Additional Mathematics Project Work 2017 Name
: Cheow Jun Hao( 5 ) Chow Wen Chong Ng Zhuo Ran Teoh Teoh Yu Heng Heng
IC
: 000130-08-1217
!hoo"
: #$#J#%# am Te&
C"a''
: 55
Tea!he : $# a*
CONTENT No# 1
Wo+'hee&
,age 1-2
3
Wo+'hee& 1: e&' Coo.*na&e ( ,a&1#1 ) Wo+'hee& 1: e&' Coo.*na&e ( ,a&1#2 ) Wo+'hee& 2: e&' ge& *n&o 'hae
/
Wo+'hee& 3: n4o &he 4oune
7
5
Wo+'hee& /: Co!onu& .e"*gh&
8-6
5
Wo+'hee& 5: hae an. 'hae a"*+e
10-11
Wo+'hee& : C"o'e en!oun&e
12-13
7
Wo+'hee& 7: Nu'e un
1/-15
8
Wo+'hee& 8:$one-m*n.e.
1-18
6
Wo+'hee& 6#1: He""o N*!e mee&*ng ou
16-21
10
Wo+'hee& 6#2: The gan. "*ne-u
22-25
11
Wo+'hee& 10: o*ng *&9 HT
2-3/
12
Wo+'hee& 11: Re;"e!&*on
35
2
3-/ 5-
CONTENT No# 1
Wo+'hee&
,age 1-2
3
Wo+'hee& 1: e&' Coo.*na&e ( ,a&1#1 ) Wo+'hee& 1: e&' Coo.*na&e ( ,a&1#2 ) Wo+'hee& 2: e&' ge& *n&o 'hae
/
Wo+'hee& 3: n4o &he 4oune
7
5
Wo+'hee& /: Co!onu& .e"*gh&
8-6
5
Wo+'hee& 5: hae an. 'hae a"*+e
10-11
Wo+'hee& : C"o'e en!oun&e
12-13
7
Wo+'hee& 7: Nu'e un
1/-15
8
Wo+'hee& 8:$one-m*n.e.
1-18
6
Wo+'hee& 6#1: He""o N*!e mee&*ng ou
16-21
10
Wo+'hee& 6#2: The gan. "*ne-u
22-25
11
Wo+'hee& 10: o*ng *&9 HT
2-3/
12
Wo+'hee& 11: Re;"e!&*on
35
2
3-/ 5-
Worksheet 1: Lets Coordinate ! Part1"1 #
$en% &escartes 'io(ra)h* o+ $en% &escartes ,h*"o'ohe Ren< =e'!a&e' wa' >on on $a!h 319 1569 *n a Hae en Toua*ne9 a 'ma"" &own *n !en&a" an!e# He wa' &he ounge'& o; &hee !h*".en# He '&a&e. &o '&u. a& &he age o; 8 a& &he Je'u*& !o""ege o; Hen* I? *n a "@!he ;o 'een ea'# The 'u>4e!&' he '&u.*e.9 'u!h a' he&o*! an. "og*! an. &he Ama&hema&*!a" a&'9B wh*!h *n!"u.e. mu'*! an. a'&onom9 a' we"" a' me&ah'*!'9 na&ua" h*"o'oh an. e&h*!'9 eu*e. h*m we"" ;o h*' ;u&ue a' a h*"o'ohe# o .*. he 'en.*ng &he neD& ;ou ea' ean*ng a >a!!a"auea&e *n "aw a& &he En*e'*& o; ,o*&*e'# =e'!a&e' "a&e a..e. &heo"og an. me.*!*ne &o h*' '&u.*e'# Fu& he e'!hewe. a"" &h*'9 Ae'o"*ng &o 'ee+ no +now"e.ge o&he &han &ha& o; wh*!h !ou". >e ;oun. *n m'e"; o e"'e *n &he gea& >oo+ o; &he wo".9B he wo&e mu!h "a&e *n Discourse on the Method of Rightly Conducting the Reason and Seeking Truth in the Sciences 9 u>"*'he. *n 137# =e'!a&e' *' !on'*.ee. > man &o >e &he ;a&he o; mo.en h*"o'oh9 >e!au'e h*' *.ea' .ea&e. w*.e" ;om !uen& un.e'&an.*ng *n &he ea" 17&h !en&u9 wh*!h wa' moe ;ee"*ng->a'e.# Wh*"e e"emen&' o; h*' h*"o'oh ween& !om"e&e" new9 h*' aoa!h &o &hem wa'# =e'!a&e' >e"*ee. *n >a'*!a"" !"ea*ng ee&h*ng o;; &he &a>"e9 a"" e!on!e*e. an. *nhe*&e. no&*on'9 an. '&a&*ng ;e'h9 u&&*ng >a!+ one > one &he &h*ng' &ha& wee !e&a*n9 wh*!h ;o h*m >egan w*&h &he '&a&emen& AI eD*'&#B om &h*' 'ang h*' mo'& ;amou' uo&e: AI &h*n+G &hee;oe I am#B "&hough h*"o'oh *' "age" whee &he 20&h !en&u .eo'*&e. =e'!a&e' ea!h !en&u ha' ;o!u'e. on .*;;een& a'e!&' o; h*' wo+h*' *ne'&*ga&*on' *n &heoe&*!a" h'*!' "e. man '!ho"a' &o !on'*.e h*m a ma&hema&*!*an ;*'&# He *n&o.u!e. Ca&e'*an geome&9 wh*!h *n!ooa&e' a"ge>aG &hough h*' "aw' o; e;a!&*on9 he .ee"oe. an em**!a" un.e'&an.*ng o; a*n>ow'G an. he oo'e. a na&ua"*'&*! a!!oun& o; &he ;oma&*on o; &he 'o"a ''&em9 a"&hough he ;e"& he ha. &o 'ue'' mu!h o; &ha& .ue &o a"*"eo' ;a&e a& &he han.' o; &he Inu*'*&*on# H*' !on!en wa'n& m*'"a!e.,oe "eDan.e ?II "a&e a..e. =e'!a&e' wo+' &o &he In.eD o; ,oh*>*&e. Foo+'#
H*' eo"u&*ona *.ea' ma.e h*m a !en&e o; !on&oe' *n h*' .a9 an. he .*e. *n 150 ;a ;om home *n &o!+ho"m9 we.en# 13 ea' "a&e9 h*' wo+' wee "a!e. on &he Ca&ho"*! Chu!hK' LIn.eD o; ,oh*>*&e. Foo+'L#
Contri,-tions and achie.ements o+ $en% &escartes He a"'o ha. an *n;"uen&*a" o"e *n &he .ee"omen& o; mo.en h'*!'# H*' ea" wo+ on o&*!' wa' &he ;*'& &o men&*on &he "aw o; e;"e!&*on9 wh*!h '&a&e' &ha& "*gh& *' e;"e!&e. a& &he 'ame ang"e a& wh*!h *& aoa!he' a 'u;a!e# He 'uo&e. &he *.ea o; !on'ea&*on o; momen&um an. "a*. ou& &he *n!*"e o; *ne&*a# He o*.e. &he ;*'& .*'&*n!&" mo.en ;omu"a&*on o; "aw' o; na&ue an. a !on'ea&*on *n!*"e o; mo&*on9 ma.e numeou' a.an!e' *n o&*!' an. &he '&u. o; &he e;"e!&*on an. e;a!&*on o; "*gh&9 an. !on'&u!&e. wha& wou". >e!ome &he mo'& ou"a &heo o; "ane&a mo&*on o; &he "a&e 17&h Cen&u# H*' !omm*&men& &o &he '!*en&*;*! me&ho. wa' me& w*&h '&*.en& oo'*&*on > &he !hu!h o;;*!*a"' o; &he .a# In 1379 he u>"*'he. h*' goun.->ea+*ng h*"o'oh*!a" an. ma&hema&*!a" &ea&*'e L=*'!ou' .e "a m<&ho.eL (&he A=*'!ou'e on $e&ho.B)9 an. one o; *&' aen.*!e' *n a&*!u"a9 La o"*! eDe''*on' *n ma&hema&*!' > =*ohan&u'9 "%hwa*Mm* an. ano*' ?*@&e9 La e!ome +nown a' &he '&an.a. a"ge>a*! no&a&*on9 u'*ng "owe!a'e a9 b an. c ;o +nown uan&*&*e' an. x 9 y an. z ;o un+nown uan&*&*e'# I& wa' eha' &he ;*'& >oo+ &o "oo+ "*+e a mo.en ma&hema&*!' &eD&>oo+9 ;u"" o; aK' an. bK'9 x 2K'9 e&!# I& wa' *n La e .e'!