ADDITIO ADDITIONAL NAL MATHEMAT MATHEMATICS ICS PROJECT WORK WORK SPM 2/2015 QUESTION 2 (Concept of triangle in daily life)
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Index No.
Info
Page
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Title
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2
Index
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Introd!ction
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Part 1# a. Concept of triangle$ %. Exa&ple$ of it$ !$e$ in daily life
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Part 2# a. E*!ilateral triangle$ %. +rea of e*!ilateral triangle$
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Part
12'1"
,
-!rter exploration
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/
0eflection
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2
INTRODUCTION
+n old o&an i$e$ to prepare a piece of traditional %lanet for er grandcild. Se ant$ to &ae !$e of to$e $&all piece$ of clot left %eind to c!t into e*!ilateral triangle$ and 3oint$ all $ape$ togeter. AIM
Te ai& of ti$ pro3ect or i$ to apply concept of triangle in daily life Objective
+t te end of ti$ pro3ect or4 yo! ill %e a%le to# i) ii) iii)
Identify te triangle. Identify &etod !$ed to calc!late te area of triangle 4 St!dy and analy$i$ te relation %eteen area and length of triangle
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P!"t 1#!$% C&'ce(t &) t"i!'*+e + clo$ed fig!re %o!nded %y tree line $eg&ent$ i$ called a triangle. + triangle a$ tree $ide$4 tree angle$ and tree 5ertice$. + triangle a$ tree $ide$4 tree angle$ and tree 5ertice$. Te tree $ide$ and te tree angle$ are called te $ix co&ponent$ of a triangle. + triangle di5ide$ a plane into tree part$ # • • •
Te part it te collection of all te point$ in$ide te triangle. Te part it te collection of all te point$ on te triangle4 or te triangle it$elf. Te part it te collection of all te point$ o!t$ide te triangle.
Te part it te collection of all te point$ in$ide te triangle i$ called te interior of te triangle4 ile te part it te collection of all te point$ o!t$ide te triangle i$ called te exterior of te triangle. ,e"te- &) ! t"i!'*+e To ad3acent $ide$ of a triangle inter$ect at a point called a 5ertex. Te pl!ral of 5ertex i$ 5ertice$. E5ery to ad3acent $ide$ incl!de an angle. Te angle i$ na&ed after te 5ertex. E-te"i&" !'*+e Te angle$ o%tained on extending te $ide$ of a triangle are called te exterior angle. +n exterior angle and te corre$ponding interior ad3acent angle of a triangle for& a linear pair. Te $!& of te &ea$!re$ of an exterior angle and te corre$ponding interior ad3acent angle i$ e*!al to 1/67. Te $!& of te lengt$ of any to $ide$ of a triangle i$ alay$ greater tan te lengt of te tird $ide.
0igt angled triangle
+c!te triangle
4
I$o$cele$ triangle
O%t!$e triangle
E*!ilateral triangle
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P!"t 1#b$ Ue &) t"i!'*+e i' .!i+ +i)e •
Triangle$ in engineering and arcitect!re +$ &entioned a%o5e4 Pytagora$8 teore& i$ an incredi%ly i&portant teory. 9oe5er4 it$ i&portance goe$ %eyond te field of p!re &ate&atic$ and $pan$ oter field$ incl!ding engineering and arcitect!re. :i$cipline$ $!c a$ te$e4 ic foc!$ a great deal !pon te $afe di$tri%!tion of eigt or force4 for exa&ple4 rely ea5ily on Pytagora$8 teore&4 ic i$ entirely concerned it triangle$. -!rter&ore4 &any %ridge$ and oter $i&ilar $tr!ct!re$ are often de$igned to incl!de triangle $ape$4 a$ te$e $ape$ are a%le to it$tand a great a&o!nt of pre$$!re (in a $i&ilar ay to arce$). ;eca!$e of te ay tat triangle$ di$per$e pre$$!re tro!go!t teir $ape4 tey are a%le to it$tand &ore pre$$!re tan a differently'$aped o%3ect (for exa&ple4 a $*!are) of te $a&e $i
•
Triangle$ in a$trono&y Principle$ of trigono&etry4 or te $t!dy of triangle$4 are !$ed idely in field$ $!c a$ a$trono&y4 $pace tra5el and co&&!nication in ay$ tat I4 a$ a non' a$trono&er4 cannot e5en %egin to !nder$tand. 9oe5er4 &y re$earc $!gge$t$ tat trigono&etry play$ a role in a$pect$ of a$trono&y $!c a$ deciding o far a%o!t te eart a $atellite di$ $o!ld %e placed.
