deber 8 mate
TRANSCRIPT
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16/36
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,K- W&D42D* D& D5%#$=2) )* D& 9&:* ;*%53 )* 2% =3&C*4$5 3*4="%#2D5 42F& 9&:* ;&F53 ;$)* A;
F :2: D&)5: %5 C&3&D*D5: ;$)*% >; F A;8 W5%:=32F& *D =3&C*4$58
B*:C2*:=&0 4!2 3( )m
IL-
X% D& /$#23& &)?2%=& :* ;2*:=3&% + 42&)3&)5:8 N$ D5: D&)5: )* D5: 42&)3&)5: !F ";$)*%+4; F >I4; 3*:C*4=$6&;*%=*E 4&D42D* *D C*3H;*=35 )*D 42&)3&)5#8
B*:C2*:=&0 +,4;
I(- X% 2% d*7"#5%5 3*#2D&3 ABCDEF E :593* *D D&)5 CD :* 45%:=32F* *7=*3$53;*%=* *D
42&)3&)5 CGHD 8 N$ :* 45%:=32F* *7=*3$53;*%=* *D =3$"%#2D5 DHEE 4&D42D* D& ;*)$)&)*D "%#2D5 !DHE8
B*:C2*:=&0 15o
@A[ (HETJHLEI S UEHF ZH DN PIOTCINI
I>- N$ 2% =3$"%#2D5 =$*%* $#2&D C*3H;*=35 G2* 2% 42&)3&)5E *%=5%4*: *: aXB.b. G2*0
&- *D "3*& )* D& :2C*3/$4$* )*D =3$"%#2D5 *: $#2&D G2* *D "3*& )* D& :2C*3/$4$* )*D 42&)3&)58
9- *D D&)5 )*D 42&)3&)5 *: ;": #3&%)* G2* *D D&)5 )*D =3$"%#2D58
4- *D "3*& )* D& :2C*3/$4$* )*D 42&)3&)5 *: ;&F53 G2* *D "3*& )* D& :2C*3/$4$* )*D =3$"%#2D58
)- D& )$%&D )*D 42&)3&)5 =$*%* $#2&D D5%#$=2) G2* D& &D=23& )*D =3$"%#2D58
B*:C2*:=&0 4-
I+-
B*:C*4=5 & D& :$#2$*%=* /$#23& :* 45%54* G2* !!"# ! !"5E !!"# ! !"5E !" ! !4; F !*: *D C2%=5 ;*)$5 )*D :*#;*%=5 !"8 .*=*3;$%* *D "3*& )* D& :2C*3/$4$* )*D =3$"%#2D5 !"#8
B*:C2*:=&0 A=9 3
4cm
2
!#
"
A B
G
E D
H
F
C
-
7/25/2019 Deber 8 Mate
17/36
!"#$%& (I )* +,
IA- N$ 2% =3$"%#2D5 *G2$D"=*35 =$*%* 6 cm )* C*3H;*=35E :2 "3*& *: $#2&D &0
&- 3 9-9
43 4- 3 3 )- 9 3 *- 12
B*:C2*:=&0 &-
I@-
X% D& /$#23& &)?2%=& :* =$*%* *D 42&)3&)5 V.X1 F :* 45%54* &)*;": G2* bV T @2E VW T >2E
*%=5%4*: *D "3*& )* D& 3*#$`% :5;93*&)&0
&- KP,L 2>
9- (LP+ 2>
4- (LPK 2>
)- JLPK 2>
*- AP+ 2>
B*:C2*:=&0 )-
I,- W5%:$)*3* D&: /2%4$5%*: )* 6&3$&9D* 3*&D f