*>e. > &wo num>e' on a "ane9 one g**ng &he o*n&' ho*Mon&a" "o!a&*on an. &he o&he &he e&*!a" "o!a&*on9 wh*!h hae !ome &o >e +nown a' Ca&e'*an !oo.*na&e'# He u'e. een.*!u"a "*ne' (o aDe')9 !o''*ng a& a o*n& !a""e. &he o*g*n9 &o mea'ue &he ho*Mon&a" ( x ) an. e&*!a" (y ) "o!a&*on'9 >o&h o'*&*e an. nega&*e9 &hu' e;;e!&*e" .**.*ng &he "ane u *n&o ;ou ua.an&'# n eua&*on !an >e ee'en&e. on &he "ane > "o&&*ng on *& &he 'o"u&*on 'e& o; &he eua&*on# o eDam"e9 &he '*m"e eua&*on y O x *e".' a '&a*gh& "*ne "*n+*ng &oge&he &he o*n&' (090)9 (191)9 (292)9 (393)9 e&!# The eua&*on y O 2 x *e".' a '&a*gh& "*ne "*n+*ng &oge&he &he o*n&' (090)9 (192)9 (29/)9 (39)9 e&!# $oe !om"eD eua&*on' *no"*ng x 29 x 39 e&!9 "o& a*ou' &e' o; !ue' on &he "ane# ' a o*n& moe' a"ong a !ue9 &hen9 *&' !oo.*na&e' !hange9 >u& an eua&*on !an >e w*&&en &o .e'!*>e &he !hange *n &he a"ue o; &he !oo.*na&e' a& an o*n& *n &he ;*gue# E'*ng &h*' noe" aoa!h9 *& 'oon >e!ame !"ea &ha& an eua&*on "*+e x 2 P y 2 O /9 ;o eDam"e9 .e'!*>e' a !*!"eG y 2 - 1 x a !ue !a""e. a aa>o"aG x 2 Q a2 P y 2 Q b2 O 1 an e""*'eG x 2 Q a2 - y 2 Q b2 O 1 a he>o"aG e&!# Ren< =e'!a&e' a"'o *n&o.u!e &he =e'!a&e'K u"e o; '*gn'9 *n h*' wo+ a oun. on &he num>e o; o'*&*e o nega&*e ea" oo&' o; a o"nom*a"# The u"e *' a"*e. > !oun&*ng &he num>e o; '*gn !hange' *n &he 'euen!e ;ome. > &he o"nom*a"K' !oe;;*!*en&'# I; a !oe;;*!*en& *' Meo9 &ha& &em *' '*m" om*&&e. ;om &he 'euen!e# The u"e o; '*gn' g*e' u' an ue >oun. num>e o; o'*&*e o nega&*e oo&' o; a o"nom*a"# I& *' no& a !om"e&e !*&e*on9 mean*ng &ha& *& .oe' no& &e"" &he eDa!& num>e o; o'*&*e o nega&*e oo&'# The u"e '&a&e' &ha& *; &he &em' o; a o"nom*a" w*&h ea" !oe;;*!*en&' ae o.ee. > .e'!en.*ng a*a>"eeDonen&9 &hen &he num>e o; o'*&*e oo&' o; &he o"nom*a" *' e*&he eua" &o &he num>e o; '*gn .*;;een!e' >e&ween !on'e!u&*e nonMeo !oe;;*!*en&'9 o *' "e'' > a mu"&*"e o; 2#' a !oo""a o; &he u"e9 &he num>e o; nega&*e oo&'
*' &he num>e o; '*gn !hange' a;&e mu"&*"*ng &he !oe;;*!*en&' o; o..-owe &em' > 19 o "e'' > a mu"&*"e o; 2#
Worksheet 1: Lets Coordinate ! Part1"2 # Name o; game: $a&e*a"' : Num>e o; "ae' D"ana&*on o; &he game >oa. :
:
The In&e""*gen& Ra!e game >oa.( 'hown *n neD& age)9 a .*!e9 / &o+en Two "ae' &o ;ou "ae' 1) The game >oa. *' !on'*'&' o; 1/D1/ o; >"o!+'# 2) D"ana&*on o; &he 'm>o" The '&a&*ng o*n&#
S F
Ru"e' o; &he game
:
1)
2) 3) /)
The ;*n*'h*ng o*n&#
Th*' 'm>o" ee'en&' &he o>'&a!"e'# You ae no& a""ow &o go &hough *& o '&e on *&# Th*' 'm>o" ee'en&' &he >oo'&e# When ou '&e on *&9 ou ae a""ow &o eo"" &he .*!e ;o &he 'e!on. &*me# Th*' 'm>o" ee'en&' &he *'"an.# When ou '&e on *&9 &he 'm>o" w*"" au&oma&*!a"" &un *n &o &he !o"ou &ha& ee'en&' ou9 hen!e ou oonen& *' no& a""ow &o go &hough *& o '&e on *&# a!h "ae *' g*en a &o+en ( .*;;een& !o"ou ;o .*;;een& "ae ) an. *& *' "a!e. a& &he o'*&*on AB a& &he e >eg*nn*ng o; &he game# The ;*'& "ae hae &o o"" &he .*!e# The ;*'& "ae hae &o wa"+ ;o a !e&a*n amoun& o; '&e'(>"o!+) a!!o.*ng &he num>e u o"" ou&# ,"ae' ae a""ow &o wa"+ *n ho*Mon&a"" 9e&*!a""o >o&h &oge&he .u*ng &he '&e'(>oD) ou ae eu*e. &o wa"+ ;o#
5) ,"ae' mu'& no& wa"+ on o '&e on &he 'm>o" A ”. In'&ea.9 "ae mu'& ;*n. ano&he wa &o wa"+ aoun. *&# ) ,"ae' &ha& '&e on &he A aga*n#
”ae a""ow &o o"" &he .*!e on!e
7) ,"ae' &ha& '&e on &he A ”w*"" au&oma&*!a"" !"am &he A ” a' h*'She own# o no oonen& !an '&e on *&# In'&ea.9 oonen& mu'& ;*n. ano&he wa &o wa"+ aoun. *&# 8) ,"ae &ha& ea!h &he AB ;a'&e'& w*n' &he game
Choose /isel* and enjo* *o-r (ame
F
S
=*!e
To+en'
Worksheet 2: Lets (et into sha)e (a) Ta>"e 1 'how' &he !oo.*na&e' o; &he e&*!e' o; &wo 'e&' o; &*ang"e'# >'ee &he a&&en 'hown an. !om"e&e &he &a>"e#
Triangle A B C " $
Set1 Coordinates of vertices ( 0,1 ) ( 2,0 ) ( 2,2 ) ( 0,2 ) ( 2,2 ) ( 2,4 ) ( 0,3 ) ( 2,4 ) ( 2,6 ) ( 0,4 ) ( 2,6 ) ( 2,! ) ( 0,# ) ( 2,! ) ( 2,10 ) ( 0,6 ) ( 2,10 ) ( 2,12 ) TA'LE 1
Triangle % ' S T *
Set2 Coordinates of vertices ( #,0 ) ( #,2 ) ( &,1 ) ( #,2 ) ( #,4 ) ( &,3 ) ( #,4 ) ( #,6 ) ( &,# ) ( #,6 ) ( #,! ) ( &,& ) ( #,! ) ( #,10 ) ( &, ) ( #,10 ) ( #,12 ) ( &,10 )
( > ) *D e!&ang"e'9 .awn on a Ca&e'*an "ane9 ae 'hown on age 3# n &he Ca&e'*an "ane9 .aw a"" &he &we"e &*ang"e' 'hown *n &a>"e 1 &o !om"e&e &he "aou& ;o &he mo.e" o; a ua&M !'&a"# ! $EE$ TO T3E 4$AP3 PAPE$ ON NE5T PA4E # ( ! ) E'e &he eDa!& "aou& !ea&*e" &o >u*". &he mo.e"# ! $EE$ TO T3E MO&EL '6LT # ( . ) Ca"!u"a&e &he &o&a" 'u;a!e aea9 *n !m9 o; &he mo.e" &ha& ou hae >u*"&#
ea o; a &*ang"e
|
O
2
20
2 1
0
21
1
O
2 1
O
2 1
O
|
1 0
2
ea o; 12 &*ang"e' O 2 × 12!m
!m
O 2/ !m
|( 0 × 0 )+ (2 × 2 )+( 2 × 1 ) −( 2 × 1 )−( 2 × 0 ) −(0 × 2)| |0 + 4 + 2 − 2− 0 − 0| |4|
!m
!m
!m