0ooftop$
Sign%oard$
Pyra&id$ 6
P!"t 2% Ei+!te"!+ t"i!'*+e In geo&etry4 an e*!ilateral triangle i$ a triangle in ic all tree $ide$ are e*!al. In traditional or E!clidean geo&etry4 e*!ilateral triangle$ are al$o e*!iang!lar = tat i$4 all tree internal angle$ are al$o congr!ent to eac oter and are eac 67. Tey are reg!lar polygon$4 and can terefore al$o %e referred to a$ reg!lar triangle$. a) e*!ilateral triangle$ ic a5e different lengt #
7
%)
x c&
x c&
x 2
If
x
x
c&
c&
2
i$ te lengt of a $ide of te e*!ilateral triangle and
h
te
1
¿ x . h
eigt ten ti$ rectangle a$ area 1
2
4 and ence4 %y
2
2 2 Pytagora$> teora& x =h +( 2 x ) . T!$4
h=
√
3 4
2
x
√ x 2 A = Terefore4 . 2 3
8
c) #i$ Met&. 1 3 4&"+! # E*!ilaretal Triangle 1 ? c& $ A
1 =
2
bh
h =√ 3 2−1.52
h =2.6 cm
1
A = ( 3 )( 2.6 )
A =3.9 cm2
2
Met&. 2 6 C&&".i'!te 7e&et"
7 6 5 4 3 2 1 0 -2
-1
0
1
2
3
4
5
6
7
8
9
10
-1 -2
+@
| ( |
2 3.6
+@
|
A x ( B y – C y ) + B x ( C y − A y )+ C x ( A y − B y ) 2
−1 ) + 3.5 ( 1−1 ) + 5 ( 1−3.6 ) 2
| 9
+@
|
(
2 2.6
|
)+3.5 ( 0 )+ 5 (−2.6 ) 2
2 unit +@ .A"
#ii$ Met&. 1 3 4&"+! # E*!ilaretal Triangle 1 ? "c& $ A
1 =
2
bh
√ −2 2
h =3.5 cm
2
h= 4 1
A = ( 4 )( 4.5 )
A =7 cm2
2
Met&. 2 6 C&&".i'!te 7e&et" 7 6 5 4 3 2 1 0 -2
-1
0
1
2
3
4
5
6
7
8
9
10
-1 -2
+@
| ( |
2
|
−1 ) + 4 ( 1−1 ) + 6 (1 −4.5 )
2 4.5
+@
|
A x ( B y – C y ) + B x ( C y − A y )+ C x ( A y − B y )
2
10
|
+@
(
2 3.5
) + 4 ( 0 ) + 6 (−3.5 ) 2
|
unit 2
+@ ,
#iii$ Met&. 1 3 4&"+! # E*!ilaretal Triangle 1 ? c& $ A
1 =
2
bh
h =√ 6 2−32
h =5.2 cm
1
A = ( 6 )( 5.2) 2
2
A =15.6 cm
Met&. 2 6 C&&".i'!te 7e&et" 7 6 5 4 3 2 1 0 -2
-1
0
1
2
3
4
5
6
7
8
9
10
-1 -2
11
+@
| ( | ( |
2 6.2
+@
2
|
−1 )+ 5 ( 1 −1 )+ 8 ( 1−6.2 )
2 5.2
+@
|
A x ( B y – C y ) + B x ( C y − A y )+ C x ( A y − B y )
2
|
) + 5 ( 0 ) + 8 (−5.2 ) 2 2
+@ 1. unit
P!"t 8 a.
