Download - Iranian 9th grade coursebook : Math
-
7/23/2019 Iranian 9th grade coursebook : Math
1/151
-
7/23/2019 Iranian 9th grade coursebook : Math
2/151
: :
:
:
:
: (
)
:
: : :
www.chap.sch.ir : :
: :
: :
:
:
:
: :
) ) : : : : :
: :
.
ISBN 978 -964 - 05 - 2435-0 9
-964-05 - 2435-0
-
7/23/2019 Iranian 9th grade coursebook : Math
3/151
.
.
.
()
-
7/23/2019 Iranian 9th grade coursebook : Math
4/151
1 ........................................................
2 ............................................. :
6 .................. :
11 ................. :
15 ..................................
:
18...............................................
19 ...............................................
:
23 ........................................... :
28 .............................:
32 .............................
33 .....................................................:
37 ................................:
:........................................ 44
49 .................................. :
53 ........................................ :
-
7/23/2019 Iranian 9th grade coursebook : Math
5/151
59 .......................................... ..........
60.........................................................:
65..........................................................:
68........................................................:
73....................................
:
78 ............................................
79..............................
:
86........................... :
90........................................ :
95 ....................................
96.......................................................... :
102.................................... :
108.................................. :
-
7/23/2019 Iranian 9th grade coursebook : Math
6/151
............................................113
:.....................114
119.................................. :
:.......................................126
130..............................................
131...........................................
:
135........................................... :
140................................................... :
-
7/23/2019 Iranian 9th grade coursebook : Math
7/151
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
-
7/23/2019 Iranian 9th grade coursebook : Math
8/151
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.1
.
.
www.mathrde.ir
-
7/23/2019 Iranian 9th grade coursebook : Math
9/151
.
.
.
) (
...
( )
-
7/23/2019 Iranian 9th grade coursebook : Math
10/151
.
.
{A = {2,3,5 B A
A
.
A
.
B
.
C
D
.
60
3
{4,6,10} :{6,10,12} :
:
.
{.{3,4
{3,3,4}
. ) ( ) (
:
1 60
15 4
12 30
206
10
2
3
5
-
7/23/2019 Iranian 9th grade coursebook : Math
11/151
.
12
)
)
)
:
A = {__,__,__,__}
A = {
,2,1,4,5}
)
________
B = {
,
,5,7}
____
)
.
A = {a,b,5,7} A
A a aA A a
.
A
4A
A
:
A = {,2,,4}
.
1 2
4 3
A
B
A
1
.
B
A
:B = {,,,} {,,,4,,2} = A 2
A
.
B
. E
A B
ab
c
fs
k
m
n
-
7/23/2019 Iranian 9th grade coursebook : Math
12/151
.
( .
).
.)
(
)
)
24 ( ( )
) (
:
4
A = {
,
}
)1
B = {4,8,12,} {3,4,5,6,7,8,9} )2
C :
b
a
-
-
)3
D = {5}
)4
E ={ } )5
F= {2,4,6,8} )6
G:
{3,a,b})7
H={2,3,5,7} {6,4,2,8})8
{0}
{} .{}
.
-
)
)
.) (
4
-
7/23/2019 Iranian 9th grade coursebook : Math
13/151
: 5
)
)
.
)
2x+8=1 )
)
)
:
A = {1,8,27,64,125} )
C = {10} )
)
)
(.
.
)
.
._______
20
1
5
)
.
________
{2,3,4,...,9}
)
.
________
A = {0, }
)
_______
A
:
A = {3,5,7,9,11}
)
.________ A
.2 3
-
7/23/2019 Iranian 9th grade coursebook : Math
14/151
.
A
:
10 12
4 2
B
B
A
.
A
B
B
A
B A
A B B A
. A = B
B
A
.
A
:
A
A
)
)
.)
{7,8,9}
{8,9,10}
.
A B B A
.A B B A
:
, , , , , , , ,( )
=
2
2 9 2 1445 4 3 25
5 3 5 2
)
____
-
7/23/2019 Iranian 9th grade coursebook : Math
15/151
D
:
D
D
.B
D
B
.
B
D = C C D
B
D
B
. B D
D
B
D
D
C
.AA :
A
D
{10,4,-6,2} D
.B AA B AB
A
.A
A
, , , , , / , , / , , ,
=
4 4 1 2 2 57 0 625 0 5 2
10 9 2 3 5 8)
.
.
B
A
____
-
7/23/2019 Iranian 9th grade coursebook : Math
16/151
.:
d {a,b,d} {a,b,c,e} )
.
{-1,0,1,3}{4,3,0,1,-1,2} )
.
A
A B
)
B
B A
B
A.
12 a
b
A
B
:
, C A , B A , A C
A
B , B
C ,
A
C
B
A
:)
(
A = {1,3,6,4} , B = {5,1,3} , C = {2, 5,1,3,6}
B A , 3 B , A B , B C , A C , 2 A
{1,4}A , 6 A , {5,6}C , 5 C , 0A
:
A = {a,b,c}
:
, {a},{b},{c},{a,b},{a,c},{b,c},{a,b,c}
:
{a,b,c,d} ) .)
.
A
B
C
-
7/23/2019 Iranian 9th grade coursebook : Math
17/151
:
{ }, , , , , . . .= 1 2 3 4 5
.
:
.
E = {2,4,6, 8 ,...}
2k
{ }E k k= 2 : k k 2k E
.
E
.
:
{ }O k k= 2 1 ):
{ }A x x= <
-
7/23/2019 Iranian 9th grade coursebook : Math
18/151
:
{ }A x | x , x=
-
7/23/2019 Iranian 9th grade coursebook : Math
19/151
.
.
F V (
.
. )
. (
{ }B x | x= 2 3
{ }A x | x= 6
:
( A { }= ( B { }=
) B A { }=
AB(. B A )
) B A { }=
.)
AB
B
A
(
B A :
A
AB . B
.
.
{ }A B x | x A, x B=
B A :
AB .B A
.
:
{ }A B x | x A x B=
:
A B
A B
A B
A B
-
7/23/2019 Iranian 9th grade coursebook : Math
20/151
B A :
:
AB
AB
{ }A , , , , ,= 1 2 3 4 5 8 { }B , , , ,= 3 4 5 6 7
{ }A B , ,= 3 4 5 , { }A B , , , , , , ,= 1 2 3 4 5 8 6 7
A B
1 2
8
6
7
3
4
5
.
{ }A B b,e= { }A B a, b, c,d,e=
B
A
:
.
)
.
A B
a
d
cb
e
A B
a
d
c b
e
A B
a
d
c b
e
A B
a
d
cb
e
)
.
(
:
V - F
F
V
.)
{ }V F = { }F V =
-
7/23/2019 Iranian 9th grade coursebook : Math
21/151
( AC ( BC ( C)A B(
( )A B(C ( 2)A B( )( 4)AB
( A B = A ( 5)A B( ( 4)A B(
B
A
:B A .
.
)
. (
.
)W - ( ) - ( ) - (
n)A( A :
.n)A( = k
k
A
.n)A( = 4
{ }A , , ,= 2 4 6 7
A B
3 4 5
2
1
C
)B
A( A - B
:
.
B
A
:
B - A
A - B
{ }A B x | x A,x B = A B
A B B A
:
{ }B c,d,k, f , s, t=
{ }A a,b,c,d,e,k =
:
{ }A B a,b,e = { }B A f ,s, t =
-
7/23/2019 Iranian 9th grade coursebook : Math
22/151
A B
1
2 3
4
59
6
7
{ }C , , , ,= 1 7 8 1011
{ }B , , , ,= 1 5 7 3 9
{ }A , , , ,= 2 4 6 8 9
:
( AB ( BC ( A C ( AB
( A - B
( C - B
()A - C( )B - C(
()AB( - C
( AA
( A
( BB
( C
:
)A - B( )AB( = A ) B-A={6,7} )
{ }(A B) (B A) , , = 12 6 (
n )A B( = 8 )
n )A - B( = n )B - A( )
A - B = B - A )
:
)
A )2 B )
)AB( )
)
.