O 2!m ea o; e!&ang"e' O × (3 × 2) O3!m *n!e ea!h >"o!+ *n &he gah ae ee'en& 2!m To&a" 'u;a!e aea O (3P2/)D2 O120!m
Worksheet 8: Enjo* the jo-rne* &eri.e '&a*gh& "*ne 4o*n' o*n& ,
( x ) 1 ,Y 1
an. o*n& U
( x ) 2 ,Y 2
# =e*e &he ;omu"a ;o &he "eng&h
,U# =*'&an!e o; URO
y 2 - y 1
=*'&an!e o; ,RO x 2 - x 1 F u'*ng ,&hagoa' Theoem
√ ( x − x ) + ( y − y ) 2
,UO
2
1
2
2
1
6se "* .*e' ;om Town &o *'*& h*' ;*en.' who '&a *n Town F an. Town C# ;&e &he *'*&9 he e&un' &o Town # 1) Ca"!u"a&e &he &o&a" .*'&an!e o; &he 4oune#
=*'&an!e o; J
=*'&an!e o; JF
O
√ (−7 − 2 ) + (− 8− 4 )
O
√ (−9 ) + (− 12 )
O
√ 225
2
2
O
√ (5 −2 ) +( 8 −4 )
O
√ ( 3 ) + ( 4 )
O
√ 25
2
2
2
2
O15
=*'&an!e o; JC O
√ (11−2 ) + (−8− 4 )
O
√ ( 9 ) +(−12 )
O
√ 225
2
2
2
2
O5
2
2
O15
To&a" .*'&an!e O J P JF P FJ PJC P CJ P J O15 P5 P 5 P 15 P 15 P15 O 70+m 2) new '&a*gh& oa. *' &o >e >u*"& !onne!&*ng Town C &o e*&he Town o Town F# Wh*!h new oa. 'hou". "* e;e ;o &he a>oe 4ouneV D"a*n#
√ (11 +7 ) +(−8 + 8 ) 2
CO O
√ (18 )
2
√ (11−5 ) +(−8 −8 ) 2
FCO O
√ ( 6 ) +(−16 )
O
√ 292
2
2
O 18 I; C *' >u*"&9 J P JF P FJ P JC P C O 15 P 5 P 5 P 15 P 18 O 58+m
2
2
O 17#088 I; FC *' >u*"&9 J P JF P FC P CJ P J O15 P 5 P 17#088 P 15 P15 O 7#088+m
C *' e;e9 >e!au'e &he .*'&an!e *' 'ho&e9 wh*!h *' 58 +m#
Worksheet 9: Cocon-t deli(ht &eri.e '&a*gh& "*ne 4o*n' o*n& ,(
x 1 9 y 1 )an. o*n& U( x 2 9 y 2 )# ,o*n& .**.e' *n&ena""
&he "*ne ,U *n a&*o m:n# =e*e &he ;omu"a ;o &he !oo.*na&e # Hen!e '&a&e &he ;omu"a ;o &he !oo.*na&e' o; &he m*.o*n& o; ,U#
&a&e &he ;omu"a ;o ;*n.*ng !oo.*na&e 9
&a&e &he ;omu"a ;o !oo.*na&e o; m*.o*n& ,U9
PX m O XQ n
$*.o*n& o; ,UO
→
(
→
n PX O m XQ →
→
n( OX - OP )Om( →
→
OQ - OX ) →
→
n OX -n OP Om →
→
OQ -m OX →
→
m OX P n OX O m →
→
OQ P n OP →
→
(mPn) OX O m OQ →
P n OP 1
→
OX O m+ n (m →
→
OQ P n OP ) 1
→
OX O m+ n Xm
() () x2 y 2
Pn
→
OX O
x1 y 1
( ) m x 2 +n x 1 m +n m y 2 +n y 1 m +n
O
(
m x 2+ n x1 m y 2 + n y 1 , m+ n m +n
)
x1 + x 2 y 1 + y 2 , 2
2
)
6se 1 )a""eh9 &he owne o; &he "an.9 wou". "*+e &o "an& ano&he !o!onu& &ee *n 'u!h a wa &he ;ou !o!onu& &ee' ae a& &he e&*!e' o; a aa""e"ogam# Whee 'hou". a""eh "an& &he ;ou&h !o!onu& &eeV how ou !a"!u"a&*on# $*.o*n& o; CO$*.o*n& o; F=
(− + + ) ( + + + + ( ) ( + ( ) (+ + ) 10 35 2
x
O
$*.o*n& o; C=O$*.o*n& o; F
5
,
2
,
10
y
40
2
30
2
x
5
2
,
2
x +5
2
35
2
30
O
35
2
O
y
25
x
x
2
35
2
2
25 50
,
x
2
,
,
y + 40 2
y + 40
=
2
)=( − = )( 5
2
−5 2
DP35O-5 DO-/0 The ;ou&h !o!onu& &ee 'hou". >e "an& a& ( -/09-10 )#
25ODP5 DO20 The ;ou&h !o!onu& &ee 'hou". >e "an& a& ( 20920 )#
2 ) ,u&e*9 a""eh' w*;e9 wou". "*+e &o "an& &wo moe !o!onu& &ee'# ( a ) ne &ee *' &o >e "an&e. >e&ween &he !o!onu& &ee' a& "o!a&*on F an. C 'u!h &ha& &he &hee !o!onu& &ee' ;om a '&a*gh& "*ne# u&he9 &he "o!a&*on o; &he &ee *' 'u!h &ha& *&' .*'&an!e ;om C *' &hee &*me' *&' .*'&an!e ;om F# Whee 'hou". ,u&e* "an& &he !o!onu& &eeV o!a&*on o; &he !o!onu& &ee9 O
O
O
( ( (
( )+ 1 (35 ) 3 (30 )+ 1 ( 40 ) , 3+ 1 3 +1
3 5
50 130 4
12
,
1 2
4
, 32
) 1 2
)
)
,u&e* 'hou". "an& &he !o!onu& &ee a&
(
12
1 2
, 32
1 2
)
#
( > ) The o&he &ee *' &o >e "an&e. 10m awa ;om &he !o!onu& a& "o!a&*on 'u!h &ha& &h*' !o!onu& &ee an. &he &wo !o!onu& &ee' a& "o!a&*on an. F ;om a '&a*gh& "*ne# Whee 'hou". ,u&e* "an& &he !o!onu& &eeV =*'&an!e o; F9 O
√ (−10 −5 )− ( 10−30 )
O
√ 15 −20
O
√ 625
2
2
2
2
O25 e& &he "o!a&*on o; &he !o!onu& &ee &ha& *' go*ng &o >e "an& O :F 10:15 2:3
( − ( 3
O
(− 10)+ 2 ( 5 ) 3 ( 10 )+ 2 ( 30 ) , 3+ 2 3+2 20
O
5
,
90 5
)
)
O( -/918 ) ,u&e* 'hou". "an& &he &ee a& ( -/918 )#
Worksheet : ;hare and share alike &eri.e
( x 1 , y 1 ) 9 F ( x 2 , y 2 ) 9 ! ( x 3 , y 3 ) an. =
( x
4
, y 4 ) ae &he e&*!e' o; ua.*"a&ea"
FC=#=e*e &he ;omu"a ;o &he aea o; &*ang"e FC# Hen!e9 .e.u!e &he ;omu"a ;o &he aea o; ua.*"a&ea" FC= an. '&a&e &he !on.*&*on eu*e.# omu"a ;o &*ang"e FC ea o; FC O aea o; &aeM*um =F P aea o; &aeM*um FC aea o; &aeM*um =C 1
1
( y + y )( x − x ) 2
O
1
1 2
2
2
P
1
2
( y + y )( x − x ) 2
3
3
-
2
( y + y )( x − x ) 1
3
3
1
O 1 2
[ ( x y + x y − x y − x y ) +( x y + x y − x y − x y )−( x y 2
1
2
2
1
1
1
2
3
2
3
3
2
2
2
3
3
1
O 1
[ x y + x y − x y − x y + x y + x y − x y − x y − x y − x 2
2
1
2
2
1
1
1
2
3
2
3
3
2
2
2
3
3
1
1
O
[− x y − x y − x y + x y + x y + x y ]
1
O
1
2
2
|
x 1 y 1
2
x2 y2
2
3
3
1
2
1
3
2
1
3
|
x 3 x 1 y 3 y 1
1
ea o; &*ang"e FCO
2
|
x 1 y 1
1
ea o; Uua.*"a&ea" FC= O
|
x2 y2
2
x 3 x 1 y 3 y 1
|
x 1 y 1
x2 y2
|
x 3 x 4 x 1 y 3 y 4 y 1
Con.*&*on eu*e.: ;ou o*n&' g*en mu'& no& >e !o""*nea# 6se The ;ou >o&he' agee. &o 'hae &h*' *e!e o; "an. eua"" among &hem'e"e'# The .e!*.e. &o .**.e *& *n&o ;ou &*angu"a "o&' o; eua" aea' a' 'hown *n =*agam 3# 1 )Ca"!u"a&e &he aea9 *n +m9 o; ea!h "o& o; &*angu"a "an.# ea o; who"e "an.