F g0
XD C*3H;*=35 )*D 3*4="%#2D5 ABCD E *% 2%$)&)*:E *: $#2&D &0
&- > 9- + 4- A )- , *- J
B*:C2*:=&0 )-
II- gk2< C534*%=&?* )*D "3*& )*D 3*4="%#2D5 bVW. *: D& )* D& C&3=* 3&F&)&h
&- >L o
9- >@ o
4- (J o
)- +L o
B*:C2*:=&0 9-
IJ- Z% 3*4="%#2D5 )* D&3#5 )5: 6*4*: :2 &%4d5 =$*%* *% :2: *7=3*;5: )5: =3$"%#2D5:
3*4="%#2D5: $:`:4*D*: *7=*3%5: )* ;&%*3& G2* :2 d$C5=*%2:& 45$%4$)* 45% *D &%4d5 )*D
3*4="%#2D58 .*=*3;$%* D&: ;*)$)&: )*D 3*4="%#2D5E :$ :* 45%54* G2* *D "3*& )* D& /$#23&
=5=&D *:2
363m 8
x
y
-
7/25/2019 Deber 8 Mate
18/36
!"#$%& (J )* +,
IK- N$"$%#*: 2% 3*4="%#2D5E &F ':5% D5: C2%=5: ;*)$5: )* :2: 3*:C*4=$65: D&)5:E *%=5%4*:
*D "3*& )* D& :2C*3/$4$* )*D =3$"%#2D5 #&'E *% ()>E *: $#2&D &0
&- @A
9- ,J
4- I>
)-
IJ*- K,
B*:C2*:=&0 4-
JL- X% *D =3$"%#2D5 )* D& /$#23& &)?2%=& Qi PP VW
N$ bV T ,2E VW T (@2E Qi T K2E bW T (@2
X%=5%4*: *D 454$*%=* C*3] p bVW- P C*3 ] p bQi- *: $#2&D &0
&-
@P+9- >@PK
4- AKP(,
)- IP@
*- IPA
B*:C2*:=&0 &-
J(- X% D& /$#23& ;5:=3&)&E ABCD *: 2% 42&)3&)5 F O *: :2 4*%=358 N$ ABE *: 2% =3$"%#2D5
*G2$D"=*35E *%=5%4*: D& 3*D&4$`%rea !DCO( )rea !ABE( )
"
#
$$
%
&
''
*: $#2&D &0
&-
1
9-3
3
4-2
3
)-2
3
*-13
16
B*:C2*:=&0 9-
J>- X% *D 42&)3&)5 ABCD E EE FE G F H :5% D5: C2%=5: ;*)$5: )* :2: D&)5:E =&D 45;5 :*
$%)$4& *% D& /$#23&8
.*=*3;$%*0
&- N$ *: G2* PQRS*: 2% 42&)3&)58
9- XD 6&D53 )* D& 3&e`%ABCD
PQRS
A
A
A B
CD
O
E
A B
CD
P
R
24cm
8cm
-
7/25/2019 Deber 8 Mate
19/36
!"#$%& (K )* +,
J+- N* d&% )$92?&)5 D5: 42&)3&)5: OABC F CDEF *% *D CD&%5 4&3=*:$&%5 &)?2%=50
XD "3*& )* D& :2C*3/$4$* )*D =3$"%#2D5 ACE *: $#2&D &0
&-ab
2
9- ab a+b( )
4- b a+b( )
)- b a!b( )
*- a2+b
2
B*:C2*:=&0 )-
JA- W&D42D* *D 6&D53 )*D "3*& 45;M% G2* :* /53;& *%=3* D&: =3*: 3*4=&: *% *D CD&%5E )&)&: C53
D&: *42&4$5%*:0
y=1!