______
)
.______B A )
.B A ______B A )
.
______
A - B
)
.
______
)AB( ))B - A )
.
A B A B A C
B AA B (A C) C
-
7/23/2019 Iranian 9th grade coursebook : Math
23/151
:
=___________________
.
A S
. n(A)
P(A)n(S)
= P)A( A
: :
.)
.
)
. (
. (
:
A
)
:
{ } { }A , , S , , , , , ; n(A) , n(S)= = = =3 6 1 2 3 4 5 6 2 6
n(A)P(A)
n(S)= = =
2 1
6 3
B : { }; B , , ; n(B)= =2 3 5 3 (
n(B)P(B)
n(S)
= = =3 1
6 2
C :
; C n( ) 0= = (
P(C) P( )= = =0
06
D :
{ }; D , , , , , S= =1 2 3 4 5 6 (
n(S)P(D) P(S)
n(S)= = = =
61
6
:
-
7/23/2019 Iranian 9th grade coursebook : Math
24/151
S .S
:
: )
{ }A ,= 3 1 )( )(
{ }B ,= 1 2
{ }C ,= 2 3
{ }D = 2
.
. S (
.
.
.S (
1
2
3
.
10987654321
A . { }S , , ,= 1 2 10 (
A .
.
.4
10 )
B C B (
C B . C
-
7/23/2019 Iranian 9th grade coursebook : Math
25/151
:
.
)
.
)
.
)
.
)
.)),,( )
)
(
.
:
.
)
.
)
.
)
: )(
)n)s( = 36 S )
.
)
.
)
.
)
.
)
)(
.
!
) )
.
:
.
-
7/23/2019 Iranian 9th grade coursebook : Math
26/151
.
.
:
.
) (
( ( .
-
7/23/2019 Iranian 9th grade coursebook : Math
27/151
.
1
:
1
3
}xN|x3{
}3 , 4 , 5 , {
2
}xw|x2{
}________{
3-3
}xZ|_____{
}________{
4
-1
}________{
}________{
x3
1 , 2 , 3 , 4 , 5
. 1
3
1
2 2
.
1
3< ? ( ab , a + b
( a > , b
( ab , a + b >
.
:
) a ( |a| = -a
( a , b
( |a| = a
) a > ( |a + b| = a + b
) a > , b > ( |a + b| = -)a + b(
:
.
)
. )
:
/2 1 4 /3 1 7 /5 2 2 /6 2 4 / /7 2 6 8 2 8
-
7/23/2019 Iranian 9th grade coursebook : Math
38/151
:
| | ( ) = = + = 1 2 1 2 1 2 2 1
:
1 2
/2 1 4 :
1( | | =2 3 :
2( | | =7 8 :
3( | |2 5 5 :
4( | | =4 3 :
:
|a + b + c|
c = -3
b= 2
a =1
2
:
| a b c | | ( ) | | / |+ + = + + = +1
2 3 2 5 22
( / )2 1 4 2 5 2+ /
. / 2 5 2 -) / +2 5 2(
| | | | ( ) ( ) + = 3 5 2 5 3 5 2 5 :
= + + =3 5 2 5 5
:
a ( ) ( ) ( ) ( )
2 2 2 2 2 2 2 23 3 6 6 7 127 325
3
a | a |=2 :
: ( ) 21 3 :
( ) | | ( ) = = = +21 3 1 3 1 3 1 3
-
7/23/2019 Iranian 9th grade coursebook : Math
39/151
:
( |)-7(2| |-7|2
|( |-8 + 5| |-8| + |5
|( |3-9| |3| - |9
: 2
|| = | |4
3=
|73- 74 |= |/
- /26 |=
:
3
( ( ) =22595 ) ( )21394 =
) ( ) + =23 10 )
( ) =22 5
: c = 122
, b= 1
4, a =/25
|a+b | + 2|a-b -c|
: 2
)| |3 5 )
| |7 5 3 )
| |+0 5
: 3
|5-12| 1 +
.
a=2
a= a=-2
|a|+a
4
|a|+a a
. a a=2 5
( ) 22 1 ( ) 21 10
:
6
-
7/23/2019 Iranian 9th grade coursebook : Math
40/151
.
!
( ) !
-
7/23/2019 Iranian 9th grade coursebook : Math
41/151
:
:.1
. :
. :
8. 4 2
.
6
.
213
..
.
.
. .
1 1
.
: 2
:
-
7/23/2019 Iranian 9th grade coursebook : Math
42/151
.
1
.)
.)
.
B
A
)(
3
.
A
B
-
7/23/2019 Iranian 9th grade coursebook : Math
43/151
)
(
..
:1
2.
.100
.
:
.90
.
.
:
.
.
3.
.)
.
)
-
7/23/2019 Iranian 9th grade coursebook : Math
44/151
.
. )
)
.
: 4
. .
:
.
:
.
.
-
7/23/2019 Iranian 9th grade coursebook : Math
45/151
.
)(
.)(
:
:
1
:
:
.
:
.:
________.:
:
.
__________
_______
.________
2
.
:
-
7/23/2019 Iranian 9th grade coursebook : Math
46/151
.ABCD :
. :
A
= ____=____=____ =90
: AB=____ , AD =____ ____=:AC
AB||____ , AD||____
BA
D C
O
:
1.
.
A
B C
D
E F
=: = :
=
2
.
EFGH ABCD 3
.EFGH ABCD
-
7/23/2019 Iranian 9th grade coursebook : Math
47/151
1
:
.A AD ABC
:AD
:
:
A A
=1 2 :
A
AD
:
ADC ADB AD D D
=1 2
BD=DC :. )(
B ABC.
.
A
2
C. A ABCD AC
ADC ABC
AC C C
=1 2 A A
=1 2
.
31
2
.
A B
D C
1
1
2
2
A
BD
C
1
1
2
2
-
7/23/2019 Iranian 9th grade coursebook : Math
48/151
AB P 4
.
)(BHP AHP
.
PB
PA
AB P
.AB
P
HA B
:
.180 :
:
.
180
60
:
180
.180
:
ABC
:
A B C A A A
+ + = + + =1 1 3 2 180
A
BC
1 2
3
-
7/23/2019 Iranian 9th grade coursebook : Math
49/151
:
.
d
a
b
12
3 4
56
78
A
B C
1
2
3
d ABC
. BC d A
A
AB A A23 2 1
C A
= 3 AC B A
= 1
A B C A A A
+ + = + + =2 1 3 180 : C
B
A
3 A
1
:
d| |BCB A
ABA B C A A A
d| |BCC A
AC
=
+ + = + + =
=
1
2 1 3
3
180
.
. :
.5000 5000
.
-
7/23/2019 Iranian 9th grade coursebook : Math
50/151
. :
: O
3 O
1
O OO O O O O O
O O
+ =
+ = + =+ =
1 2
1 2 3 2 1 3
3 2
180
180
1
2 3
O
.
:
.
.
ABC
:
180
60 C
B
A
1
A A A A
+ = = = =1 2 2 1180 180 180 60 120
B C
+ = + =60 60 120 A B C
= +2
.
2 :
.
.
.
C
A
B
1
2
-
7/23/2019 Iranian 9th grade coursebook : Math
51/151
Q P :
.
S T :
.
N M :
.
.3
.
.
ABCD )
.ABCD
. . ABCD (
.ABCD
. .
ABCD )
.
ABCD
. 4
:
.
:
.
.
N M
QP
TS
-
7/23/2019 Iranian 9th grade coursebook : Math
52/151
.
