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Solution: +HUH x x
Solution: +HUH x RU x x
RU x
RU x
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RU x RU x î
RU x
RU x RU x
RU x
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([DPSOH 6ROYH [ î x
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RU x î x RU x ± î x î
RU î x î î RU x ± î x î
RU x î RU x ± x
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RU x î D ± D
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RU D RU D
RU x RU x RU x
x RU x
RU x ? x = 3,2
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RU \ \
RU \ \
RU \ ± \
RU \ ± \
RU \ ± \ ± \
RU \ \ ± ± \ ±
RU \ ± \ ±
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(LWKHU RU
\ ± \ ±
RU \ RU \
RU x RU \
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RU x
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Solution: xx \y x x+-y2y
x - y)(x - y) - (x + y)(x + y)
(x + y)(x - y)
x2 xy - xy + y2 x2 xy + xy + y2
x2 - y2
x xy xy yx xy y
x x x y y y
([DPSOH 6LPSOLI\ x + y x - y y - x
Solution: x y x - y y - x
+
xx yy xx yy y x [
x - xy x x y
x x xy
x x y
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([DPSOH 6LPSOLI\ x x x x x x
Solution: x x x x x x
x x x x x x x x x
x x x x x x x x x
x x x x x x
x x xx x x
x x x
x x x
x xx x
([DPSOH 6LPSOLI\ x x x x ି࢞࢞ା࢞
Solution: x x x x ଵି௫ଶమ௫ା௫ర
x x x x xx xx x x x
> x x xx @ > x xx x@ x x x
x x x x x x
x xx x x x x
x x x x x x
x x x x x xx xx x
x x xx x x x x
x x x x
xx x
([DPSOH 6LPSOLI\ x x x y y y x xx y y y x yx y y
Solution: x x xy y y x x x y y y [ yx y y
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x y xx x yx yy y xx yx y xy xy y [ yx y y
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x xx y y [ yx y y
x x x yy y
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x xx yy y x x x y xyy y
x xx yy y
([DPSOH 6LPSOLI\ a b b a b b a a bb a a bb
Solution: ab b a bb a abb a a bb
E D D EE D E ED E a a bb a a bb
ab ba b a b b a ab b a a bb
a a bb a a bb a a bb
ab a a bb a abb a b a a bb
a b a ba b a b ab a a bb
a a bb a a bb
a b a a bb a ab b a b
a b aa bb a b
a aE îb a
a a bb
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([DPSOH 6LPSOLI\ x a x a x a x a
Solution: x a x a x a x a
x a x a x a x a
x x aa x x a a x -xa a x xa a
x x D x xD xD D
x x a x x a
x xx aa x x ax a
x x a a x xx a a x
x a a xx a
(xDPSOH 6LPSOLIy m m m m m
Solution: m m m mm
$OJHEUD m m m m m mm
m m m m mm
mm m mm
mm mm mm mm
m m m m m m
m m m m m
m m
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x x x x xx xx
xx x x x x xx
x x x x xx
xx x x xx
x x x x
xx x x
x
(xDPSOH 6LPSOLIy x a x x a x x a x x a
Solution: x a x xa x xa x xa
x a x xa x xa x x a
x a x xa x xa x a x x a
x a x xa xx xa a x a x
x a x xa xx aa xx a x
x a x xa x xa ax xa
x a x xa x x a x xa a
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x a x xa x xa
x a x xa x a x x a
x a xx x a a x ax
x a xx a a xx ax
x a x xa xa xa
x a x ax xx aa x a
