC ng dng vi DCA. T suy ra: DC2 = DH . DAc) Cho AB = 10cm, AE = 8cm. Tnh EC, HC.

Bai Tap Chuyen de Toan 8

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