:
)
(
AB A B
AC A C ABC A B C
A A
=
=
=
.
A
C
B
A
C
B
.
1
:
45
60
75
4
44 3 4
?
?
?
40
40
40
??
3
3
.
2(
.)
:
60
?
b
13
8
?
?
a
138
60
b
a
1513
8
120
13
7
8
-
7/23/2019 Iranian 9th grade coursebook : Math
53/151
.
:
.
.
:)( OC=OD OA=OB :
)( O
2 O
1 .AD=BC
OBC OAD
:
OA OB
OC OD OBC OAD AD BC
O O
=
= =
= 1 2
O
A C
D B
2 1
.
)A
(
.
10 )B
(
C
D )BC=6( 6
.
.AC=BD
.
m10A B
CD
D C
BA
-
7/23/2019 Iranian 9th grade coursebook : Math
54/151
BD AC . D
C
1
AC=BD :
.AC=BD
ADB
ACB
N ABCDM
ADM ABN
.CB CD
:
1
AD = AB =____=____ , BN =____
A
=____, B
=____ , DM =____
: ADM ABN
ADM
ABN
)1(
2.
. ADM ABN
A
B
C
M N
D
.
.
AB
CD
-
7/23/2019 Iranian 9th grade coursebook : Math
55/151
.
:
!
:
.
:
.
.
A
B
CD
1
1
2
2
:
AB || CD, B D
BD
BD
=
=
=
1 1
)(
)(
. CBD ABD
AD =____ :
B D
=1 1
AB =____ : B D
=2 2
( )
)
(
)(
AC
_____
___
__ __
_____
____ ____
-
7/23/2019 Iranian 9th grade coursebook : Math
56/151
.
.
___________
1
.
.OB = OD
OA = OC :
( .
2
!(
.AM ABC 3
AM
AMC
AMB
BC AM A
MBMA M 4
.
( . :
).B A M
d
5
d B A AB
.OH=OH
)AH=BH
(
:OH=OH
OA OB
O O OAH OBH OH OHAH BH
=
= ==
1 2
)(
)(
)(
)(
A
O
B
CD
A
B CM
M
A
O
B
A
H
O
d
B1
2
H
-
7/23/2019 Iranian 9th grade coursebook : Math
57/151
.
.
.
B
A
:
.
A .
.
B
B
.AH=BH
:
. 1
.
. 2
.
A
M
d
B
H
H
-
7/23/2019 Iranian 9th grade coursebook : Math
58/151
)( )( 3
BH
AH MA=MB AB M .
AH=BH
:d
MA=MB , H H
= = 90
AH=BH
. 4
.
.
:
)( ( (MA MB
H H AH BH
( )
=
= = ==
90
.
CD
AB
. CD AB 1
. CD AB
.CD AB
.
3
.
.
O
B
A
D
C
O
B
A
D
C
____ __ __
___ ___
O
B
A
D
C
-
7/23/2019 Iranian 9th grade coursebook : Math
59/151
HO
B
A
D
C
H
AO
B
C
D
OA OD( )
H H OAH AH
( )OH OH AH AB CD
= =
= = = = = =
2 2
AB=CD
AB CD= 1
: 2
AB CD
BC BC
AB BC CD BC
=
=
=+ = +
AC=BD 3
B
A D
C
ABCD
Q
P
N
M
MN=PQ :
AD BC O 2
.
BC
AD
A M
NQ
B
CPD
___
___ ___
______
___
_____
CD AB 4
D A .)OH=OH
(
CD
AB
:
-
7/23/2019 Iranian 9th grade coursebook : Math
60/151
N M ABC 3
.BM=NC BC
.
AMN
AF
BE
ABCD
4
B1
A1
AF. BE
5
:
.MB=MC
A
B CM N
11
D F E C
A B
A
B CD
M
A B
CD
1
1
H
H
ABCD
BD C A CH
AH
. D1
B1 .
BCH
ADH
CH
AH
:
.______
______
-
7/23/2019 Iranian 9th grade coursebook : Math
61/151
.
:
.
.
-
7/23/2019 Iranian 9th grade coursebook : Math
62/151
: 1
:
))1
))2
.
100,000 1 . 2
100,000.
.
)1(
)(
-
7/23/2019 Iranian 9th grade coursebook : Math
63/151
. %50 . 3
.
)
).
.
. 1
-
7/23/2019 Iranian 9th grade coursebook : Math
64/151
__2
1
.
.
..
.
A
D E
F
B C
86
10
5
4 3
2
1 2
3
-
7/23/2019 Iranian 9th grade coursebook : Math
65/151
: 2
A = 12
B =
10
C = 30 ABC
A =
3
2 B
=
3
6 C
=
7
2ABC
.
. 1
A B C
ED
F
HG
LK
JI
MN
O P
-
7/23/2019 Iranian 9th grade coursebook : Math
66/151
2
3
.
3/5
.
1:200
4
5
6
x+7 10 x-1 DEF8 5 ABC4 7
.
x
)
(
ABC 8
A
B C
A
B C
-
7/23/2019 Iranian 9th grade coursebook : Math
67/151
/1019 ,000,000,000,000,000,000
.
.
.
( )
-
7/23/2019 Iranian 9th grade coursebook : Math
68/151
:
= )-5(=5
=
43 81
4 256
=
51 1
2 32
.a=
a
-
)
(
:
:
:
=5
1 1
32 2
=4
1 1
16 2
=3
1 1
8 2=
2
1 1
4 2
1
2124816
222222021222324
(
)
)
.)
: (
-= -= -5=
: n a
n
na
a
= 1
a , nN
:
) = =22
1 17
497
)
= = =
4
4
2 1 1 81
163 162
813
(
= = =
2
21 1 1 25
15 1
255
( ( )( )
= =
3
3
1 1282
-
7/23/2019 Iranian 9th grade coursebook : Math
69/151
-
7/23/2019 Iranian 9th grade coursebook : Math
70/151
am* an= am+n: a n m
.
= = = =6
4 6 6 6 4 2
4 4
1 33 3 3 3 3
3 3
+ = = = =
5 2 7
5 2 5 2 71 1 1 12 2 2
2 2 2 2
( ) ( ) ( )
= = =
3 5 8
3 5 81 1 12 2 2
2 2 2
:
53* 5-7=
)
(
a
n
m
:
am* an= am+n
:
23*2-5*2-4=23-5-4=2-6
)2x-1(*)3x4(*)4x3(=24x-1+4+3 =24x6 )x0(
)b , x, y0( :
5-7*510= )-4(-9*)-4(-1=
=
4 93 3
8 8( ) ( ) =4 22 2
b-2*b-3=x x
y y
7 11
-
7/23/2019 Iranian 9th grade coursebook : Math
71/151
.
.
(
2
1
9
3
6
(
1
1
33
1
3
(
1
1 1
1
12
3 4
7
)
+
+1 1
1
1 1
3 4
3 4
7
(
2
2
5
5
1
25
) ( )
2
3
3
2
8
: n m b a
mm n m n
n
aa a a
a
= = ; mm
aa
= 1
;mm
m
a a
bb
=
;
)am(n=amn ; )ab(m= am.bm ; a=
)
=3
5
7
7
=( -*5-
)
=
3
3212
3
)
=
12
2
5
(
= =
8 10 8 10
4 6 4 6
2 5 2 5
2 5 2 5
)
( )x .y .z x .y .z
x .y .z
= =5 2
5 2 2 7 1 3
2 7 3 x,y, z
-
7/23/2019 Iranian 9th grade coursebook : Math
72/151
.-
.
:
=
> ) = 2 2
66
( -=- ( -> -
.
)
10
3127
3 ( )/(-* 5-
(
4 415 45
14 28
( )-5-(-
.