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x x-aa xx a
x ax - ax a
x a x xa a
x a
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Solution: +HUH
x \ ««««««««« L
x \ «««««««««« LL
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D E F G H I x x J x x K L
M యξ N DE D O x\ ] \]
D E F G H ]x J x\ K x\ L ED
I x \
D E F G H I J K L
D E F I
G H M D
D E F G H I J K L x x \
D E
D D E E x F DD G PP
D E F G H I J K [
Exercise - 4.2.2 (Rationalization)
D E F G H I J K
D E F G H ± I J
K x x L M
D E F G
D E F D ED E
D E x DDx F
D E F
D x E D D F x
Exercise - 4.2.3 (Radical Equation)
D E F G H I
H
D E F G H I J K
D E F G L ଶଵହ I J K L
L M
D E F G H I
H I J K
D E D F G H
D E F G
D D E F G
$OJHEUD DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ
Exercise - 4.3.1 (Indices)
D E F G H I J K
L N O P Q R S
M Q
D E F
D x \ E x \ F DE G H
x \
I DF
E
J x\ K D E L x M N \x O x\
P ξͷ ܽయమ ξܾ Q ଶ௫௬మమ R ௫௬మ S D E
D E F G H I J K xD E F L
D E F K x D E F
G D P Q S H I J
L M N O
D E F
I
G x D E D F H
D x x D Q E ሺሺଶ௬ିି௬ሻሻమ F G ሺ௫ି௫యሻೣ
D x Q E x S T F \ x \
\ \ x
G H I
Exercise: 4.3.2 (Exponential Equation)
D E F G H r I J K L
D E F G H I J K L M N O
D E F G H
I J K L
D E F G H I J
K L M N O
Exercise - 4.4 (Simplification) E \ x x \\ G \ x x \\
D DD E E F P PP K xx
J x \ \ ]
H D DE DE E I S S G x x
N ሺ௫௫ା௬௬ሻ K
L x x x M D E F x
J P P Q Q
D D D E E x x \\
H I x x \ \
L DE M
F P PP P G
D D D E x x x
ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ $OJHEUD
H D D D I x x J x x x K \ \\ \
E P P I x x
D E \ \ \ F G
D x x x H x x x \ \ \
G [[ [ E F D DD E E E
D E D D D I x x x
D x x x x F
H x xx I F x x x
D x x E D D J x x x G
D D E x F G D
D x x x\ \ x x x \\ \x \ E D D E K x
G D D
H x x \ I D D
F x x G x H x D x D
F P P QQ G x x
J D E E K x x \
D D E x F x G x D
H x x I x xx J x K DD E E
L x M ଵ N O P ଵିξ ௫௫మ
ሺ௬ାଶሻమ
Exercise - 4.5 (Simultaneous Equation)
D x \ E x \ F x \ G x \
J x \ K x \
H x \ I x \
L x \ M x \
D E F G
D E F
D E
D P E P P
D 5V 5V E 5V 5V
D \HDUV \HDUV E \HDUV \HDUV F \HDUV \HDUV
D \HDUV \HDUV
E \HDUV \HDUV F \HDUV \HDUV G \HDUV \HDUV H \HDUV \HDUV
I \HDUV \HDUV J \HDUV \HDUV
D \HDUV \HDUV E \HDUV \HDUV F \HDUV \HDUV
I \HDUV \HDUV
G \HDUV \HDUV H % 6
$OJHEUD
DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ
D E F G H
D E F G H I
D E F G
D NP KU NP KU E 3 0 F NP KU NP KU G NP KU NP KU
D 5V 5V E 3HQV ,QN SRWV F WLFNHWV HDFK FDWHJRU\ G H GD\V
D \HDUV \HDUV E \HDUV \HDUV F \HDUV \HDUV G H
Exercise - 4.6 (Quadratic Equation)
D E F G H I
J K L
D E F G H I
D E F
D E F G
D E F G H
D E F G H
D E F
D \HDUV E \HDUV F \HDUV \HDUV G \HDUV \HDUV H \HDUV
D \HDUV E \HDUV \HDUV F \HDUV \HDUV G \HDUV \HDUV
D E F G H
D E F
D P P E P F P P G P P
D FP FP FP E FP FP F FP FP FP G FP FP
H FP FP FP I FP FP FP
D E F P P G P P
ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ $OJHEUD
Chapter Geometry
5
Objectives
At the end of this chapter, the Students will be able to
• construct the triangles and quadrilateral having equal areas.