x y= 2
x+1
2x!5=0
B*:C2*:=&075
8u2
J@- X% D& /$#23& &)?2%=& :* =$*%* *D 42&)3&)5 BDEF F :* 45%54* &)*;": G2* AB = 5cm E
BC= 2cm E *%=5%4*: *D "3*& )* D& 3*#$`% :5;93*&)&E *% ()>E *: $#2&D &0
&-19
8
9-
9
80
4-3
10
)-14
3
*-40
9
B*:C2*:=&0 9-
x
yD
A
C
B
E(a, b)
FO
-
7/25/2019 Deber 8 Mate
20/36
!"#$%& >L )* +,
J,- W5%:$)*3* *% D& /$#23& &)?2%=& *D =3$"%#2D5 ABC F D&: 3*4=&: L1
F L2
C&3&D*D&: *%=3* :HE
:*C&3&)&: > 4;8 N$ *D "3*& )* D& :2C*3/$4$* )*D =3$"%#2D5 ABC =$*%* 2% 6&D53 )* (L 4;>E
*%=5%4*: D& D5%#$=2) )*D :*#;*%=5 AB *: $#2&D &0
&- @ 4;
9-
(L 4;
4- >8@ 4;
)- J 4;
*- > 4;
B*:C2*:=&0 9-
JI- .*D =3&C*4$5 $:`:4*D*: ABCD :* 45%54* G2* D& )$/*3*%4$& *%=3* D&: D5%#$=2)*: )* :2:
9&:*: *: $#2&D & 6 cm F G2* :2 C*3H;*=35 *: $#2&D & 38 cm8
XD "3*& )* *:=* =3&C*4$5E *% cm2
E *: $#2&D &0
&- 15 7 9- 15 3 4- 6 7 )- 6 3 *- 9 7
B*:C2*:=&0 &-
JJ- W&D42D* *D "3*& )* D& 3*#$`% :5;93*&)&E )&)5 G2* *D "3*& )*D d*7"#5%5 3*#2D&3 *: $#2&D &
>@J 4;>8
B*:C2*:=&0 (>K 4;>
JK-
ABCD *: 2% 42&)3&)5 )* "3*& $#2&D & +, 4;q )5%)*!"
!" T + F XE 1E rE f :5% C2%=5:
;*)$5:8 .*=*3;$%* *D "3*& )* D& 3*#$`% :5;93*&)&E :$ _nOs *: 2% 42&)3&)5 F D5: =3$"%#2D5:
:5% 45%#32*%=*:8
B*:C2*:=&024 cm
2
L1
L2
A B
C
A B
CD
12cm
-
7/25/2019 Deber 8 Mate
21/36
!"#$%& >( )* +,
KL- O& 3*#D& )* 4533*:C5%)*%4$& )* D& /2%4$`% )* 6&3$&9D* 3*&D *: f x( ) = log x( ) 8
!&3& *D 35;95 ;5:=3&)5E 4&D42D*0
&- N2 C*3H;*=358
9- N2 "3*&8
B*:C2*:=&0 Per = 4 101 uE A= 20 u2
K(- N*& D& /2%4$`% f : )*/$%$)& C53 f x( ) = 3sen 2x 8
XD "3*& )* D& :2C*3/$4$* )*D =3&C*4$5 ;5:=3&)5 *% D& /$#23&E *% u2
E *: $#2&D &0
&- + 9- , 4- J )- (L *- (>
B*:C2*:=&0 9-
x
y
0
-4
-3
-2
-1
0
1
2
3
4
f
x
y
f
-
7/25/2019 Deber 8 Mate
22/36
!"#$%& >> )* +,
K>- X% D& :$#2$*%=* /$#23& M F N :5% D5: C2%=5: ;*)$5: )* )5: )* D5: D&)5: )*D 42&)3&)5
ABCD 8
XD "3*& )* D& :2C*3/$4$* )*D =3&C*4$5 NMBD*+$#E *% u2
E *: $#2&D &0
&- (8@ 9- (8I@ 4- + )- 2 2 *- 3 2
B*:C2*:=&0 &-
K+- N*& D& /2%4$`% f :!! 0{ }"! 42F& 3*#D& )* 4533*:C5%)*%4$& *:0
f x( ) =10log x
3
!