-
5-
21
2
)-(
)-(
-
--
.
)/(> )/(> )/(
25
4 > )/(
23
4< )/5(
.
)
3 3
5 8
2 8
3 3
2 2
)
12
2
3
-
7/23/2019 Iranian 9th grade coursebook : Math
73/151
.
105
105
104
104
1000
1000
100
100
10
10
/
2
9/3
. 300000000 2
.
.
= 600 0000000060000 =
= 60000
.
:
/ / /
/
=
= =
=
8
8 5 13 14
5
300000000 3 10
3 10 3 6 10 10 8 10 1 08 10
360000 3 6 10
.
.)(
:
-
7/23/2019 Iranian 9th grade coursebook : Math
74/151
:
=/*5 =/*
=/ * =/*
5/ =/5 *
/.
: 0/000007
0/000007 = 710-6
/
.
1/610-3
a 10n
.n 1a > 10
0/0000127
= 1/27
10- 123= 1/23102 :
0/0137 = 1/37 10-2 29000= 2/9104
: 1
24000 = 10000000 =
0/00
= 0/000061 =
1404 = 0/127
=
:
2
/2 * 10-3= 7/304 * 10-=
2/28 * 108 = 9/4612 * 109=
6/02 * 10-2= 1/1 * 104=
-
7/23/2019 Iranian 9th grade coursebook : Math
75/151
: 1
)
5 5
5 5
3 10 25
4 15
)
1 2
4
8 4
12
8
2
1/02 * 10-= 0/00000102 /9 * 10-1=0/9
4/3 * 10
3
= 4300
7/004 * 10
-2
=0/7004
6/18 * 107= 61800000 8/2
70 * 104= 82
70
.
5
3
. /5 4
.1/3* 1071/4 * 109
:
6
2 * 10-7 * 4 * 109 /5 -4
___________
5 -19
6/287 * 1089/17 * 107 7
.
.
3
8
/ * < / / < / *
: 9
/5- , / , 5/5- , /-
-
7/23/2019 Iranian 9th grade coursebook : Math
76/151
:
)-(= ( ) =25 5 =
21
7
=
22
3
( ) =25
=
21
7= )-(=
.
-
.
-
)
(
:
9
9
. =9 3 = 9 3
:
552
3
2
3-33
6149
169
) (
7 7 . 2
3
2
3
4
9
. =0 0 .
b b b
.
.
b
.
02
35
1
2
1
5-33-22
64-88
)
(
:
-
7/23/2019 Iranian 9th grade coursebook : Math
77/151
8 .= )(
. =3 8 2 8 3
= 3 8 2
-
-
)-(=-
.
. 8
27
:
( ) =3 38 =3
1
8 =3125 =3 27
.
b3 b
.
:
=81 =24 ( ) =24 =3 1
=3 27
125 =3 36 =3
8
1000 ( ) =33 7
: x | x |=2
( ) =26 =28
=
23
5
( ) =
2
1 2 ( ) =
2
2 9
=
21
13
x y+2 2
.
x y+ =2 2 .)x( y ( x
-
7/23/2019 Iranian 9th grade coursebook : Math
78/151
:
b
a
ab a b= a ab b
=
:
.b a
a b3 33 ababb3ba3a
2*5=10101000512528
1
8
27
27-8
ab a b= 3 3 3 : b a
:b
a a
b b=
3
3
3
.
+ = +3 3 38 27 8 27
.
.4 5
-
7/23/2019 Iranian 9th grade coursebook : Math
79/151
:
49
16,
1
81, , , ,
:
, , - , 1
216
,
( ) = 21 1 ( ) = 33 1 1 ( ) | | = =25 5 5 ( ) = 33 5 5
= 49 7
256 16
/ /=1 44 1 2
( ) =21 1
= 3 64 4
:
3 125 36 3 1 81 381
3 3 325 5
-
-
:
= =3 3
4 16 = =
33 33 2 5 4 15 8
= =33 3 3
128 64 4 2 = 33 3
20 4
= =
3
3
3
125 125
64 64
= = =3
33
3
54 5427
22
___
___
___
___
______
-
7/23/2019 Iranian 9th grade coursebook : Math
80/151
a
:
a x <
= = =2 2
90 2 3 5 3 10 3 10 :
150 , 80 , 24 ,3 2125
( ) = 33 2 2 = 3 34 4
:
=3 32 16 3 4
=8 5
10
=
3 3
3
18 60
5
-
7/23/2019 Iranian 9th grade coursebook : Math
81/151
. 2 6
.
.
x x ( )+ =2 2 22 6 :
________
:
x= x= :
. ________
2 3 12 ________
.
________= + + + = =2 3 2 3 2 3 2 3 4 2 3 8 3
:
7 2 3 2
+ =7 2 3 2 10 2
=7 2 3 2 4 2 :
:
+ = + =12 9 3 2 3 9 3 11 3
.
32 7 2
32 2 5
:
x
x
:
2xya35
2
332
2
3 7
3x2b22 524 5
5x7a1
52
103
38 28 7
6 2xy4b72 10 35 22 5
+39 2 22
11 7 2 5
-
7/23/2019 Iranian 9th grade coursebook : Math
82/151
.
.
) +72 32 18
:
).(
____272 6 2 6= =
____ ____232 4 4= =
____
18 3 2= =
____ _______ _______
72 32 18 6 5 2 + = + = :
( + + = + + 332 3 33 3
50 24 81 5 2 2 3 3 3
____ ____ ____ ____ ____3 3 3
5 2 3 5 5= + + = +
. ( )+48 3 2 :
) ( )+ = + = + 2 2 248 3 2 48 3 48 2 4 3 4 3 2
( )= + = +24 3 4 6 12 4 6
) ( ) ( ) ( )+ = + = + = +248 3 2 4 3 3 2 4 3 3 2 12 4 6
.
( )+ 12 27 48 3 :2
+ = + =12 27 48 2 3 3 3 4 3 3
1( .(
-
7/23/2019 Iranian 9th grade coursebook : Math
83/151
.
( +98 50 128 ( + +1 4
481 9
( +27 12 75 48
( + 3 3 35 2 3 54 4 128 ( ( )( )+ 2 3 3 2 3
20
2
:
2
= = =20 20 2 20 2
10 222 2 2
.
.
( = =5 5 5 3
62 3 2 3
( = =3 2
3 3 3 2
2 2 5
5 5 5
) = = =4 4 4 3 2
2 2 2 2
3 3
) = =3 3
3 2
2 7 2 14
22
(x
xx x(x )
= =
>
2 2 2
0
(
z(z )
=
3
3 2
3
5
0
-
7/23/2019 Iranian 9th grade coursebook : Math
84/151
.
(3
6
2
(2
32
)
12
6
) x3
5
3
)x(
.
1
( + +2 50 32 2 72 )3 227
( ( )( ) +2 5 10 2
( + 8 128 50
(
3 27
64
( 2 48 3 27
x x22 .x>0 2
3 5
.
a
a
h
.
.a
ABCD Q P N M
ABCD .
MNPQ
aa
h
a
2
a
2
MA B
Q
D P C
N
-
7/23/2019 Iranian 9th grade coursebook : Math
85/151
:< = > 6
+5 4 +5 4 4 +2 2
3 2
3
11
3
11 +2 23 4 5
: 7
) = 10 ) =2 6 ) =1
3) =8 2
(
=
52
64
2
)( )
=
2
2
123
4 3
)m m
m
m
=
6 2
)
( )
=
3
3
39 27
4
. 8
)5
2 3)
a3 2
2)
2
7
. x ( x )=2 2 9
x0
)
.
)
.
)
.
.
10-9
.
.
-
7/23/2019 Iranian 9th grade coursebook : Math
86/151
a
b
c
a = b +c
2 2 2
.