12
%.
(x²)
+rea (+)
2
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No.
Dengt (x c&)
1
B r a p
of +rea (+) again$t lengt (x) and Brap of +rea (+) again$t x are plotted %elo.
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Graph of Area against Length 9000000 8000000 7000000 6000000 5000000 4000000 3000000 2000000 1000000 0 0
500
1000
1500
2000
2500
3000
3500
4000
4500
Graph of Area against Length 9000000 8000000 7000000 6000000 5000000 4000000 3000000 2000000 1000000 0 0
5000000
10000000
15000000
20000000
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c) Brap of +rea (+) again$t lengt (x) $o$ an exponential grap ile te grap of +rea (+) again$t x $o$ a linear grap. Ti$ $o$ tat en te +rea of grap increa$e$ te 5al!e of te lengt $*!are increa$e$ proportionally. Te %a$ic property of exponential$ i$ tat tey cange %y a gi5en proportion o5er a $et inter5al. In ti$ ca$e4 te lengt of $ide$ a$ canged %y a con$tant proportion a$ te cange$ d!e to te $ided of triangle. d) x
A x ²
2.2 . "./ .1 A. 1.A 1/. 21., 2/.1 6. ./ A.6 /.1 122.1 1A.6 1/,. 2.1 2A.1 26." A." "11." ""., /.6 1,6.1 1/2A.
6." 6." 6." 6." 6." 6." 6." 6." 6." 6." 6." 6." 6." 6." 6." 6." 6." 6." 6." 6." 6." 6." 6." 6." 6."
A
A A Te 5al!e of x ² $o$ al&o$t $i&ilar 5al!e ic i$ 6." ile te x $o$ a con$tant increa$e in te 5al!e$. Ti$ i$ d!e to te for&!la craced earlier in P+0T 2 (%). Te √3
for&!la of area i$ + @
4
x ²
√3 t!$ en e di5ide %y x4 te 5al!e left i$
4
ic i$
15
√3 e*!al to 6."4 ile en di5ide %y x4 e ill get
4
x
4 ere all te re$!lting
5al!e are 6." &!ltiply %y te 5al!e of x.
4"te" E-(+&"!ti&' a) Det te $i
@ 1,A2.1
+pproxi&ately 1,A2 n!&%er of triangle$ are re*!ired to for& a traditional %lanet it $i
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Re)+ecti&' Tro!g ti$ pro3ect4 I a5e &anaged to# 1. +c*!ire effecti5e &ate&atical co&&!nication to!g oral and riting and to !$e te lang!age of &ate&atic$ to expre$$ &ate&atical idea$ correctly and preci$ely. 2. Increa$e intere$t and confidence a$ ell a$ enance ac*!i$ition of &ate&atical noledge and $ill$ tro!g application of 5ario!$ $trategie$ of pro%le& $ol5ing. . :e5elop noledge and $ill$ tat are !$ef!l for career and f!t!re !ndertaing$. ". 0eali
REFERENCE 1. 2.
ttp$#en.iipedia.orgindex.ppF$earc@triangleGinGe5eryGdayG!$e &ate&atic$ reference %oo$
APPRECIATION I o!ld lie tan te folloing for elping and g!iding &e in acco&pli$ing ti$ pro3ect # 1. 2. .
Hy teacer. Hy friend$. Hy parent$.
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+::ITION+D H+T9EH+TICS P0OECT JO0K SPH 2261
Na&e # Syl5ia Dee ia Jern Scool # SHK 9eng Ee +nga Biliran # PC61,+1"
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