• show the interrelation between the triangle and quadrilateral .
• prove and verify the properties related to area of triangle and quadrilateral .
• prove and verify the properties and theorems related to arc, chord and angle
of a circle.
• show the interrelation of the properties related to arc and angle of the circle.
• solve the problems related to arc and angle of the circle.
Teaching materials:
&KDUW SDSHU ÀDVK FDUGV JHRPHWULF ER[ SHQFLOV GL൵HUHQW FRORU VLJQ SHQV DQG
board markers, paper cuters, thread, triangular and circular solid objects etc.
C
O B
AD
$UHD RI 7ULDQJOH DQG 4XDGULODWHUDO $ l %
E
&
$UHD RI UHFWDQJOH
%
,Q WKH DGMRLQLQJ ILJXUH $%&' LV D UHFWDQJOH 2SSRVLWH VLGHV RI WKH &
UHFWDQJOH DUH HTXDO
L H $% '& l OHQJWK '
$' %& E EUHDGWK
7KH DUHD RI D UHFWDQJOH LV FDOFXODWHG DV WKH SURGXFW RI LWV OHQJWK DQG EUHDGWK
6R DUHD RI WKH UHFWDQJOH $ OHQJWK î EUHDGWK
$% î %&
$UHD RI WKH UHFWDQJOH $ l î E
$UHD RI WULDQJOH $
$& LV WKH GLDJRQDO RI WKH UHFWDQJOH $%&' :H NQRZ WKDW WKH
GLDJRQDO $& ELVHFWV WKH UHFWDQJOH LQ HTXDO DUHD
6R '$%& '$'& UHFWDQJOH $%&' LQ DUHD '
$UHD RI '$%& '$'& OHQJWK î EUHDGWK EDVH î KHLJKW
:H FRQFOXGH WKDW
,I WKH EDVH DQG KHLJKW RI D WULDQJOH DUH JLYHQ
7KH DUHD RI WKH WULDQJOH LV EDVH î KHLJKW
$V LQ JLYHQ ILJXUHV
$ $ $
K K
K
&E % % '& % E& '
E
$UHD RI '$%& $UHD RI '$%& $UHD RI '$%&
$ %& î $% $ %& î $' $ %& î $'
EK EK EK
$JDLQ
,I WKUHH VLGHV D E DQG F RI D WULDQJOH $%& DUH JLYHQ WKH DUHD RI WKH WULDQJOH LV
V V D V E V F ZKHUH V LV WKH VHPL SHULPHWHU RI WKH WULDQJOH $%&
L H 6 D E F +HURQ¶V IRUPXOD
ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ *HRPHWU\
$UHD RI 6TXDUH
ǡ Ǥ
ͻͲι
Ǥ α α α α
ǡ
Ǥ
ǡ
DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ
α
α α ʹ
ȋ Ȍ α ʹ
ǡ
'
α ξܤܣଶ ܥܤଶ
α ξܽଶ ܽଶ
ǡ α ξʹܽଶ
ǡ α ξʹ Ǥ
ǡ α ξʹ Ǥ
? α ξௗଶ
ȋ Ȍ α
α ξௗଶ ξௗଶ
α ଵଶ ʹ
α ଶଵ ʹ
*HRPHWU\
$ 1'
$UHD RI SDUDOOHORJUDP 0 &
+HUH $%&' LV D SDUDOOHORJUDP ZKHUH $% &' DQG $%
&' $' %& DQG $' %& 7KH GLDJRQDO $& ELVHFWV WKH
SDUDOOHORJUDP LQ HTXDO DUHD
L H '$%& '$'& LQ DUHD
'UDZ SHUSHQGLFXODUV $0 DQG &1 RQ %& DQG $' UHVSHFWLYHO\ %
1RZ DUHD RI WKH SDUDOOHORJUDP $%&' $UHD RI '$%& DUHD RI '$'&
? $UHD RI SDUDOOHORJUDP $%&' %& î $0 $' î &1
%& î $0 %& î $0 >$0 &1 DQG $' %&@