ea o; ea!h &*angu"a "an. O5
÷ /
O1/ +m
|
O
2
8 10 0
2 2
10
6 0 2
1
O
2 1
O
2 1
O
|
1 0
2
|0 +12 +0 +20 − 4− 80−60 −0| |−112| ( 112)
O5 +m
2 ) =e&em*ne &he !oo.*na&e o; o*n& # e& &he !oo.*na&e o; O ( a9> )
|
1 0
10
2 2
|
0 2
0
10 0
|
a0 b2
|
a0 b2
O1/
O28
10>P2a-20O[28 10>P2a-20O28 10>P2a-20O -28 Who"e eua&*on *' mu"&*"*e. > 39 'o aO-30>P1// (1) aO-30>-2/(2)
u> (1) &o (3) -30>P1//O-2>P32 -28>O-112 >O/ when >O/ aO-30(/)P1// aO2/ aO/ ( /9/ )
|
1 10
8
2
6
|
0
10
8
0
6
|
a 10 b0
|
a 10 b 0
O1/
O28
0P8>-a-10>O[28
aO-2>P32 (3)aO-2>P88 (/)
u> (1) &o (/) -30>P1//O-2>P88 -28>O-5 >O2 when > O 2 aO-30(2)P1// aO8/ a O 1/ (1/92 )
u> (2) &o (3) -30>-2/O-2>P32 -28>O5 > O -2 when > O -2 aO-30(-2)-2/ aO3 aO ( 9-2 )
u> (2) &o (/) -30>-2/O-2>O88 -28>O112 > O -/ when > O -/ aO30(-/)-2/ aO-1// a O -2/ ( -2/9-/ )
*n!e &he !oo.*na&e o; o*n& D *' 0\D\10 an. *' 0\\109 The !oo.*na&e o; *' ( /9/ ) #
Worksheet <: Close enco-nter The +orm-la 1 ) '&a*gh& "*ne a''e' &hough o*n&
( x
1
, y 1 ) an. o*n& F ( x 2 , y 2 ) 9 &a&e &he ;omu"a ;o
&he ga.*en& o; &he '&a*gh& "*ne#
y 2 − y 1 a.*en& O
2 ) The ga.*en&' o; &wo '&a*gh& "*ne' ae
x 2− x
m 1 an.
1
m2 e'e!&*e"# &a&e &he !on.*&*on ;o
&he &wo '&a*gh& "*ne' &o >e: ( a ) aa""e" The ga.*en& o; &he &wo "*ne' mu'& >e &he 'ame9 wh*!h mean'
m1 O
m2 #
( > )een.*!u"a The o.u!& o; &he ga.*en& o; &he &wo "*ne' mu'& >e -19 wh*!h mean'
m1 m2 O -1
6se a) R*&a >eg*n' ;om "o!a&*on (12918) an. .*e' a"ong a '&a*gh& oa. wh*!h a""ow' he &o >e a"wa' eu*.*'&an& ;om &he o"*!e '&a&*on an. &he ;*e '&a&*on a& an &*me# F !on'*.e*ng ga.*en&' on"9 .e&em*ne &he "o!a&*on on &he oa. when R*&a *' neae'& &o &he e&o" '&a&*on# e&9 ,o"*!e '&a&*on O 9 ,e&o" '&a&*on O F9 *e '&a&*on O C9 ,o'*&*on o; R*&a O =9 ,o'*&*on neae'& &o &he e&o" '&a&*on O $ a.*en& o; C
+1 − 4− 4
−3
5
O
4
6
O
a.*en& o; =$ ( M DM )O-1
M DM O
−8
4 3
−3 O
4
a.*en& o; CO a.*en& o; F$
−3 O
4
ua&*on o; F$
b−2 a +3 O
−3 4
/>-8O-3a-6 />P3aO-1 (1)
ua&*on o; =$
b−18 4 = a−12 3 3>-5/O/a-/8 3>-/aO （2）
( 1 )mu"&*"e > 39 ( 2 )mu"&*"e > / 12>P 6aO-3 12>-1aO2/ 25aO-27
−27 aO
25
−27 When aO
25
−27 3>O/(
25 42
3>O
25 14
> O
25
¿
P
−27 The o'*&*on on &he oa. when R*&a *' neae'& &o &he e&o" '&a&*on *' (
> ) I; R*&a .*e' w*&h an aeage 'ee. o; 80+m &a+e' &o ea!h &h*' neae'& "o!a&*on# =*'&an& o; =$ O
O
O
√(
√ √
12
+
27 25
106929 625
) +( ²
+
−
18
190096 625
11881 25
109
O
5
O21#8 21.8
T*me O
80
D 0
O0#2725 D 0
O1#35m*nu&e'
14 25
)
²
h
−1
25
14
9
25
¿
#
9 !a"!u"a&e &he &*me9 *n m*nu&e'9 'he
Worksheet 7: N-rser* -n ;tate &a&e &he ;omu"a ;o ;*n.*ng &he eua&*on o; a '&a*gh& "*ne g*en *n &he ;o""ow*ng: 1 ) a.*en& an. &he -*n&e!e& 2) a.*en& an. one o*n& on &he "*ne 3 ) Two o*n&' on &he "*ne /) The D-*n&e!e& an. -*n&e!e& 1 O m(D D1)
6se 1
1 ) The ga.*en&' o; >oun.a*e' F an. FC ae
2
an. -1 e'e!&*e"# *n. a"" &he eua&*on'
o; a"" &he ;*e >oun.a*e' o; &he nu'e >e.# a.*en& o;
−0 0− 3
a.*en& o; =
5
O
−0 6 −3
−2 7−6
O
2
O2
2
O
−5 O
a.*en& o; C=
O
3
4
3
ua&*on o; C= ua&*on o;
y −5 x −0 O
−5 3
ua&*on o; =
y −0 x −3 O
2
y −4 x −7 O2
3
-/O2D-1/ 3-15O-5D
−5 O
3
x + 5
ua&*on o; F
y −5 1 = x −0 2 2-10OD 1
O
2
x + 5
3O2D- 2
O
3
x -2
ua&*on o; FC
y −7 =−1 x −4 -7O-DP/ O-DP11
O2D-10
2 ) Ca"!u"a&e &he aea o; &he nu'e >e. > u'*ng &wo me&ho.'9 one o; wh*!h mu'& >e *n&ega&*on# Method one Coo.*na&e o; ,o*n& F
ea
1
x + 5 (1) 2
O
O
O-DP11 (2) 'u> (1) &o (2) 1 2
x + 5 =− x + 11
2
76
3
0
2 5
7
42
0
5
2 1
x O
|
4
1
O
O 3
|
1 0
2
|16 +14 + 15−20 −49 −24 −6|
|−54|
O27un*&
O/ When DO/ 1
O
2
( 4 )+ 5
O7 !oo.*na&e o; F *' F( /97 )
Method t/o ea 4
O
3
6
7
∫ AB +∫ BC −∫ AE−∫ DE−∫ CD 0
4
O
7
4
1
0
7
3
6
3
∫ 2 x + 5 dx+∫− x +11dx −∫ 0
4
0
−5 3
6
2
7
∫ 3 x −2 dx −∫ 2 x −10 dx