"#
$
%&
N$ ABD *: 2% =3$"%#2D5 *G2$D"=*35E 4&D42D*0
&- *D C*3H;*=35 )*D 35;95 ABCD 8
9- *D "3*& )* D& :2C*3/$4$* )*D 35;95 ABCD 8
B*:C2*:=&0 &- >A2E "# $"%
x
y
x
y
-
7/25/2019 Deber 8 Mate
23/36
!"#$%& >+ )* +,
KA- N$ *D =3&C*4$5 !"#$ *: $:`:4*D*: F X *: *D C2%=5 ;*)$5 )*D :*#;*%=5 !"E )*=*3;$%* *D
"3*& )* D& :2C*3/$4$* )*D 42&)3$D"=*35 !"#$8
B*:C2*:=&0 A=10 3 u2
K@- X% D& /$#23& &)?2%=& :* 45%54* G2* ADLI *: 2% 42&)3&)5E BCFE F GHKJ :5%
=3&C*4$5: $:`:4*D*:8 N$ cmDI 25= F cmBC 3= E *%=5%4*: D& :2;& )* D5: C*3H;*=35:
)* D&: 3*#$5%*: :5;93*&)&:E *% ()E *: $#2&D &0
&- 5 2 +12
9- 8 1+ 2( )
4- 25! 2
)- 18+ 2
*- 4 4+ 2( ) B*:C2*:=&0 *-
K,- N*& D& /2%4$`% f :!"! )*/$%$)& C53 " #( )= #!" !"0
W&D42D* *D "3*& )* D& :2C*3/$4$* )*D =3&C*4$58B*:C2*:=&0 >IP> 2
>
x
y
F
CA
B D
E
I L
J
K
G
H
-
7/25/2019 Deber 8 Mate
24/36
!"#$%& >A )* +,
@A87 #BEMDN\HEHNMBF S MTEMDOI
KI- W&D42D* D& D5%#$=2) )* D& :$#2$*%=* /$#23&0
B*:C2*:=&0 m!5.12
KJ- XD "3*& )* 2% 4H342D5 45% )$";*=35 , 4;E *: K#4;>8
&- a*3)&)*35 9- 1&D:5
B*:C2*:=&0 &-
KK- N$ *D C*3H;*=35 )* 2%& 4$342%/*3*%4$& *: +L#4;E :2 )$";*=35 ;$)* >L 4;8
&- a*3)&)*35 9- 1&D:5
B*:C2*:=&0 9-
(LL-X7$:=* 2%& 4$342%/*3*%4$& )* 3&)$5 ,F C*3H;*=35 &=&D G2* &;95: :*&% %M;*35: 3&4$5%&D*:8
&- a*3)&)*35 9- 1&D:5
B*:C2*:=&0 9-
(L(-W&D42D* *D "3*& )* D& C&3=* :5;93*&)&E :$ *D 3&)$5 )*D 4H342D5 ;&F53 ;$)* , 4; F *D 3&)$5 )*
D5: 4H342D5: C*G2*t5: ;$)* > 4;8
B*:C2*:=&0 20! cm2
(L>-N* =$*%*% )5: C5D*&: 45% 3&)$5: > F A 4; )* D5%#$=2) 3*:C*4=$6&;*%=*E =&D 45;5 D5
;2*:=3& D& /$#23& &)?2%=&8 O5: 4*%=35: )* D&: C5D*&: :* *%42*%=3&% & @ 4; )* )$:=&%4$&8
Z%& 4533*& CD&%& 2%* *7=*3$53;*%=* D&: )5: C5D*&: C&:&%)5 C53 D5: C2%=5: b F V8
X%=5%4*: D& D5%#$=2) )*D :*#;*%=5 AB *7C3*:&)5 *% 4;E *: $#2&D &0
&- 21
9- 29
4- +
)- K
*- 27
B*:C2*:=&0 9-
-
7/25/2019 Deber 8 Mate
25/36
!"#$%& >@ )* +,
(L+-W&D42D* *D "3*& F *D C*3H;*=35 )* D& /$#23& )* 45D53 %&3&%?&8
(LA-X% bD*?&%)3H&E d&4* ;": )* >ELLL &t5:E *D ;&=*;"=$45 X3&=`:=*%*: 3*&D$e` *D :$#2$*%=*
*7C*3$;*%=50
uWD&6` )5: 6&3$DD&: )* $#2&D D5%#$=2) ]C*3C*%)$42D&3*: & D& =$*33&- 2%& *% bD*?&%)3H& F 5=3&
*% N$*%&E JLLl; &D :238 X:C*3` *D ;5;*%=5E & D5 D&3#5 )*D )H& *% *D G2* D& 6&3$DD& 4D&6&)&