-
7/23/2019 Iranian 9th grade coursebook : Math
87/151
.
.
7 , x , x, a x z 3 23 ,1
5xy , x, z ,
2
7
.
x
1 , 3x, x2 , |x| , x+x , y3 , +x
-3 xy x y27
2 xy
.3x3x
:
( )- x7 x( = )-8 x3(=- x3
( ( x y) ( ) .(x ) . y x y= =2 3 3 2 3 3 6 32 2 8
3 3 27
3( ( x ) ( x ) 3 2 2 31
33
=
( ( a b)(ab)( a c )
=2 2 51 2
2 7
( ) xy()- xy(=
( )xy()3xy3( + xy3)- x3y(=
a
ax3y
. y 3 x a
.
+3 =
y
x
:
-
7/23/2019 Iranian 9th grade coursebook : Math
88/151
.
x
yy
x
2+4=642a , x , ya x y3 2 43
5x2y2z4
-12xu
3
5
.
-5 x3y2
3x2y3
)
(
: .
x-x+ , x-x ,
ax y axy axy 2 22 3
3 2 , 3x
x
- xy3+ xy-
3. y
. x y .
)
(
.
:x 3
( 3x+ -x + x3= x3+3x-x+
( -3bxy3+axy-bx3y
( x y xy x y + 2 2 3 31
2 3 42
.
x
( -a-3ax+x-]a+ax-)3a-8ax([=
-
7/23/2019 Iranian 9th grade coursebook : Math
89/151
-( )x+ x()x3 x +( =
( )x+x()x+x+( =
()x-x+( )x+x3-(=
:
x
)x+3(2x2+6x+96xx2x
-20
5
3
2
.
x
.
x+x+9
)x+3(
)x+3(=)x+3()x+3(=
.
.
)x+3(=x+x+9
. x=.3x-3 =x+
.x
.
.
( )a+(=)a+()a+(=a+a+a+=a+8a+
-
7/23/2019 Iranian 9th grade coursebook : Math
90/151
( )x+(=) + () + ( = + + + =x+x+
( )a+b(=) + () + (= + + + =a+ab+b
)x+( =x+x+
3
:
x x
x x
)x+(
x+
:
( x ) ( x) x+ = + +2 2 25 2 5 2 5 2 2
b
a
)a+b(= a+ab+b .
= )a+b(= s+s3+s
.
( )x-(=)x-()x-( =
=( )3x-x(
=( )a-b(
.
S1 S3
S3 S2
a
a
b
b
-
7/23/2019 Iranian 9th grade coursebook : Math
91/151
.
a(b+c) = ab+ac
. 1
) (2x+1)2= =) (4a+3b)2
) (x ) =2 21
2
) ( xy x ) =3 21
2 2
) ( )+ 22 3 3 = ( ( ) 25 2 2 =
. 2
( b ) a b+ = + +2 2 2 4
3 4 9
=2a = 4a2=(2a)2
=2(2a)(3b2) = 12ab2
(2a+3b2)2=4a2+12ab2+9b4 :
) (1+b)
2
=1+ +b
2
)
(xy ) = +
21 1
2 4
( ( ) xx
= +2 4
4
1 (x0) (-) (
2=36x2-12xy+
:
(a+b)2= a2+2ab+b2 : b a
(a-b)2= a2-2ab+b2
-
7/23/2019 Iranian 9th grade coursebook : Math
92/151
)a+b( a+ab+b
:
a+ab+b=)a+b(=)a+b()a+b(
.
( x+x+9=)x+3(=)x+3()x+3(
)x( )3(
=( x-x+=) - (
=( n-n+=) - (
( 8ax+axy +8ay=a )x+xy+9y( =a) + (2=
)
.
.
(=a
ab+ac
:
ab+ac=a)b+c(
)) )..(
:
( 8x+x= x.x +x.3
)
.
.
(=x
= x) x + 3 ( )
(
( a-8a3=
( 7x-x3+x=
= ( xy-xy+xy
-
7/23/2019 Iranian 9th grade coursebook : Math
93/151
.
( ( m) ( m) ( m) ( m) 2 3 2 31
5 2 22
)( 7a3-b3+c3-)a3-9b3-c3
)( )xm-()xm- ])( x-])y-x(-)y-
.
2
(
(
. 3
( )y-3x(= =( )-3a-a(
) ( x ) 21
83
= =( )/7(+)/7()3/3(+)3/3(
.
( )x+y(-)x-y(= 4 x y
( a (a )aa
+ = + 2 2
2
1 12 )a(
.
( x3+8x+8x ( 3a3b-ab3+a3b3
( a)x+(+b)x+( ( a3-a+a
( xy-xy+
( x+3x3+9x
.
)a+b(=a+ab+b-b b
(a b )
(a ( b))
+ =
2
2
3m2n
3m2n
2m2n
3m2n
2m2n
5m2n
3xy
4xy
5xy
-
7/23/2019 Iranian 9th grade coursebook : Math
94/151
. 1
=:)a+b+c(2 )a+b+c()a+b+c( =
:))a+b(+c(2= )a+b(2+2)a+b(c +
.
)a+b-c(2=
.
2
)a+b()a-b(=
)a+b()a-b(=a2-b2 b
-
7/23/2019 Iranian 9th grade coursebook : Math
95/151
.
1( )10-x()x+10( =)10-x()10+x( =100-x2
2( )-y-2z( )-2z+y( = )____- y( )-2z+y( = 4z2-_____
3( )-7y+t()t+7y( =
4( )-4y-2z()2z-4y(=
5( )x-2y+5( )x+2y-5( = [x- )2y-5(] [x+)2y-5(]=
.
4x2-)7-3y(2
.
)2x-7+3y()2x+7-3y(
.
)A2-B2=)A+B()A-B
.
1( x2-9= )x+3() - (
2( y z2 41
44
= ) + () - (
3( )2x+1(2- y2=[)2x+1(-____][) (+y]
4( 1-)3a+z(2
=[1-) (][
+) (]=) ()1+3a+z(
5( )2x+1(2-)3x+4(2=[) (-) (][) (+) (]
=)-x-3() + (
6( x4-y4=)x2+y2() - (
=)x2+y2()x+y() - (
-
7/23/2019 Iranian 9th grade coursebook : Math
96/151
.
.
1
( )x+2()x+5(=x2+5x+2x+10=x2+7x+10
2+5 2*5
( )x+9()x-4(=x2+9x-4x-36=x2+5x-36
.
)x+a()x+b(=x2+) (x+______
.
1
2
b a
x2+)a+b(x+ab=)x+ ()x+ (
( x
2
+7x+10=)x+ () + (
( x2+7x+12=)x+ () + (
(=( y2+y-6 + () - (
(=( y2-y-6 + () - (
( y2+5y+6=) + () + (
B A
.
.
98*102=)100-2(*)100+2( =______=
497 * 503 =______*______=
)1001(2= )1000+1(2=______
-
7/23/2019 Iranian 9th grade coursebook : Math
97/151
x2+10x-24 3
.
:)x+6()x-4(
:)x+12()x-2(
:)x+6()x+4(
:)x-12()x+2(
. 1
( ( x)( x) +1 1
4 4) )3x+y-z()3x+y+z(
( )5x+4( )5x+3(
( )x-1()x+1()x2+1(
)
(z )(z ) +3 3 )( )x-2( )x+2( )x+3
. 2
( )xy-z()xy+z(=-z2
( )x+a()x-b(=x2+.-
) ( )( ) y+ = 21
5 54
-( )x2+()x2-5(= x4+2x2
.