x + 5 dx −
3
6
[
O
1 4
1
OX(
+ 5 x
x
4
] [ 4
2
P
0
( 4)
2
P5(/))-0] P X(
] [ 7
2
− x
+ 11 x
−( 7 ) 2
2 2 ( ( 6 ) −2 (6 ))−( (3 ) −2 (3 )) 2
6
−¿ 33 2
2
6
] [ 3
0
2
+ 11( 7 )¿−(
−( 4 ) 2
2
-
2 6
]
6
2
x
− 2 x
3
[ x −10 x ] 2
-
7 6
5 + 11( 4 ))¿−[( − ( 3 ) + 5 (3 ))−0 ] 2
6
2
6
O2/P
-
4
− 5 x + 5 x
2
] - X(
2
( 7 ) −10 ( 7 )¿−( ( 6 ) −10 (6 ))¿
15
-
2
-3-3
O27un*&
Worksheet =:Mone*>minded 1 ) oa. 4un!&*on *' &o >e >u*"& a"ong &he h*ghwa !onne!&*ng *""age F an. ?*""age U &o &he h*ghwa# The eua&*on o; &he h*ghwa *' O2DP1# =e&em*ne &he "o!a&*on o; &he oa. 4un!&*on 'o a' &o m*n*m*Me &he !on'&u!&*on !o'&# E'e &wo me&ho.'# Method one e&9 &he 4un!&*on &o *""age UO$ &he 4un!&*on &o *""age ,ON ?*""age UO U ?*""age ,O, a.*en& o; $NO2
a.*en& o; $NO2
−1 a.*en& o; $UO ua&*on o; $U9
y −7 −1 = x −8 2 2-1/O-DO8
2
−1 a.*en& o; N,O ua&*on o; N,9
y + 1 −1 = x − 4 2 2P2O-DP/
2
1
¿− x +¿ 2
1
11
Jun!&*on o; *""age U9
2
1
11 （1）O2DP1 （2）
'u> (1) &o (2) 2 x
+ 1=
−1 2
2
Jun!&*on o; *""age ,
1
¿− x +¿
¿− x +1
¿− x +1 2
(1)
O2DP1 （2）
'u> (1) &o (2)
x +¿ 11
5
x =¿ 10 2
1
2DP1 5 2
¿− x +1 2
x =0
DO/ when DO/ O2(/)P1 O6
DO0 when DO0 O2(0)P1 O1
Jun!&*on o; *""age U *' ( /96 )
Jun!&*on o; *""age , *' ( 091 )
Method t/o e&9 &he 4un!&*on &o *""age UO$ &he 4un!&*on &o *""age ,ON ?*""age UO U ?*""age ,O, ua&*on o; $N9 O2DP1 ga.*en& o; eua&*on $N O 2 ga.*en& o; eua&*on $N In e!&o ;omO
ua&*on o; $N9 O2DP1 ga.*en& o; eua&*on $N O 2 ga.*en& o; eua&*on $N In e!&o ;omO
(1 ) 2
"e& DO1 O2(1)P1 O3 o*n& O ( 193 )
(1 ) 2
"e& DO1 O2(1)P1 O3 o*n& O ( 193 )
() 1
,o*n& *n e!&o ;om O
() 1
,o*n& *n e!&o ;om O
(1 )
→
MN 9 O
ua&*on o; e!&o
3
(1 )
→
MN 9 O
ua&*on o; e!&o
3
2
() 1
&P
2
() 1
&P
3
(1)
3
ua&*on o; N,
(1)
1
¿− x + 1 2
ua&*on o; $U9 1
¿− x +¿
2
¿−
ga.*en& o; eua&*on $U
2
(− ) 1
o*n& O (/9-1 )
2
(− )
1
4
,o*n& *n e!&o ;omO
o*n& O ( 897 )
()
,o*n& *n e!&o ;om O
ua&*on o; e!&o NP , r =
7
(− ) 2
→
MQ , O
ua&*on o; e!&o
1
→
8
1
'P
2
1
s P
(2)
(1 )
(2)
2
u> (1) &o (2)
( ) ( ) (− ) ( ) 1
1
&P
3
8
2
O
1
'P
( ) (− ) (− ) 1
&P
3
=
2
s P
1
t +¿ 1 2 t +¿ 3
&P1O2'P8
) ( O
+¿ 8 −s +¿ 7 2s
)
2&P3O-'P7
4
1
( +¿+¿ )=(− +¿−¿ ) t 2 t
1
2s
4
s
1
3
7
&P1O2'P/
2&P3O-'-1 1
(
(− ) (− )
u> (1) &o (2)
8
2
2
2
(− )
7
1
ga.*en& o; eua&*on N, *n e!&o ;omO
1
ga.*en& o; eua&*on $U *n e!&o ;omO
()
¿−
ga.*en& o; eua&*on N,
11
&O2'P3 (1) 'u>(1) &o (2) 1
2'P3O-
2
s −2
&O-
2
s −2 (2)
4
1
−1 &O2'P7 (1)
&O
2
'O-2 when 'O-2
s + 2 (2)
−1
(
2'P7O
2
s+2
(
(−2 )+¿ 8 −(−2 )+¿ 7
2
0
O
1
Jun!&*on o; *""age , *' ( 091 )
'O-2 When 'O-2 2
) ()
(−2)+¿ 4 −(−2 )−¿ 1
'u> (1) &o (2)
) () 4
O
9
Jun!&*on o; *""age U *' ( /96 )
2) '!hoo" *' &o >e >u*"& a' ;a awa ;om &he h*ghwa a' o''*>"e an. *& mu'& >e 5+m ;om ea!h o; &he &wo *""age' , an. U# =e&em*ne &he "o!a&*on o; &he '!hoo" > 'o"*ng '*mu"&aneou'"# ( *n. one "*nea eua&*on an. &wo non- "*nea eua&*on ) *nea eua&*on
y −3 −1 = 2 x −6
Non- "*nea eua&*on (1) e& &he "o!a&*on o; '!hoo"O UO5
√ ( x −8 ) +( y −7 ) ² 2
2-O-DP
−1 O
2
x + 6
DO-2P12 (1)
a ) &wo non-"*nea eua&*on' (2)-(3) DP-1D-1/P88O0 - DP-8DP2-8O0 -8D-1P6O0 -D-2O-12 -DO2-12 DO-2P12 (/) u>(/) &o (3) (-2P12 )P-8(2P12 )P2-8O0 1//-/8P/8P-
2
√ ( x − 4 ) +( y + 1 ) ² 2
O5
2
( x −8 ) +( y −7 ) =25 D-1DP/P-1/P/6O25 DP-1D-1/P88O0 --(2)
>) one "*nea an. one non "*nea eua&*on 'u> (1) &o (3) (-2P12 )P-8(2P12 )P2-8O0 1//-/8P/8P6P1P2-8O0 5-30P/0O0 -P8O0 (-/)(-2)O0 When O/ DO-2(/)P12 DO/
Non- "*nea eua&*on (2) e& &he "o!a&*on o; '!hoo"O ,O5
2
O5
2
( x −4 ) + ( y +1 ) =25 D-8DP1PP2P1O25 DP-8DP2-8O0 (3)