*% bD*?&%)3H& %5 =26$*3& :5;93&8 X% *:* ;5;*%=5E &D#2$*% *%4&3#&)5 )*=*3;$%` *%
N$*%&E G2* D& ;*)$)& )*D "%#2D5 /53;&)5 *%=3* D& 6&3$DD& F *D 3&F5 )* D2e *3& )* !!8 W5%
*:=5: )&=5: F :2 &;CD$5 45%54$;$*%=5 )* D& #*5;*=3H& F *D "D#*93&E X3&=`:=*%*: 4&D42D` D&
D5%#$=2) )* D& 4$342%/*3*%4$& =*33*:=3* 45% 2% *3353 )* (o8v
.$92?* 2% #3"/$45 C&3& 3*C3*:*%=&3 *D *7C*3$;*%=5 )* X3&=`:=*%*: F *:=$;* D& D5%#$=2) )*
D& 4$342%/*3*%4$& =*33*:=3*8 n2:=$/$G2* 4D&3&;*%=* :2 3*:C2*:=&8B*:C2*:=&0 41,142.86 km
@A88 (IOTCINIG S MBEMDN\HEHNMBFG
(L@-N* =$*%* 2% 42&)3&)5 )* D5%#$=2) )* D&)5 a] $%:43$=5 *% 2%& 4$342%/*3*%4$& )* 3&)$5 r]
*%=5%4*: :* 42;CD* G2* .2
2ar =
&- a*3)&)*35 9- 1&D:5
!&3& D&: :$#2$*%=*: )5: C3*#2%=&: 45%:$)*3* D& 4$342%/*3*%4$& ;5:=3&)& *% D& /$#23&&)?2%=&8
(L,-XD =3$"%#2D5"%-*: :*;*?&%=* &D =3$"%#2D5 $#-8
&- a*3)&)*35 9- 1&D:5
B*:C2*:=&0 &-
(LI-N$ D& D5%#$=2) )*D &345"$*: $#2&D & >#4; F D& D5%#$=2) )*D 3&)$5 )* D& 4$342%/*3*%4$& *: +
4;E *%=5%4*: *D "%#2D5 $%:43$=5"%$;$)*2
3
!8
&- a*3)&)*35 9- 1&D:5
B*:C2*:=&0 &-
-
7/25/2019 Deber 8 Mate
26/36
!"#$%& >, )* +,
(LJ-X% D& /$#23& &)?2%=&E *D =3$"%#2D5 *: 3*4="%#2D5 * $:`:4*D*: F :2 4&=*=5 ;$)* a 2%$)&)*:8
.*D 6
-
7/25/2019 Deber 8 Mate
27/36
!"#$%& >I )* +,
((>-X% D& /$#23& &)?2%=&E D& 42*3)& AB *: 45%#32*%=* 45% *D 3&)$5 r )* D& 4$342%/*3*%4$& F D&
42*3)& BC *: $#2&D & r 28 O& ;*)$)&E *% 3&)$&%*:E )*D "%#2D5 *% *D 6
-
7/25/2019 Deber 8 Mate
28/36
!"#$%& >J )* +,
((,-N$ 2% 42&)3&)5 42F5 D&)5 ;$)* .2%$)&)*: :* 4$342%:43$9* & 2%& 4$342%/*3*%4$& 42F5 3&)$5
;$)* , 2%$)&)*:8 O& 3*D&4$`% *%=3* *:=&: D5%#$=2)*: *:0
&- L = 2r
9- L = 2r
4-
2L = r
)- 2L = r
*- L = 3r
B*:C2*:=&0 9-
((I-N* =$*%* 2% =3$"%#2D5 *G2$D"=*35 F =3*: 4$342%/*3*%4$&:
G2* :5% 45%#32*%=*: )* 3&)$5 $#2&D & ( 2%$)&)E D&: 42&D*:
*:="% $%:43$=&: *% *D =3$"%#2D5 =&D 45;5 :* ;2*:=3& *% D&
/$#23& &)?2%=&8 .*=*3;$%* *D "3*& )* D& :2C*3/$4$* )*
)$4d5 =3$"%#2D58
((J-O& D5%#$=2) )* D& 4$342%/*3*%4$& ;5:=3&)&E 42F5 4*%=35 *: O E ;$)* 8!cm8 N$ *D d*7"#5%5
$%:43$=5 *: 3*#2D&3E *D "3*& )*D 4H342D5 :5;93*&)5 *% D& /$#23& &)?2%=&E *% cm2
E *: $#2&D &0
&-!