î %& î $0 %& î $0
?$UHD RI D SDUDOOHORJUDP EDVH î KHLJKW E î K
$UHD RI UKRPEXV
+HUH $%&' LV D UKRPEXV $& DQG %' DUH LWV GLDJRQDOV ZKLFK $ '
&
ELVHFW HDFK RWKHU DW ULJKW DQJOH DW 3
'1
L H $3 &3 %3 '3
1RZ $UHD RI UKRPEXV $%&'
DUHD RI '$%' DUHD RI '%&' S
%' î $3 %' î &3 %
%' $3 &3 %' î $&
G î G ZKHUH G DQG G DUH GLDJRQDOV RI WKH UKRPEXV
? 7KH DUHD RI WKH UKRPEXV RQH KDOI RI WKH SURGXFW RI LWV GLDJRQDOV
$
$UHD RI WUDSH]LXP
+HUH $%&' LV D WUDSH]LXP LQ ZKLFK $' %& $' DQG %& DUH
LWV EDVHV $& LV WKH GLDJRQDO RI WKH WUDSH]LXP
'UDZ $0 A %& DQG &1 A $1 ZKHUH $' LV SURGXFHG WR 1
1RZ $UHD RI WUDSH]LXP $%&' %0 &
DUHD RI '$%& DUHD RI '$'&
î %& î $0 $' î &1 > $0 &1@
%& î $0 $' î $0
$0 %& $'
ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ *HRPHWU\
? $UHD RI D WUDSH]LXP î KHLJKW î VXP RI WKH EDVHV
2U $UHD RI WUDSH]LXP î KHLJKW î VXP RI WKH RSSRVLWH SDLU RI SDUDOOHO VLGHV
$
$UHD RI TXDGULODWHUDO
+HUH $%&' LV D TXDGULODWHUDO DQG %' LWV
GLDJRQDO 'UDZ $1 A %' DQG &0 A %' % 0 '
1RZ DUHD RI TXDGULODWHUDO $%&' 1
DUHD RI '$%' DUHD RI '%&'
%' î $1 %' î &0 &
%' $1 &0
? $UHD RI D TXDGULODWHUDO RQH KDOI RI WKH SURGXFW RI D GLDJRQDO DQG VXP RI WKH
SHUSHQGLFXODUV GUDZQ IURP WKH RSSRVLWH YHUWLFHV WR WKH GLDJRQDO
8. Area of a kite :
ǡ
α α ǡ
Ǥ
α ' Ϊ '
α ଶଵ Ϊ ଵଶ
αଵଶ ȋ Ϊ Ȍ
α ଵଶ ȋ Ϊ α Ȍ
α ଶଵ ͳ ʹ
Ǥ α ଶଵ
*HRPHWU\ DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ
:RUNHG 2XW ([DPSOHV
([DPSOH )LQG WKH DUHD DQG SHULPHWHU RI WKH VTXDUH ZKRVH OHQJWK RI GLDJRQDO LV
Solution: FP $ '
+HUH % &
/HQJWK RI D GLDJRQDO RI WKH VTXDUH FP
$UHD RI WKH VTXDUH $ GLDJRQDO
î FP
î FP
FP
1RZ OHQJWK RI D VLGH RI WKH VTXDUH l $
ξ͵ ܿ݉ଶ 3 6
FP
? 3HULPHWHU RI WKH VTXDUH l
= î FP FP
([DPSOH ,Q WKH JLYHQ ILJXUH 3456 LV D UKRPEXV 35
DQG 46 DUH LWV GLDJRQDOV ,I 34 FP DQG 2
35 FP ILQG WKH DUHD RI UKRPEXV 3456
Solution: +HUH 45
3456 LV D UKRPEXV 35 DQG 46 DUH LWV GLDJRQDOV ZKLFK DUH ELVHFWHG DW ULJKW DQJOH DW
2 ,Q ULJKW DQJOHG '324
34 FP 32 35 î FP FP
? 42 34 32
FP FP
FP FP FP FP
1RZ DUHD RI WKH UKRPEXV 3456 35 î 46
35 î 42 > 46 42 62@
=
î FP î FP FP
([DPSOH ,Q WKH DGMRLQLQJ ILJXUH :;<= LV D :
TXDGULODWHUDO :$ DQG <% DUH
SHUSHQGLFXODUV RQ WKH GLDJRQDO ;= RI WKH %
$
TXDGULODWHUDO ,I $% $; %= FP