3
( a2-8a+15
( x2-13x+36
( x2+x+1__4
)
x2-12x+36
)
x2+10x+24 ( )x+y(2-9
( x2-2x-8 )
bx2-5bx-50b
( 4ax2-a )
x4-5x2+4
b
a
a=b
4
5
.
x2
ax
bx
x + a
x + b
ab
x
x
a
b
-
7/23/2019 Iranian 9th grade coursebook : Math
98/151
.
b
a
.b a
< >
.b a
a b a b b a
p
a=b+p
p
a < b b a
.a=b+p p
.
( x=y+4x b( b a x
1>2>5 :
:
a b
a x b< 3
7 9 21 27
( ) > 1
.p q2)p-1( =2q-3 ( .a ba-b=1 )
.a b
a b
= 32
)
.u v
u-v=-2
)
:c, b, a 2
)ac
b2>0 )
a
bc
-
7/23/2019 Iranian 9th grade coursebook : Math
102/151
.65 85 7
4500.
(. 232(
.
..
.
)(
.
-
7/23/2019 Iranian 9th grade coursebook : Math
103/151
.
.
)
(
-
7/23/2019 Iranian 9th grade coursebook : Math
104/151
.
.
: n2n
, , , , , n
, , n
1 2 3 4
2 4 6 8 2
..
0 1 1/5 2 3 3/5 4 5x )(
0 2y )
(
x
x
y
y
x
:
-
7/23/2019 Iranian 9th grade coursebook : Math
105/151
y
x
y
x
1
)x(
)y(
1
2
3
4
100
x
y=______
xy y x
.
0 0/5 1 1/5 2 2/5 3x
)(
0 0/25 1
y
(
)
0
0
:
y=-x+
x
y
=
=
1
x
y
=
=
2
x
y
=
= 3
x
y
=
= 4
x
y
=
= 0
.
y= x=
-
7/23/2019 Iranian 9th grade coursebook : Math
106/151
. 2
.
0 1 -1 2 -2)
( x
2)
( y
0
2
x
y
.
.
x-y= y=x-
x
y
=
=
x
y
=
=
x
y
=
=
x
y
=
=
x
y
=
=
x
y
=
=
x
y
=
=
x
y
=
=
x
y
=
=
x
y
=
=
.
y=ax+b
.
y
x
.
b=
a =
y=x+
.
-
7/23/2019 Iranian 9th grade coursebook : Math
107/151
.
y=-x+ y x=3
2
)
0
0
(
y=x
y=x
a
y=ax
. y=x y=-x
.y=ax
.
0
0
1
2 (
0
0
1
1 )
0
0
3
1
y=x
.
-
7/23/2019 Iranian 9th grade coursebook : Math
108/151
. y x= +1
42
.
2
1)
.)
.
-
)
60
20
6
3
6
2
)
(
.
y=x-
y x
y
=
=
2 1
2 2 1
. y x= +1
22
-
.y=x+
-
7/23/2019 Iranian 9th grade coursebook : Math
109/151
.y
x .
.
y=/8x+
.
(
1
3
0
0
)
1
1
2
3)
0
1
1
4
.y=-x+
. y x= +3
45
6
.
y x= +1
22
7
.-1
2
3
.-2
.
-
7/23/2019 Iranian 9th grade coursebook : Math
110/151
1
.
=( y1
2x ) y=x ) y=3x ) y=-x ( y=-2x
.
x
x
2
.
y=2x-1 , y=2x , y=2x+3
2 x
.
x
a . a y= ax+b
b .
.
.-3 2 y = -3x+2
:
y=x
-
7/23/2019 Iranian 9th grade coursebook : Math
111/151
. 1
y=2x-4 y x= 2
3y = -3x+1
: 2
.-1 -2 (
.31
2(
.
0
4
y=2x +1
)
.
1
22 3
y ax b y x b b b y= + = + = + = =
2 2 2 1
2 2 1
1
.
2
B A
.
y
x
A
B
O
y
Hx
AH
OH
BH
OH
-
7/23/2019 Iranian 9th grade coursebook : Math
112/151
y = x - d3
.
B
A
.
d EBEA
= = =2 21
D
C
.
.
.
)1( .
)2( .
)3(
d1
A
B
C
D
F
E
d2
A
B
C
D
F
E
0 4d
.
.d
d
EB
EA= =
0
1
d
.d.1
. y=x-
-
7/23/2019 Iranian 9th grade coursebook : Math
113/151
2
1
2
0
.
.
:
2
:
.x=
.ax+by = c
)
a= , b= , c=
a=- , b= , c=
.c b a (
-x+y= a= b= c=
y=x+ a= b= c=
.
c
b
a
x=
)
ax by c x+ = =
2
.
A =
B
=
C =
D =
A B C D
-
7/23/2019 Iranian 9th grade coursebook : Math
114/151
.x = - y=
.
y-x = 6 x-y = 8 x-y =
.
x =
y = -
.
2
1x
.
c
b a
ax by c+ =
.
y x y x y x = = + = + 4 8
2 4 8 2 4 8
2 2
y=x+
x-y=6
x+y-9=
-
7/23/2019 Iranian 9th grade coursebook : Math
115/151
.
1
1y-x= 6
.7
=y y
x x
2 1
2 1
.
4
1
3
2
8
x2
y2
x1
y1
xA
y
1
1
xB
y
2
2
a
b b
.
.y=ax+b
.
-
7/23/2019 Iranian 9th grade coursebook : Math
116/151
2000
3000
1
y x .
.x y
3000
x ))y .
.
.
1/5
y=2x-1 2
.
y = -x +2
.
.
:
y
x
-
7/23/2019 Iranian 9th grade coursebook : Math
117/151
x-6y=
.
x-y=
.
)
)________________________
)
2x - y =
x - 6y = 8
.
x y
x y
=
+ =
1
3
x y
x y
=
+ =
2 3
2 3
-
7/23/2019 Iranian 9th grade coursebook : Math
118/151
.
(
x yx y
= + =
3
4 2 6(
x yx y
= + =
3 5 1
2 3 7(
x yx y
+ = + =
3 2 50
2 2 35
1kg1kg 1kg 1kg
1kg1kg
1kg 1kg
2kg
2kg
:
(
=
+ ==
2 2
5 5
7 7
(
x x
x xx x
=
+ ==
2 2
3 3
(
_______________
.
.
(
{x yx yx
=+ =+
=
1
3
2 4
)
{ {x y x yx y x yx
= =
+ = + =+
=
2 2 3 4 2 6
2 4 2 4
5 10
x= x+y= x= x+y=
y y+ = =2 3 1 +y =
y y= =2 2 1
:
2
1
:
2
1
y x
.
=
=
=
-
7/23/2019 Iranian 9th grade coursebook : Math
119/151
.
x y
y x
=
=
2 3 5
1 2
3 3
y : (
.)
x- ) (=
.
.
..
:1
:x
:x-
=)x+x-(=
:2
:x
:y
{ {y x x y(x y) x yy y
= =
+ = + =
=
2 3 2 3
2 24 2 2 24
2 3 24
:3
:x :y
y x
(x y) (x x )
=
+ = + =
2 3
2 24 2 2 3 24
-
7/23/2019 Iranian 9th grade coursebook : Math
120/151
.
(
x y
x y
=
=
3 7
2 7 15(
x y
x y
=
+ =
3 6
12 8
3
.
(
(x y) y
x ( x y)
+ =
=
2 3 4
3 2 2 7(
x y
x y
=
+ =
1 1 1
2 3 6
4
.y x
22x-y-=x+y-
x+y= x-y=
.
23
ay = ax +
a=-
a=
a=
.
.
6
.
( )
.
6
x y
x y
=
=
2 3 7
4 6 5
.
.
.
6
7
7
.
-
7/23/2019 Iranian 9th grade coursebook : Math
121/151
.
.
.