3
9-
2!
3
4-4!
3
)- 2!
*- 4!
B*:C2*:=&0 4-
((K-X% D& /$#23& &)?2%=&0 bW *: *D )$";*=35E bV T , ;E VW T A ;U .X *: C&3&D*D5 & VWU .X T ( ;8
X%=5%4*: *D "3*& )*D p b.X *: &D "3*& )*D p bVW 45;50
&- (P+,9- (P(,
4- (PA
)- (P>L
*- (P>@
B*:C2*:=&0 9-
O
-
7/25/2019 Deber 8 Mate
29/36
!"#$%& >K )* +,
(>L-R E S F T :5% D5: 4*%=35: )* D&: :*;$4$342%/*3*%4$&: G2* &C&3*4*% *% D& /$#23&8
ABCDEF*: 2% d*7"#5%5 3*#2D&38 W&D42D&3 *D 6&D53 )*D "3*& )* D& /$#23& :5;93*&)&8
B*:C2*:=&0 24 3! 6( ) cm2
(>(-.*=*3;$%* *D "3*& )* D& 3*#$`% :5;93*&)&E :$ *D 42&)3&)5 =$*%* D5%#$=2) )* D&)5 a 8
(>>-O& /$#23& ;5:=3&)& *: 2%& :*;$4$342%/*3*%4$& )* D5%#$=2) )* 3&)$5 r F )* 4*%=35 O 8 N$ B
*: 2% C2%=5 )* =&%#*%4$& F ECBEDBAD === E *%=5%4*: *D "3*& )* D& 3*#$`%
:5;93*&)&E *%u2
E *: $#2&D &0
&-2
5r
! 9-
2
6r
! 4-
2
8r
! )-
2
12r
! *-
2r!
B*:C2*:=&0 9-
(>+-
W&D42D* *D "3*& )* D& e5%& :5;93*&)&E :&9$*%)5 G2* W *: *D 4*%=35 )* D& 4$342%/*3*%4$& )*3&)$5 WjE G2* Ww *: *D 4*%=35 )* D& 4$342%/*3*%4$& )* 3&)$5 Wwj F G2* D& D5%#$=2) )* *:=&
MD=$;& *: )* J#4;8
B*:C2*:=&02
24 cm!
-
7/25/2019 Deber 8 Mate
30/36
!"#$%& +L )* +,
(>A-O& D5%#$=2) )*D 3&)$5 r )* 2%& 4$342%/*3*%4$& $%:43$=& *% 2% =3$"%#2D5 *G2$D"=*35 *:
r =3
6lE )5%)* l *: D& D5%#$=2) )*D D&)5 )*D =3$"%#2D58
&- a*3)&)*35 9- 1&D:5
B*:C2*:=&0 &-
(>@-X% D& /$#23&E D& D5%#$=2) )*D :*#;*%=5 bV *: )* > 4; F D& )* D5: :*#;*%=5: 45%#32*%=*: VW
F W. *: )* ( 4;8 O&: 4236&: :5% =5)&: :*;$4$342%/*3*%4$&:8 W&D42D&3 *D "3*& )* D& e5%&
:5;93*&)&8
B*:C2*:=&011!
8cm
2
(>,-
W&D42D* *D "3*& :5;93*&)& )* D& /$#23&E :&9$*%)5 G2* a=
1m 8
B*:C2*:=&02
m!