:$ <% FP ILQG WKH DUHD RI WKH ;
TXDGULODWHUDO :;<= <
+HUH
Solution:
$% FP ? $% FP
$; FP ? $; FP
ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ *HRPHWU\
%= FP ? %= FP
:$ FP ? :$ FP
<% FP ? <% FP
;= $; $% %=
FP FP FP FP
$UHD RI WKH TXDGULODWHUDO :;<= ;= :$ <%
î FP FP FP
FP î FP
FP
([HUFLVH
)LQG WKH DUHD RI D UKRPEXV ZKHUH WKH OHQJWK RI WKH GLDJRQDOV DUH FP DQG FP
,I WKH DUHD RI D VTXDUH LV FP ILQG WKH OHQJWK RI LWV GLDJRQDO
,I WKH DUHD DQG EDVH RI WKH SDUDOOHORJUDP DUH FP DQG FP UHVSHFWLYHO\ ILQG LWV DOWLWXGH
&DOFXODWH WKH OHQJWK RI D VLGH RI UKRPEXV ZKRVH OHQJWK RI GLDJRQDOV DUH FP DQG FP
UHVSHFWLYHO\
7KH DUHD RI WKH UHFWDQJOH LV FP ,I WKH OHQJWK RI D VLGH LV FP ILQG WKH OHQJWK RI LWV
GLDJRQDO
,I WKH SHULPHWHU RI D VTXDUH LV FP ILQG WKH OHQJWK 3 6
RI LWV GLDJRQDO 1
,Q WKH JLYHQ ILJXUH 3456 LV D SDUDOOHORJUDP ZKHUH
30A45 DQG 31A56 ,I 30 FP 45 FP DQG
31 FP ILQG WKH OHQJWK RI 34
40 5
$ '
)
,Q WKH DGMRLQLQJ ILJXUH $%&' LV D SDUDOOHORJUDP
ZKHUH $(A%& DQG $)A&' ,I %& FP $( FP
DQG $) FP ILQG WKH OHQJWK RI $%
%( &
*HRPHWU\ DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ
,Q WKH JLYHQ ILJXUH '()* LV D WUDSH]LXP LQ ' + (
ZKLFK '( *) *+A'( ,I *+ *) '( FP
FDOFXODWH WKH DUHD RI WUDSH]LXP '()*
* )
$
$& DQG %' DUH WZR GLDJRQDOV RI WKH NLWH $%&' ,I
$& %' FP ILQG LWV DUHD
%2 '
& '
,Q WKH JLYHQ ILJXUH $%&' LV D WUDSH]LXP LQ ZKLFK $
$' %& DQG '0A%& ,I &' FP $' &0
%0 FP FDOFXODWH WKH DUHD RI WUDSH]LXP $%0'
% 0 &
3 6
,Q WKH DGMRLQLQJ ILJXUH 3456 LV D TXDGULODWHUDO
DQG 46 LV D GLDJRQDO 5; DQG 3< DUH ;<
SHUSHQGLFXODUV RQ 46 ,I 3< FP 5; FP 4
34 FP DQG 6< FP ILQG WKH DUHD RI WKH
TXDGULODWHUDO 3456
5
$
,Q WKH DGMRLQLQJ ILJXUH $%&' LV D TXDGULODWHUDO % ( '
LQ ZKLFK '( DQG %) DUH SHUSHQGLFXODUV RQ $& )
,I WKH DUHD RI TXDGULODWHUDO $%&' LV FP DQG
OHQJWK RI GLDJRQDO $& FP ZKDW ZLOO EH WKH &
VXP RI WKH OHQJWK RI %) DQG '(" $
,Q WKH JLYHQ ILJXUH $%&' LV D TXDGULODWHUDO LQ %3 '
ZKLFK GLDJRQDOV $& DQG %' LQWHUVHFW DW ULJKW &
DQJOH DW 3 ,I WKH DUHD RI TXDGULODWHUDO $%&' LV
FP %' FP DQG $3 FP FDOFXODWH WKH
OHQJWK RI &3
ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ *HRPHWU\
7KHRUHP $ '
The diagonal of the parallelogram bisects it in equal area.