6P1P2-8O0 ( /9/ ) 5-30P/0O0 The "o!a&*on o; &he -P8O0 '!hoo" 'hou". >e (-/)(-2)O0 ;u&he &han *""age , When O/ Whenan. O2*""age U9 'o &he "o!a&*on o; &he '!hoo" DO-2(/)P12 DO-2(2)P12 DO/ DO8 *' ( 892 )# ( /9/ ) ( 892 ) The "o!a&*on o; &he '!hoo" 'hou". >e ;u&he &han *""age , an. *""age U9 'o &he "o!a&*on o; &he '!hoo" *' ( 892 )#
Worksheet ?"1: 3ello Nice meetin( *o- &e+ine ,o*n& ,( D9 ) "*e' on a !a&e'*an "ane# =e;*ne &he o'*&*on e!&o o; o*n& , e"a&*e &o &he o*g*n# →
P O
() x y
O
x
y
*P
4
@-estion →
→
e'e!&*e"# I& *' g*en &ha&
OJ =
( ) −t 6 −3 t 4
→
OK O
9
( ) −2 t −4 −t 9
→
→
OJ ,OK an. OL
The o'*&*on e!&o' o; Johan9 %a''*m an. a&*; e"a&*e &o &h o*g*n ae →
an. OL O
(
− 4 t 4 t −12 2
whee & *' &he &*me *n 'e!on.'# ( a ) I.en&*; &he name' o; '&u.en& 9 '&u.en& F9 an. '&u.en& C# →
OJ =
( ) −t 6 −3 t 4
When & O 0
→
OK O When &O0
( ) − 2 t − 4 − t 9
→
OL O When &O0
(
− 4 t 4 t −12 2
)
)
→
OJ = →
OJ =
(
−( 0) 6 −3 (0 ) 4
)
(
OK O
() 4
→
OL O
(− )
OK O
→
9
9
→
6
)
−2 ( 0 ) −4 −( 0 )
→
OL O
→
)
(− ) 2
→
4
(
−4( 0) 4 ( 0 )−12 2
12
→
OJ *' Johan
OK *' %a''*m
OL *' a&*;
Johan *' '&u.en&
%a''*m *' '&u.en& C
a&*; *' '&u.en& F
( > ) *n. &he eua&*on o; &he a&h &a+en > ea!h o; &hee '&u.en&'# Johan (&u.en& ) When &O0 When &O1 →
OJ =
() 4 6
(
%a''*m (&u.en& C) When &O0 When &O1
)
→
−(1 ) OJ = 6 −3 ( 1 ) →
→
OJ =
4
→
()
→
OK O
OL O
OL O
(− )
(
(− )
(
−2 ( 1 ) −4 −( 1)
4
3
→
OK O 9
3
a&*; (&u.en& F) When &O0 When &O1
9
)
2
12
→
→
(− )
−6 =3 3 −4 3
ua&*on o; Johan9
y −6 =3 x −4
7
5
−5 + 4 − 1 = −2 7 −9
-O3D-12 O3D-
1
O
2
ua&*on o;%a''*m9
y + 4 x − 9 O
1 2
2P8OD-6 2OD-17 1
O
2
x −
a.*en&9
− 8 + 12 = 4 − 2− 2 − 4
a.*en&9
17 2
( )
OL=
OK O a.*en&9
)
−4 ( 1) 4 ( 1 ) −12 2
O -1 ua&*on o; a&*;
y + 8 =−1 x + 2 P8O-D-2 O-D-10
−2 −8
( ! )=e&em*ne whe&he Johan9 %a''*m an. a&*; w*"" mee& one ano&he a& a !ommon "o!a&*on > u'*ng a e!&o me&ho.# I; &he mee&9 .e&em*ne &ha& "o!a&*on > u'*ng &wo me&ho.'9 one o; wh*!h mu'& a e!&o me&ho.# Method one →
→
→
OJ O OK
→
OJ O OL
( )(
−t = 9−2 s −4 − s 6 −3 t 4
)
( )( − t = 6 − 3 t 4
−4m 4 m − 12 2
/-&O6-2' 2'-&O5(1) -3&O-/-' '-3&O-10(2) when (2) *' mu"&*"e > 2 2'O&-20(3) u> (1) &o (2) &-20-&O5 5&O25 &O5 when &O5 2'O(5)-20 'O5
/-&O2-/m /mO&-2(1) -3&O/m-12 /mO18-3&(2) u> (1) &o (2) &-2O18-3& /&O20 &O5 when &O5 /mO18-3(5)
when &O5
when &O5
(
→
OJ O
3
mO
) ( )
−5 6 −3 ( 5 ) 4
O
−1 −9
OK O
4
→
OJ O
(
) (−− )
−5 6 −3 ( 5 ) 4
1
O
9
3
When 'O5 →
)
(
When mO
) ( )
−2 ( 5 ) −4 − 5
9
O
−1 −9
4
( ) 2
→
OL O
4
3
−4( ) 4
(−− ) 1
O
3
( )−12
9
4
n'we: Johan9 %a''*m an. a&*; w*"" mee& one ano&he9 &he w*"" mee& a& ( -19-6 )
Method T/o O3D-(1) 1
O
2
x −
u> (1) &o (2) 3D-O
17 2
1
2
x −
17 2
u> (1) &o (3) 3D-O-D-10 /DO-/ DO-1 when DO-1
u> (2) &o (3) 1 2
x −
17 2
O-D-10
(2)
5 2
x =
−5
O-D-10(3)
2
O3(-1)- O-6 ( -19-6 )
3 2
x =
−3 2
DO-1 DO -1 when DO-1 when DO-1 O3(-1)- O-(-1)-10 O-6 O-6 ( -19-6 ) ( -19-6 ) n'we: Johan9 %a''*m an. a&*; w*"" mee& one ano&he9 &he w*"" mee& a& ( -19-6 )
( . ) I; V x an. V y ae &he ho*Mon&a" an. e&*!a" e"o!*&*e' o; a '&u.en& e'e!&*e"9 eDe''9 ;o ea!h '&u.en& V x an. V y *n &em' o;
an.
Hen!e9 ;*n. &he e'u"&an& e"o!*& ? o; ea!h '&u.en& *n &he ;om D 4og' &he ;a'&e'&# Johan ('&u.en& ) →
OJ =
(
%a''*m ('&u.en& C) →
→
OK
OL
6
O
O
( ) ( − 2 t − 4 − t 9
DO/-&
dx =− dt
DO6-2&
V x =−
Re'u"&a n& e"o!*& O
√ (−1 )
2
− 4 t 4 t −12 2
DO2-/&
dx =− dt
dx =− dt
V x =−
V x =−
Re'u"&a n& e"o!*& O
Re'u"&a n& e"o!*& O
√ (−2 )
√ (−4 )
O
√ 10
a&*; ('&u.en& F)
O
√ 5
2
O
2
! ~
> u'*ng .*;;een&*a&*on# ~
P ! &o .e&em*ne who
√ 32 n'we: a&*; ('&u.en& F) 4og' &he ;a'&e'& >e!au'e &he e'u"&an& e"o!*& *' &he h*ghe'& wh*!h *'
√ 32
#
Worksheet ?"2: The (rand line>-) →
The o'*&*on e!&o' o; am9 ,au" an. $a&*n e"a&*e &o &he o*g*n ae
→
OM e'e!&*e"# I& *' g*en &ha&
→
(
−2 O"= −6 t + 9 2 t
)
→
9
( )
−t + 5 OP = −t −2
→
O" 9
OP 9 an. →
an.