L
L+ +
150
2150
-
7/23/2019 Iranian 9th grade coursebook : Math
122/151
2
3.4
.
x+4 x
: x
x+4
x
x=
+
2
4 3
3x = x+8 x=8 =
. x
x+ 4
.
v
v v
v v+1 2
1 2
2v
:
.
v
k
2
2
k v
a b+
2
b a
:
x
x x
+3 22 5
5 2 1
x
x
+
5
1
a
4
2
5
x 3
4
x
y
x x
xy
+2 3 1
9
x
1
x +
10
2
x
x
+2
3 7
xy
(x y)
2
2
x3
1
a
b
x3+x-
( .(
xy
x
x y+
|x-y|
x
1
2
:
-
7/23/2019 Iranian 9th grade coursebook : Math
123/151
x
7
1
x+6
3
ah
2
x+3
5
x2
25
| x | | y |
x
+
x y
x
+2
1
x
+
5
3 1
x31
mn n
n
+
2
5
4 a
x
+
3
2
:x
x
+
5
3
-5-101
27-2x
x
x
+
5
3
x
x
+
5
3x=3
.8
0
.
x x
(x )(x )
+
+
27 1
1 2:
:
:
)x-()x+(=
:
-
7/23/2019 Iranian 9th grade coursebook : Math
124/151
(x ) x
(x ) x
= = + = =
1 0 1
2 0 2
.
x = -
x =
.)(
) y
y y=
3
5 2
18 3
60 10
)(x ) (x )x x
x x
+ ++ + =+ +
2
2
3 36 9
4 3 (x ) (x )+ +1 3
x
x+=+
3
1
)x +8 5
2)
x
x
+7)
b
b
+
2 1
2 1
)x
x +23
4
(x
x 2 1)
a
a a
+
+25
5 6
3
4.
3
418
246
89
12
36
48
:
= =
36 3 12 3
48 4 12 4
ac a
bc b=
(b , c ) 0 0
:AC
BC
AC A
BC B= )B C C B )A
-
7/23/2019 Iranian 9th grade coursebook : Math
125/151
:
)m
m
216
4)
m
m
+
+
6 18
7 21
)a a
a a
+
2
2
5 14
2
)x y
y x
4 4
a ax
a
+
)a ax
a xa
+= + )
a ax a( x)x
a a
+ += = +
11
.
)x
ab25
3
)y
y( y )
2
2 6)
P
P P 22
12
(
x
x
+2 5
(
x
x
+
21
5
(
a
a
+
+
3
2 1
:
)x
x x
+23
5 6
)x x
x
+
+
24 8
12 24)
x
x x
2
2
24
12 6
( y y yy y
+
3 2
2
2 3
( tt
+
4
2
1
1
(x y
x y
2 216 9
8 6
)y (y )(y )
y (y )
+ = =
+ +
29 3 3
3 9 3 3
(
ab b
a b =
7 4
2 3
8 2
20
(
b b
b (b )
= =
5 5
15 5
-
7/23/2019 Iranian 9th grade coursebook : Math
126/151
)a b
ab
4 2
8
6
4
)a
a a
+ 2
2 8
2 8
. -1 1 3
(y
y
+
2 3
2 3
(y
y
2 3
3 2
)y
y
+
+
2 3
3 2)
y
y
+
2 3
2 3
.
4
5(
a
a
+
2
54(
a
a
+
+
2
53(
a
a
2
5
2(
a
a
+
2
5
1(
a
a
+
2
5
10(
a
a
+
+
2
59(
a
a
2
5
8(7(a
a
2
5
6(
a
a
2
5
5
)z
z z z
=
+3
1)
x
x x x=
2
3
3 6
( ( )y +
=3 2 1
5 5
(( )(x )
x(x )(x )
= +
51
2 5
. z(x y)
t
+ 6
)z
(x y)t
+ )zx y
t
+) z(x y)
t +
1
(x y
zt
+
(zx zy
t t+
(zx
yt
+
-
7/23/2019 Iranian 9th grade coursebook : Math
127/151
.
:
a c ac
b d bd =
a c a d ad
b d b c bc = =
)b , d ( )b , c , d(
.
:
.
.
) xy z yz
xx z y =
1 23 3
2 2 2
1 3
5 16 2
38 15
)x x x
x x x
+ + =
2
2
3 3
2 15
x
x (x )
+
2
3 (x )=
5
)x x x x ( )( )
( )( ) (x )(x ) xx x x x
= =
+ + + + + +
2
2 2
6 3 18 6 1
3 4 412 36 7 12
)x x x y
xy xy xy
= =2 2 3
3
4 8 4
3 3 8 6
(a a a a a a a
aa a a a a a
+ + =
+ +
2 2 2
2 2 2
4 5 3 2 4 5 4
44 4 3 2
(a )(a ) a a
a( ) ( )( )
+ = =
1 5 4 5
-
7/23/2019 Iranian 9th grade coursebook : Math
128/151
.
(
a a a
a a
+
+
2
2
6 3
3 4
( a b ab aba (a b)
+
+
2 2
2
3
)x x x
x x
+ + +
+ +
23 2 1
2 5 )
x x
xy y
4
2 5
4 8
3 9
:
a c a c
b b b++ =
a c ad bc
b d bd++ =
) b ( )b , d(
.
.
.
)x x x x
x x x
+ + + + = =
+ + +
3 7 2 3 3 7 2 3
2 2 2
+ =3 4 7
5 5 5
)x x x ( x ) x
x x x x
+ + + = = =
+ + + +
3 7 2 3 3 7 2 3 10
2 2 2 2
=
(a a a (a )
aa a
+ + =
+
2 2 2
2 2
20 2 20 2
24 4
+ =7 9
10 5 10
a a (a a ) = = =
2 22 4 16 2 2 8
(a )( )
(a )(a )
= =
+
2 4
2 2
)a a (a )( ) ( a )
a a(a ) a(a )
+ + + + = =
+ +
1 3 2 1 3 2
2 2
=
= =
-
7/23/2019 Iranian 9th grade coursebook : Math
129/151
.
)x y
x y y x+
2 2
)x x
+
6 4
)x x
xx
+
+
2
2
2 16 4
24
)x
x x x x+
+ +2 27
2 4 3
(
x x(x y) y(x )y(x y) x(x y) xy(x y)
+ + ++ = = 2 2
2 1 2 3 1
3 3
.
)
(
)x( .
(
x ( )
x x ( )( )x xx x( )( ) xx xx ( )
x xx x
= = = = + + +
22
2 2
22
2 2
1 6 1 61 1
64 3 4 3 14 3
1 1
)
x xx x x x x xx x x
x x x x xx x x
+ = = =
+ +
2
2 2 22 2
2 2 2 2
2 2
1 6 61
6 4 3 6
4 3 4 31
x
2
x x +2 4 3
= = =
(
-
7/23/2019 Iranian 9th grade coursebook : Math
130/151
)a( )( )
(a ) aa a a a
a (a )a( )( )
a a a a
+ + +
= = =
+
+ +
2 3 2 3
2 1 31 1
2 3 2 3 2 3 1
1 1
(
) a a
a a
+ =
+
2 3
1
2 3
1
a a-
) ( .
)
nn
n m
m
n m
=
+
2
2
2 21
)
y x
x y x y
x y
x y x y
+ =
++
.
5
26
.
.
= x+5
=( )2
x + 3
x + 5
-
7/23/2019 Iranian 9th grade coursebook : Math
131/151
=) () (
= =5
26
(x ) x
( )( x ) x
+ +
= = + +
25 5 5 5
4 2 3 26 4 6 13
7x=35x=5
=
=
)x
-
7/23/2019 Iranian 9th grade coursebook : Math
132/151
.