*LYHQ $& LV WKH GLDJRQDO RI WKH SDUDOOHORJUDP $%&'
7R SURYH '$%& '$'& $%&' LQ DUHD % &
Proof :
6WDWHPHQWV 5HDVRQV
,Q '$%& DQG '$'&
L $% '& 6 L %HLQJ RSSRVLWH VLGHV RI WKH SDUDOOHORJUDP
LL %& $' 6 LL %HLQJ RSSRVLWH VLGHV RI WKH SDUDOOHORJUDP
LLL $& $& 6 LLL %HLQJ WKH FRPPRQ VLGH
? '$%& # '$'& %\ 6 6 6
$UHD RI '$%& DUHD RI $UHDV RI FRQJUXHQW WULDQJOHV DUH HTXDO
'$'&
$UHD RI '$%& DUHD RI )URP VWDWHPHQW
'$'& DUHD RI $%&'
Proved
7KHRUHP
Parallelograms on the same base and between the same parallel lines are equal in area.
*LYHQ $) %& DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ
$%&' DQG (%&) DUH VWDQGLQJ RQ WKH VDPH EDVH %& DQG
EHWZHHQ WKH VDPH SDUDOOHOV $) DQG %&
7R 3URYH $UHD RI $%&' DUHD RI (%&)
*HRPHWU\
Proof :
6WDWHPHQWV 5HDVRQV
,Q '$%( DQG ''&)
L $% '& 6 L 2SSRVLWH VLGHV RI $%&'
LL %$( &') $ LL %HLQJ FRUUHVSRQGLQJ DQJOHV DV $% '&
LLL $(% ')& $ LLL %HLQJ FRUUHVSRQGLQJ DQJOHV DV (% )&
? '$%( # ''&) %\ 6 $ $ RU $ $ 6
'$%( ''&) LQ DUHD $UHDV RI FRQJUXHQW WULDQJOHV DUH HTXDO
WUDSH]LXP $%&) ''&) %\ HTXDO D[LRP
WUDSH]LXP $%&) '$%( :KROH SDUW D[LRP
$UHD RI $%&' DUHD RI (%&)
Proved
7KHUHIRUH SDUDOOHORJUDPV RQ WKH VDPH EDVH DQG EHWZHHQ WKH VDPH SDUDOOHO OLQHV DUH HTXDO LQ DUHD
$ (' )
$OWHUQDWH 0HWKRG
Construction: 'UDZ (0 A %& %0 &
Proof:
6WDWHPHQWV 5HDVRQV
$UHD RI $%&' %&u(0 )RUPXOD RI WKH DUHD RI SDUDOOHORJUDP
EDVH u KHLJKW
$UHD RI (%&) %&u (0 )RUPXOD RI WKH DUHD RI SDUDOOHORJUDP
EDVH u KHLJKW
$UHD RI $%&' $UHD RI (%&) )URP VWDWHPHQW DQG
Proved
&RUROODU\
3URYH WKDW WKH DUHD RI D SDUDOOHORJUDP DQG WKH DUHD RI UHFWDQJOH RQ WKH VDPH EDVH DQG EHWZHHQ WKH
VDPH SDUDOOHO OLQHV DUH HTXDO
*LYHQ 5HFWDQJOH $%&' DQG SDUDOOHORJUDP %&() DUH VWDQGLQJ RQ
WKH VDPH EDVH %& DQG EHWZHHQ WKH VDPH SDUDOOHO OLQHV $( DQG %&
7R SURYH $UHD RI UHFWDQJOH $%&' $UHD RI SDUDOOHORJUDP %&()
*HRPHWU\
ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ
3URRI 5HDVRQV
6WDWHPHQWV L %RWK DUH ULJKW DQJOH
,Q ' $%) DQG ' &'( LL EHLQJ RSSRVLWH VLGHV RI D SDUDOOHORJUDP
L %$) &'( 5 LLL VDPH DV UHDVRQ LL
LL %) &( + %\ 5+6 IDFW
LLL $% &' V $UHD RI FRQJUXHQW WULDQJOHV DUH HTXDO
$GGLQJ WUDSH]LXP %&') RQ ERWK VLGHV RI
' $%) # '&'(
VWDWHPHQW
$UHD RI '$%) $UHD RI '&'(
$UHD RI '$%) $UHD RI WUDSH]LXP :KROH SDUW D[LRP XVLQJ LQ VWDWHPHQW
%&') $UHD RI '&'( $UHD RI 3URYHG
WUDSH]LXP %&')
$UHD RI UHFWDQJOH $%&' $UHD RI
SDUDOOHORJUDP %&()
7KHRUHP
The area of a triangle is equal to half of the area of a parallelogram on the same base and between
the same parallel lines.
'$%& DQG '%&( DUH VWDQGLQJ RQ WKH VDPH EDVH $ )' (
*LYHQ
%& DQG EHWZHHQ WKH VDPH SDUDOOHOV $( DQG %&
7R 3URYH $UHD RI ' $%& DUHD RI '%&(
&RQVWUXFWLRQ 'UDZ &)__%$ WKURXJK WKH YHUWH[ & %&
Proof :
6WDWHPHQWV 5HDVRQV
$%&) LV D SDUDOOHORJUDP 2SSRVLWH VLGHV EHLQJ SDUDOOHO
L H $% )& DQG $) %&
'$%& $%&) LQ DUHD 'LDJRQDO ELVHFWV D SDUDOOHORJUDP LQ HTXDO DUHD
$%&) '%&( LQ DUHD %RWK SDUDOOHORJUDPV DUH RQ WKH VDPH EDVH
?'$%& '%&( LQ DUHD DQG EHWZHHQ WKH VDPH SDUDOOHO OLQHV
)URP VWDWHPHQWV DQG
7KHUHIRUH WKH DUHD RI D WULDQJOH LV HTXDO WR KDOI RI WKH DUHD RI D SDUDOOHORJUDP RQ WKH VDPH EDVH
DQG EHWZHHQ WKH VDPH SDUDOOHO OLQHV
1RWH 7KLV WKHRUHP FDQ EH SURYHG E\ XVLQJ WKH IRUPXOD RI DUHD RI WULDQJOH DQG SDUDOOHORJUDP
7KHRUHP
Triangles on the same base and between the same parallel lines are equal in area.
*LYHQ '$%& DQG ''%& DUH VWDQGLQJ RQ WKH VDPH EDVH
%& DQG EHWZHHQ WKH VDPH SDUDOOHOV $' DQG %&
7R SURYH $UHD RI '$%& DUHD RI ''%&
&RQVWUXFWLRQ 'UDZ &( __ %$ WKURXJK WKH YHUWH[ &
*HRPHWU\ DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ
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Prime Mathematics 10
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