( )
OM =
7 2 3
− 2 t t −5
whee & *' &he &*me *n m*nu&e'#
( a ) *n. &he !oo.*na&e' o; ,au" an. $a&*n when &he ae *n &he "o!a&*on 'hown *n =*agam 8# In &he .*agam &he !oo.*na&e o; am *' ( 093 )#
(
) ()
−2 −6 t + 9 2 t
0
O
3
2&-2O0 2&O2 &O1 when &O1 Coo.*na&e o; ,au" O
(
−( 1 )+ 5 −( 1 )− 2
Coo.*na&e *; $a&*n
)
O
2 3
(− ) 4
O
( ) ( ) −2 ( 1 )
7
3
(1 )−5
5
O
The !oo.*na&e o; ,au" O ( /9-3 )
−13 3
−13 The !oo.*na&e o; $a&*n O ( 59
3
)
( > ) n &he 'ame aDe'9 .aw &he "o!* o; &he &hee men# Name am →
O"=
When &O0 →
(
−2 −6 t + 9 2 t
)
( ) = (− )
−2 O"= −6 t +9 →
O"
2 t
2
9
Coo.*na&e O ( -296 ) →
,au" →
( ) −t + 5 −t −2
OP =
( ) =( ) −
−t + 5 OP = −t −2 →
OP
5
2
Coo.*na&e O (59-2 ) $a&*n →
( )
OM =
− 2 t
7 2 3
→
( ) 7
OM =
2 3
t −5
→
− 2 t t −5
(− )
OM =
7
5
Coo.*na&e O ( 79-5 )
When &O1 →
( ) =( )
−2 O"= −6 t + 9 →
O"
2 t
0 3
Coo.*na&e O ( 093 ) →
( ) =( ) −
−t + 5 OP = −t −2 →
OP
4
3
Coo.*na&e O ( /9-3 ) →
( ) ( )
OM =
7 2 3
→
OM =
− 2 t t −5
5
−13 3
When &O2 →
( ) =( ) −
O"= →
O"
−2 −6 t + 9 2 t
2
3
Coo.*na&e O ( 29-3 ) →
( ) =( ) − −t + 5 −t −2
OP = →
OP
3
4
Coo.*na&e O ( 39-/ ) →
( ) ( )
OM =
− 2 t
7 2 3
→
OM =
t −5
3
−11 3
Coo.*na&e O ( 59
−13
Coo.*na&e O ( 39
−11 )
3
3
¿
( The "o!* *' .aw on &he gah ae on neD& age ) ( ! ) & a !e&a*n &*me9 &he o'*&*on o; am9 ,au" an. $a&*n "*e on a '&a*gh& "*ne# ( * ) =e&em*ne &h*' u'*ng &wo me&ho.'# Method One e&9 amO ,au"O, $a&*nO$ & 1
a.*en& ,
a.*en& $
− 3 − 3 = −3 4−0 2
−13 3
−0
− 4+3 =−1 3 −2
2
−11 3 3
=
−22
3
15
5
−2
+3
−4
−11
+3 =
−2
3
3
3
− 3 + 9 =−2 1− 4
−5 + 9 =−2 2− 4
3
−13
−3
5
a.*en& ,$
=
−4 3
+4
−3
=#
−3 + 5 =−2 1 −2
n'we: & &O39 am ,au" an. $a&*n ae on &he '&a*gh& "*ne9 >e!au'e "*ne ,9 "*ne $ an. "*ne ,$ hae &he 'ame ga.*en&# Method t/o →
→
→
"P =OP −O" =
(−− −+ )−(− −+ )=(− −+ ) t t
5
2
→
→
PM =OM −OP =
2 3
t − 15
a.*en& , Oa.*en& ,$ 5
t −3 −11 = 3 −3 t + 7 2−t 5 t
3 t 7
2
6 t 9
5 t
11
( )( )( ) − 2 t
7
→
2 t
− − t + 5 = − t − 2
2
5 3
− t
t − 3
35
10&-22-5&P11&O-5&P 1 3
3
t P6&-21
t O1
&O3 n'we: & &O39 am ,au" an. $a&*n ae on &he '&a*gh& "*ne#
( ** ) *n. &he eua&*on o; &he '&a*gh& "*ne# When &O39 !oo.*na&eO( /9-6 )
y + 9 x − 4 O -2 P6O-2DP8 O-2D-1
( *** )*n. &he a&*o o; &he .*'&an!e >e&ween am an. ,au" &o &he .*'&an!e >e&ween am an. $a&*n a& &ha& &*me# When &O3 Coo.*na&e o; am *' ( /9-6 )9 Coo.*na&e o; ,au" *' ( 29-5 )9 Coo.*na&e o; $a&*n *' ( 19-3 ) =*'&an!e >e&ween am an. ,au"
=*'&an!e >e&ween am an. $a&*n
O
√ ( 4 −2 ) +(−9 + 5 ) ²
O
√ ( 4 −1 ) +(−9 +3 )
O
√ 4 + 16
O
√ 9 +36
O
√ 20
O
√ 45
2
"am∧ Pa$% 20 2 = √ = "am∧ Martn √ 45 3
2
2
2
n'we: The a&*o >e&ween am an. ,au" &o &he .*'&an!e >e&ween am an. ma&*n *'
3
Worksheet 10: Lo.in( it 3OT; 1 ) =*agam 6 'how' a '!a"e .aw*ng o; &wo aa""e"ogam'9 an. F# ( a ) Co &he &wo aa""e"ogam' ( Re;e &o &he gah ae a& neD& age ) ( > ) n ea!h aa""e"ogam9 .aw 'u*&a>"e !oo.*na&e aDe' &ha& w*"" ena>"e ou &o '&a&e &he !oo.*na&e' o; ea!h e&eD o; &he aa""e"ogam ea'*"# ,aa""e"ogam ( 29/ ) 9 F( 1/96 ) 9 C( 17913 ) 9 =( 598 )
,aa""e"ogam F ( 592 ) 9 F( 1797 ) 9 C( 13910 ) 9 =( 195 )
( ! ) Hen!e > !on'*.e*ng *n&ege !oo.*na&e' on"9 !a"!u"a&e a o''*>"e aea ;o ea!h aa""e"ogam# ea o; ,aa""e"ogam
|
1 2
O
2 4 1
O
2
14 9
17 5 13 8
|
ea o; ,aa""e"ogam F
2 4
|18 + 182 +136 +20 −56−153 −65 −16|
O
|
17
13 1
5
2 2
7
10 5
2
1
O
|
1 5
2
|35 +170 + 65 +2 −34 −91−10 −25|
#
1
O
2
1
|66|
O
O33 un*&
2
|112|
O5 un*&
2 )( a ) =*agam 10 'how' &wo '&a*gh& "*ne'9 ,U an. R9 *n&e'e!&*ng a& &he o*g*n 9 an. &he "*ne DO2 .awn on a Ca&e'*an "ane# The ga.*en&' o; ,U an. R ae e'e!&*e"# how &ha&
m1 an.