)x y(x y)
x y x y
+
2
2 2 4 4
)x y
ax bx by ay
+ ++
7 9
)a b a b
a b a b
2 2 3 3
2 2)
x xx
x
+
+
24 2
22
. 3
a
a
+2
2
5
a +2 3
3
a b
a
+2 2
2
a a
a
2
5
. x
y 4
)x
y
+
+
3
3)
x
y
3
3 )
x
y
3
3
)x
y
3
3 (
a x
a y
3
3
x
x +
+
3
5
)x
x
+
+
3
5)
x
x
+
3
5)
x
x
+
3
5)
x
x
+
3
5
6
)a a
a(a )(a )a
+ += = + 2
5 55
5 525
)a a
(a )(a ) aa
+ += =+ 2
5 5 1
5 5 525
.A B
C
2 2
2 C=ab B=a+bA=a-b
-
7/23/2019 Iranian 9th grade coursebook : Math
133/151
. 8
( ..(
(a b a b
b a ab
=
(x
xx =
13
7
20
)a b a b
=7 7
5 5 5
)a b
b a
=
1
(a b a b
=
+
1 1
) a b a ba b
= +
2 2
(
ca cb a b
c cd d
+ +
=+
(
a
cba b
c
=
. 9
.)(
.
)
a a
aa
aa
+
2
2 1
1
) x y x y
x y
(x y)
+
2 2
2
1 2
9
)x
xx x x+
++ + 2 22 1 2
12 1 1
:
.
a
a
+
2
7
)
.a
a
+
2
7)
.x
. x-9
x2 - x - 12_________
x-4
A = x2 -9
-
7/23/2019 Iranian 9th grade coursebook : Math
134/151
) 1
)
.
x y x
yx y=
5 3
2 2
14 7
2
a xz a z
x z x
=
2 4 2 3
6 5
18 2
27 3
2
:
d
c
b
aa b c a b c
d d d d
+ += + +
:6 x3-8x+
x x x xx x
+= + = +
3 3
312 18 2 12 18 2 12 3
6 6 6 6 3
:
.
)a a a a a a
aaa
+ = + = +
4 3 4 3
2
2
2 5 8 2 5 8 1 2
24
)x yz xy x y z
x y z
+ = +3 2 2 2
2 2
14 6 3
2
x z
y= +
7 3 3
2
)y y y
( y y y) ( y )y
+ + =
3 2
3 2 2
2
8 4 128 4 12 4
4
y= + = + 2 1
-
7/23/2019 Iranian 9th grade coursebook : Math
135/151
.
)a b c
ab
7 4
6
21
28
)x y z xyz
x z
+6
2
24 2 3
3
-
26 8
24 3
2
:
. 3 8 6
:
+ =
-
7/23/2019 Iranian 9th grade coursebook : Math
136/151
x22 x x
x
+ 2
7 15 5
2 x x
x
10 2
3 15
- +
x- 3x-
. x- 3x-
x
x=
33 :
.x x x
x x x
x
+
2
2
2 7 15 5
2 10 2 3
3 15
x 3 15
0
.
.
.
1
)
x x x x
x x
x x
x
+ +
+ +
+
+
3 2 2
3
2
2
4 3 7 2
4 8 3
3 9 7
3
1
)
x x x
x x
xx
+ + +
+ +
2
2
5 24 8
8
3 24
3 24
0
-
7/23/2019 Iranian 9th grade coursebook : Math
137/151
..
x x x x + +4 2 210 3 2 19 3 2
x x x x
x x x
x x
x
x
+
+ + +
+
+ +
4 2 2
4 2 2
2
2
10 3 2 19 2 3
10 15 5
12 2 19
12 18
2
.
) x x x x + 3 26 19 16 4 2 ) x x x + +3 12 8 6
.
)x y z
xz
2 3 7
5
2
18
)a y a y xy
y
+
3 4 2
2
2 15
5
)( )x3- )x-( )( )3y-y-4()3y-4
( x x x x x x+ + + +5 4 3 22 5 2 2 2 3 3
x x x x + + + 4 6 2 33 4 5 1 .
.x3+x+8x 3
. x+4 x
.a 4x+3 x3+3x-x+a 4
.
2x
2
-9x+9
5x+3 x-3 x-3 x+3
:
)x+ ()x-3(+x-
=__________________
=x4-3x+x-9
x-
.x-3
-
7/23/2019 Iranian 9th grade coursebook : Math
138/151
..
.
.
.
-
7/23/2019 Iranian 9th grade coursebook : Math
139/151
:.
.
: 1
.
.
.
_________________
______
.______._________________
2
.
.
.
.
.
RO
:
-
7/23/2019 Iranian 9th grade coursebook : Math
140/151
.
R .
.R
:
: :
.
.
.
)
)
.
______
)
)
.R
. V R= 34
3R
.10 1
.
)
.
)
.
)10. 2
-
7/23/2019 Iranian 9th grade coursebook : Math
141/151
1
.
.
!
))
.
R 2
(. (
____________:
)____________:
R
)
S=4R2:R
R
R
)) 1
.
10
.10 2
.
-
7/23/2019 Iranian 9th grade coursebook : Math
142/151
.12800 1
.
)
.)
.)( )
.
1/648/000 )
.
2
.
60
.
100
.
3
24
m
60cm
24cm
24cm
x
-
7/23/2019 Iranian 9th grade coursebook : Math
143/151
.
.
.
.
.
.
.
: :
:: :
.
ABCDE
OH
.
OABCDE.
:
1
.
.)(
: :
:
:
:
)
2
B
A
CD
E
H
O
H
O
A
B
C
D
-
7/23/2019 Iranian 9th grade coursebook : Math
144/151
ABCD
)A.
B
C
D
( N M
NABCD
MABCD
)( )2(
)3( )4(
.)2( ))1
O
H
< OH
V
V
.)4( ))3
O
H
=OH
SABC < SA
B
C
V V
(
)
)
.
O
A B
C
H
O
A B
C
H
O
A B
C
H
O
A B
C
H
MN
A
C
B
D
A B
D C
-
7/23/2019 Iranian 9th grade coursebook : Math
145/151
E
C
.
D E F
ABC
. D B D
.
ABC OABC
OH ACB
=
90 BC =10cm AC =6cm
. 5cm
.
A B
O
C
H
BDE ABD ABED 1
CBAD
CBED
2
ABC
DEF
3
DABC
CDEF
4
4 2 5
h : S
V Sh=1
3
C
D
A
E
B
C
E
B
D
C
B
D
A
AB
C
E
F
D
C
E
F
D
-
7/23/2019 Iranian 9th grade coursebook : Math
146/151
ABC
AC CBS cm
= = = 2
2
V Sh cm= = = 31 1
3 3
.BC M 10cm
OBC
OM
)
OBM (
12cm
)
.OM OBM
MH OMH (
OH
OMH
)
.
.OABCD (
.
.
.
O
A
C
B
MH
D
___ ___
______
___ ___
___
-
7/23/2019 Iranian 9th grade coursebook : Math
147/151
:
V Sh R h= = 21 1
3 3
h
R
10cm
.12cm
.
cm12
cm10
12cm
4cm
6cm
5
6
1
.10
4cm 2
.
8cm
12cm
cm
3
6cm
-
7/23/2019 Iranian 9th grade coursebook : Math
148/151
1
.
:
. 2
5
8
4
3
4 3
10
AB
AB
.
)
(
)
(
10
4
4
5
A
O
B
-
7/23/2019 Iranian 9th grade coursebook : Math
149/151
1
.
a
2
.
1
:
:
.
2
.
: :
3
.
a
3
10
3
10
20
10
-
7/23/2019 Iranian 9th grade coursebook : Math
150/151
. 5 cm
5
1
.
2
( ) )
. 3
4r o
r
r
-
7/23/2019 Iranian 9th grade coursebook : Math
151/151
a
a
a a
a
.
.
.
1
a
a
a
a
a