(>I-
W&D42D* *D "3*& :5;93*&)& )* D& /$#23&E :&9$*%)5 G2* & T A;8
(>J-
W&D42D* *D "3*& *7&4=& )* D& e5%& :5;93*&)& :$0 D& 4236& $%=*3$53 *: &345 )* D&4$342%/*3*%4$& )* 4*%=35 jU D& 4236& *7=*3$53 *: &345 )* D& 4$342%/*3*%4$& )* 4*%=35 jw U D&:
3*4=&: Vj F bj :5% C*3C*%)$42D&3*: F *D :*#;*%=5 jb ;$)* A ;*=35:8
B*:C2*:=&02
16 m
-
7/25/2019 Deber 8 Mate
31/36
!"#$%& +( )* +,
B*:2*D6& D5: > :$#2$*%=*: *?*34$4$5:0
(>K-N$ *D "3*& :5;93*&)& /2*:* )* (@#4;>F *D "%#2D5 bE D&: )5: =*34*3&: C&3=*: )*D "%#2D5 VE
4&D42D* D& D5%#$=2) )*D 3&)$58
B*:C2*:=&0 @4;
(+L-N$ D& 3*D&4$`% *%=3* D5: "%#2D5: /2*:* D& ;$:;& G2* *% &- F *D 3&)$5 ;$)$*:* (L4;E g42"%=5:
4*%=H;*=35: 42&)3&)5: ;*)$3H& *D "3*& :5;93*&)&h
B*:C2*:=&02
60 cm!
(+(-X% D& /$#23& ;5:=3&)&E *D =3$"%#2D5 ABC *:=" $%:43$=5 *% D& 4$342%/*3*%4$& )* 4*%=35 O E *D
C*3H;*=35 )*D =3$"%#2D5 ABC *:E *% 2%$)&)*:E $#2&D &0
&- +L
9- AL
4- @L
)- ,L
*-
JL
B*:C2*:=&0 )-
@A86 0BCDEFG MBEMDOFEHG
(+>-X% D& /$#23& &)?2%=& D&: )5: 4$342%/*3*%4$&: 45% =&%#*%=*:8
N$ D& D5%#$=2) )* 4&)& 4$342%/*3*%4$& *: $#2&D & 4!cmE
*%=5%4*: *D "3*& )* D& :2C*3/$4$* )*D 3*4="%#2D5 ABCD E
*% cm2
E *: $#2&D &0&- 4 9- 8 4- 16 )- 32 *- 64
B*:C2*:=&0 )-
(++-X% D& :$#2$*%=* /$#23&E PR *: *D 3&)$5 )*D 4H342D5E *D 42&D
*: 45%#32*%=* 45% 2% D&)5 )*D 3*4="%#2D5 *&'+8 N$ *D
4H342D5 =$*%* 2% "3*& $#2&D & 4! cm2
F *D 3*4="%#2D5
=$*%* 2% "3*& $#2&D & 8 cm2
E *%=5%4*: *D C*3H;*=35 )* D&
3*#$`% :5;93*&)&E *% ()E *: $#2&D &0
&- !+8 9- !+10 4- !+12 )- 2!+8 *- 2!+12
B*:C2*:=&0 &-
A B
CD
x 3x
2x 2
A
B CO
-
7/25/2019 Deber 8 Mate
32/36
!"#$%& +> )* +,
(+A-N* =$*%*% )5: 4$342%/*3*%4$&: 45%4E *: $#2&D &0
&-!
4
9-!
3 4-
!
2
)- ! *- 3!
B*:C2*:=&0 &-
(+@-XD "3*& 45;M% *%=3* *D d*7"#5%5 3*#2D&3 F *D 4H342D5 ;5:=3&)5: *% D& /$#23&E )&)5 G2* D&
&C5=*;& )*D d*7"#5%5 3*#2D&3 *: $#2&D &3 3
2m E =$*%* 2% 6&D53E *% )
>E )*0
&-9
4
9- 3
4-
81
4
)-27
16
*-
B*:C2*:=&0 9-
(+,- X% D& /$#23& &)?2%=&E AB*: 2% )$";*=35 )* D& 4$342%/*3*%4$& F ;$)* 4 cm 8 XD 6
-
7/25/2019 Deber 8 Mate
33/36
!"#$%& ++ )* +,
(+I-N* 45%54* G2* O *: *D 4*%=35 )* D& 4$342%/*3*%4$& ;5:=3&)&E OA =10cm F
m !CBA( ) = 30o E *%=5%4*: *D "3*& )*D :*#;*%=5 4$342D&3E *% ()>E *: $#2&D &0
&-
50 !
3+
3
2
!
"#
$
%&
9- 50 !
3+
1
2
!
"#
$
%&
M^ 50 !
3!