m2
m1 m2=−1
e& &he -!oo.*na&e o; &he o*n& &ha& *n&e'e!& &he "*ne DO2 >e / ,o*n& O ( 29/ )
When &he o*n& ( 29/ ) *; o&a&e 60^ !"o!+w*'e9 &he new o*n& *' ( /9-2 )#
a.*en& o; ,U9
a.*en& o; R9
−0 =2 2−0
−2− 0 −1 = 4 −0 2
4
m 1 m2=( 2)(
−1 2
) m1 m2=−1
('hown)
( > ) how &ha& *& *' o''*>"e ;o &he &wo "*ne' 2D-(mP1)O5 an. 3mOD-1 &o >e aa""e" >u& *mo''*>"e &o >e een.*!u"a# 2D-(mP1)O5
2D-(mP1)O5
(mP1)O2D-5
(mP1)O2D-5
2
2
5
O m + 1 D -
O m + 1 D -
m+ 1
5
m+ 1 ga.*en& o; &h*'
ga.*en& o; &h*' eua&*on *'
m1 m2=−1
2
eua&*on *'
m+ 1
( + )( )=− 2
m
*n!e >o&h "*ne' ae o''*>"e &o >e aa""e"9 2
3m
3m
2O-
−3 m
+ 3 m + 2= 0 2
2a
O
1 5
2
2
=−1
− b& √ b − 4 a' m=
5mO1
mO
+3 m
3m
3m
mOmP1
2
1
3m
1
2
1
m+1 O
1
−( 3) & √ (3 ) −4 ( 3 )( 2 2 (3 ) 2
( *& *'
'hown &ha& &he'e &wo "*ne' ae o''*>"e &o >e aa""e" )
O
−3 & √ −15 6
O no an'we ( *& *' 'hown &ha& &he'e &wo "*ne' ae *mo''*>"e &o >e een.*!u"a)
3 ) *n. &he een.*!u"a .*'&an!e o; &he o*n& ( 893 );om &he "*ne O2D-3 e& "*ne CF >e O2D-3 e& &he een.*!u"a o*n& ;om &he "*ne CF >e ( 893 ) e& &he !oo.*na&e o; o*n& F on &he "*ne CF >e ( D9 ) O2D-3 (1) a.*en& o; "*ne CF O 2
−1 a.*en& o; "*ne F O ua&*on o; "*ne F9
y −3 −1 = 2 x −8 2-O-DP8
2
u> (1) &o (2)
−1 2D-3O
2
x + 7
/D-O-DP1/ 5DO20 DO/ when DO/ O2(/)-3 O5 FO( /95 ) =*'&an!e o; F O
√ ( 4 −8 ) +( 5 −3 )
O
√ 16 + 4
2
2
2O-DP1/
−1 O
2
O
x + 7 (2)
√ 20
O/#/72 un*&'
/ ) ( -197 ) an. F( 7913 ) ae &wo o*n&' on &he Ca&e'*an "ane# ,o*n& , moe' 'u!h &ha& &he o.u!& o; &he ga.*en& o; , an. ,F *' a"wa' -1 ( a ) *n. &he eua&*on o; "o!u' , e& !oo.*na&e o; , >e ,( D9 ) (ga.*en& ,)(ga.*en& ,F)O-1
y −7 y −13 ¿ ( x + 1 ( x −7 ¿=−1 -13-7P61O-DP7D-DP7 DP-D-20P8/O0 ( > ) The D-!oo.*na&e o; o*n& U *' -2 an. *& "*e' on &he "o!u' o; ,# Ca"!u"a&e &he aea o; &*ang"e UF *n &wo wa' Method one When DO-2 (-2)P-(-2)-20P8/O0 -20P100O0 (-10)O0 O10 ,o*n& U O ( -2910 )
Method t/o When DO-2 (-2)P-(-2)-20P8/O0 -20P100O0 (-10)O0 O10 ,o*n& U O ( -2910 )
ea o; &*ang"e UF
U
1
O
2 1
O
2 1
O
2
|
|
−1
7
− 2 −1
7
13
10 7
O
√ (−1 +2 ) +(7 −10 ) ²
O
√ 10
2
|−13 + 70−14 − 49 +26 +10| FU
| | 30
O 15 un*&
O
√ (7 +2 ) +( 13 −10 )²
O
√ 90
2
ea o; &*ang"e UF 1
O
O
√ 10 ¿ ( √ 90 )
2
(
1
√ 900
2
O15 un*& 5 ) ,o*n& , moe' on a Ca&e'*an "ane 'u!h &ha& *& *' a"wa' a& a !on'&an& .*'&an!e ;om o*n& C( 0912 )# =*agam 11 'how' a& o; &he "o!u' o; o*n& , wh*!h a''e' &hough o*n& ( -898 )
an. o*n& F( /9/ ) ( a ) *n. &he eua&*on o; "o!u' ,# C O
√ (−8 −0 ) +( 8−12 ) ²
O
√ 80
,C O
2
√ 80
√ ( x − 0 ) +( y − 12) ² 2
DP-2/P1//O
FC O
√ ( 4 −0 ) +( 4 −12 ) ²
O
√ 80
O
√ 80
80
DP-2/P/O0
2
( > ) Ca"!u"a&e &he aea o; 'ha.e. eg*on# a.*en& o; C
a.*en& o; FC
−8 0 +8
12
O
O O-2
1
O
2 1
(
−4 0 −4
12
2
¿
(-2)O-1
'*n!e "*ne C an. "*ne FC ae een.*!u"a an. hae &he 'ame "eng&h w*&h a "o!u' , a''*ng &hough9 *& *' a 'e!&o o; a !*!"e w*&h 60^# ea o; &aeM*um C 1
O
2
( 8 +12 ) ( 8 )
O80 un*&
ea o; &aeM*um CF 1
O
2
( 4 + 12 ) ( 4 )
O32 un*&
ea o; &he 'e!&o o; !*!"e 22
O(
7
¿
(
√ 80
1
) (
4
¿
O2#857 un*&
ea o; 'ha.e. eg*onO 80P32-2#857 O/6#1/3un*& ) =u*ng &he !o". 'ea'on9 &he &h*!+ne'' o; a 'hee& o;;*!e *n a "a+e *' 1#5m# When &he wea&he ge&' wame9 &he 'hee& o; *!e me"&' a& a !on'&an& a&e# ;&e 2 wee+'9 &he &h*!+ne'' o; &he 'hee& o; *!e *' 1#2m# ( a ) I; *' &he &h*!+ne''9 *n m9 o; &he 'hee& o; *!e a;&e D wee+'9 eDe'' *n &em' o; D# O+DP! when DO09 O1#5 !O1#5 when DO29 O1#29!O1#5
1#2O2+P1#5 +O-0#15 O-0#15DP1#5 ( > ) '+e&!h &he gah aga*n'& D ( e;e &o &he gah ae a& neD& age ) ( ! ) How man wee+' ae nee.e. ;o &he 'hee& o; *!e &o me"& !om"e&e" O-0#15DP1#5 When O0 0O-0#15DP1#5 0#15DO1#5 DO10 10 wee+' ae nee.e. ;o &he 'hee& o; *!e &o me"& !om"e&e"#
Worksheet 11: $e+lection *'& o; a""9 I am g"a. &ha& I am g*en a !han!e &o .o &h*' mean*ng;u" o4e!&# Though &h*' o4e!&9 I .o no& 4u'& "ean moe a>ou& wa' &o 'o"e &he ue'&*on' a>ou& !oo.*na&e geome&9 >u& a"'o .u*ng &he &*me I am &*ng &o !om"e&e &h*' o4e!&9 I *moe. m'e"; *n !omu&e '+*""'# Fe'*.e' &ha&9 >e!au'e I !hoo'e &o .o &h*' o4e!& 'o"o9 I am g"a. I "ean& how &o manage m &*me we""# a'& >u& no& "ea'& I mu'& &han+ m &ea!he an. mo&he &ha& a*. me *n &h*' o4e!&# W*&hou& &he* gu*.an!e9 I &h*n+ I m*gh& no& a>"e &o ;*n*'h &h*' o4e!& on &*me w*&h ua"*&# =u*ng &he o!e'' when I am &*ng &o ;*n*'h &h*' o4e!&9 *& em*n.' me a>ou& &hee ea' ago9 I emem>ee. *&' &he e ;*'& &*me I '&a& &o "ean a>ou& !oo.*na&e geome&# I a"wa' &hough& &ha& &h*' !ha&e *' 4u'& a"" a>ou& &he D an. aD*'9 'ome o*n&'9 'ome ;omu"a9 'ome "*ne' an. a' *; I &hough& *& *' on" u'e;u" *n 'o"*ng ma&hema&*!' ue'&*on an. *& *' &o&a"" u'e"e'' *n ou .a*" "*;e# Fu& '*n!e a;&e I wo+ on &h*' o4e!&9 I ;oun. ou& &ha& !oo.*na&e geome& *' a!&ua"" e u'e;u" an. *& *' eewhee *n ou .a*" "*;e# om a&-!a;&9 a!h*&e!&ue9 mo.en e"e!&on*! .e*!e'9 'a&e""*&e' an. een *n *.eo game'9 !oo.*na&e *' 4u'& a ma4o a& &ha& !an& >e 'eaa&e awa ;om *&# I wa' amaMe. > &he a>*"*& o; human &ha& *' a>"e &o &an';om 'u!h a '*m"e *.ea > Ren< =e'!a&e' *n&o 'ome&h*ng &ha& *' gea& an. >ene;*!*a" &o human# o9 &o en. &h*' o4e!&9 hee *' a oem wha& I &h*n+ a>ou& !oo.*na&e geome&#
+at is te reason for a -oint +/ location and a -lace ere it is, -oint and case. A line fro e to /o is straigt and not far.