3
2
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#$
%
&'
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3!1
2
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#$
%
&'
*- 50 !
3
! 3
4
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#$
%
&'
B*:C2*:=&0 4-
(+J-W5%:$)*3* *D #3"/$45 &)?2%=58 N$ *D "3*& )* D& :2C*3/$4$* )*D =3$"%#2D5 .!j ]D& 3*#$`%
:5;93*&)&- *: )* @P> 4;xE *%=5%4*: *D "3*& )* D& :2C*3/$4$* )*D C5DH#5%5 bVW. *:0
&- @ 4;x
9- (L 4;x
4- >L 4;x
)- (>8@ 4;x
*- (I8@ 4;x
B*:C2*:=&0 9-
(+K-XD "3*& )* D& C&3=* :5;93*&)& *: $#2&D &0
&-3!r
2
4
9- !r2 1
4! 3
2
"
#$$
%
&''
4-3!r
2
64
)-3!r
2
16
*-3!r
2
8
B*:C2*:=&0 )-
-
7/25/2019 Deber 8 Mate
34/36
!"#$%& +A )* +,
(AL-N* $%:43$9* 2% =3$"%#2D5 )* D&)5: &E 9E 4 *% 2%& 4$342%/*3*%4$& )* 3&)$5 3E *%=5%4*: *D "3*&
)* D& :2C*3/$4$* )*D =3$"%#2D5 *: $#2&D &0
&- &9 P >43
9- >&94 P 3
4- A&9 P 43
)- &94 P A3*- &94 P >3
B*:C2*:=&0 )-
(A(-X% *D #3"/$45 &)?2%=5 D5: &345: QiE i!E !k =$*%*% D& ;$:;& D5%#$=2) F ! 5*: *D 4*%=35 )* D&
4$342%/*3*%4$&U D& ;*)$)& )*D "%#2D5 !Bk *: $#2&D &0
&- RP>
9- RP+4- RPA
)- R
*- RPJ
B*:C2*:=&0 4-
(A>-.&)5 *D 4H342D5 )* 3&)$5 rE D& )$:=&%4$& )*D 4*%=35 j & D& G2* :* )*9* 29$4&3 2% C2%=5 rE
C&3& G2* D& :*4&%=* =3&e&)& )*:)* )$4d5 C2%=5 )*=*3;$%* 2%& 42*3)& bV =&D G2* D&D5%#$=2) )* bV :*& $#2&D & D& D5%#$=2) rb * $#2&D & 3 *:0
&- r 3
9- 2r 3
4- r 2
)- 2r 2
*- 3r 3
B*:C2*:=&0 &-
(A+-N$ 7 T +2E F T K2E )*=*3;$%* *D "3*& )* D& C&3=* :5;93*&)& 8
N
-
7/25/2019 Deber 8 Mate
35/36
!"#$%& +@ )* +,
(AA-O&: 4$342%/*3*%4$&: ;5:=3&)&: *% D& /$#23& :5% 45%4
-
7/25/2019 Deber 8 Mate
36/36
(AJ-N$ D5: D&)5: )*D 42&)3&)5 bVW. ;$)*% A 4;8 X%=5%4*: *D "3*& )* D& C&3=* :5;93*&)& )* D&
/$#23& *: $#2&D &0
&- (, 4;>
9- J#4;>
4- A# 4;>
)-
>#4;>*- (,#4;
>
B*:C2*:=&0 &-
(AK-O& 4$342%/*3*%4$& 45% 4*%=35 *% !*:=" $%:43$=& *% *D 42&)3&)5 !"#$E &)*;": !"F !"
:5% :*;$4$342%/*3*%4$&: =&%#*%=*:8 W&D42D* *D "3*& )* D& 3*#$`% :5;93*&)& :$ !" ! !4;8
B*:C2*:=&0 16 3!!
3
!
"#
$
%& cm
2
(@L-XD d*7"#5%5 )* D& /$#23& *: 3*#2D&3 F :2 &C5=*;& =$*%* D5%#$=2) !!!"8 .*=*3;$%* *D
C*3H;*=35 )* D& 3*#$`% :5;93*&)&8
B*:C2*:=&08a
99+ 3!( ) cm