Lesson 1 math 13-1

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Lesson 1.1 POLYGONSLesson 1.2 TRIANGLES

Lesson 1.3 QUADRILATERALS

Week 1 and Week 2

MATH13-1So!d Mens"#a$!on

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A polygon !s a %osed &ane '("#e$)a$ !s *o!ned +, !ne se(en$s.

A &o,(on a, aso +e de'ned as a"n!on o !ne se(en$s s"%) $)a$/ !0ea%) end&o!n$ !s $)e end&o!n$ o on,$o se(en$s !!0 no $o se(en$s

!n$e#se%$ e%e&$ a$ an end&o!n$ and!!!0 no $o se(en$s !$) $)e saeend&o!n$ a#e %o!nea#.

1.1 POLYGON

Ree#en%e/ So!d Mens"#a$!on/ Unde#s$and!n( $)e 3-D S&a%e +,

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PARTS O4 A POLYGONSide or Edge

Exterior Angle

Vertex

Diagonal

Interior Angle


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TYPES O4 POLYGONRegular PolygonIn a #e("a# &o,(on5 a an(es a#e e6"a and

a s!des a#e o $)e sae en($). Re("a#&o,(ons a#e +o$) e6"!an("a# ande6"!a$e#a.Equiangular Polygon

A &o,(on !s e6"!an("a# ! a o !$s an(es a#e%on(#"en$.Equilateral Polygon

A &o,(on !s e6"!a$e#a ! a o !$s s!des a#ee6"a.Irregular PolygonA &o,(on $)a$ !s ne!$)e# e6"!an("a# no#

e6"!a$e#a !s sa!d $o +e an !##e("a# &o,(on.Ree#en%e/ So!d Mens"#a$!on/ Unde#s$and!n( $)e 3-D S&a%e +,

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TYPES O4 POLYGONConvex PolygonE7e#, !n$e#!o# an(e o a %on7e &o,(on !sess $)an 189:. I a !ne !s d#an $)#o"() $)e%on7e &o,(on5 $)e !ne ! !n$e#se%$ a$ os$

$o s!des.


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TYPES O4 POLYGONConcave Polygon

A %on%a7e &o,(on )as a$ eas$ one !n$e#!o#an(e $)a$ eas"#es o#e $)an 189:. I a !ne!s d#an $)#o"() a %on%a7e &o,(on5 $)e !ne

a, !n$e#se%$ o#e $)an $o s!des.


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NAMING O4 POLYGON


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NAMING O4 POLYGON


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NAMING O4 POLYGON


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NAMING O4 POLYGON

4o# n"+e#s #o 199 $o ;;;5 e%ons$#"%$ $)e nae o $)e &o,(on +,s$a#$!n( !$) $)e e' o# $)e

)"nd#eds d!(!$ $aken #o $)e onesd!(!$ !n"s $)e )e%$a5> $)en o%eed as +eo#e.


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E?AMPLES


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We sa, $)a$ $o &o,(ons a#e similar! $)e!#%o##es&ond!n( !n$e#!o# an(es a#e %on(#"en$ and $)e!#%o##es&ond!n( s!des a#e o&o#$!ona.

@, #a$!o and o&o#$!on5

SIMILAR POLYGONS


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T)e a$!$"de a o $)e $#!an(e !s %aed

$)e apothem.T)e an(e B $)a$ !s o&&os!$e $)e +aseo $)!s $#!an(e !s %aed $)e central

angle.

PROPERTIES O4 A REGULARPOLYGON


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Pe#!e$e#/ PC ns

en$#a An(e/ B C 39:Fn

A&o$)e/

nC n"+e# o s!des



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A#ea/

AC Pa

N"+e# o D!a(onas/

In$e#!o# An(e/

S" o In$e#!o# An(e/

S. I. A.C 189:n 20



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J15 &. 8/4!nd $)e a#ea o a #e("a#nona(on !$) a s!de $)a$ eas"#es 3"n!$s. Aso 'nd $)e n"+e# o

d!a(onas and $)e s" o !$s !n$e#!o#an(es.

ANS/AC KK. s. ".5 DC 25 S. I. A.C

129:

E?AMPLES


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K. 4!nd $)e s" o $)e !n$e#!o# an(e o a #e("a#

$#!a%on$aka!$e$#a(on. ANS/ K9:

. Nae ea%) &o,(on !$) $)e (!7en n"+e# o s!des. Aso 'nd $)e%o##es&ond!n( n"+e# o d!a(onas.

a0 2 +0 181 %0 d0 K3

11. T)e n"+e# o d!a(onas o a #e("a# &o,(on !s 3K. 4!nd $)ea#ea o $)e &o,(on ! !$s a&o$)e eas"#es 19 %. ANS/ 32K.1%2

12. T)e n"+e# o d!a(onas o a #e("a# &o,(on !s K. 4!nd&e#!e$e# o $)e &o,(on ! !$s a&o$)e eas"#es 8 !n.

13. T)e s" o !n$e#!o# an(es o a #e("a# &o,(on !s 129: . 4!nd$)e a#ea o $)e &o,(on ! $)e &e#!e$e# !s K %. ANS/ 1K. %2

1.1 E?ERISES &&. ;-11


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HOMEWOR 1.1

1.1 E?ERISES/ Js 1K5 15 1;5 215 235 2K &&. 11-12


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Similar Triangles:

o##es&ond!n( an(es a#e %on(#"en$ and $)e%o##es&ond!n( s!des a#e o&o#$!ona.

Sae s)a&e5 d!e#en$ s!e5 d!e#en$ eas"#een$+"$ !n o&o#$!on.

1.2 TRIANGLES


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LINES ONNETED WITH

TRIANGLESAn altitude o a $#!an(e !s $)e !nese(en$ d#an #o a 7e#$e o $)e $#!an(e&e#&end!%"a# $o $)e o&&os!$e s!de.

A median of a triangle !s $)e !ne se(en$%onne%$!n( $)e !d&o!n$ o a s!de and

$)e o&&os!$e 7e#$e.

An angle bisector o a $#!an(e !s $)e !ne

se(en$ )!%) d!7!des an an(e o $)e$#!an(e !n$o $o %on(#"en$ an(es and )asend&o!n$s on a 7e#$e and $)e o&&os!$e s!de.


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A perpendicular bisector of aside o a $#!an(e !s $)e !nese(en$ )!%) ee$s $)e s!de a$

#!()$ an(e and d!7!des $)e s!de !n$o$o %on(#"en$ se(en$s.

LINES ONNETED WITH

TRIANGLES


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rthocenter !s $)e &o!n$ o!n$e#se%$!on o $)e $#!an(esa$!$"des.

T)e centroid !s $)e &o!n$ o!n$e#se%$!on o $)e $)#ee ed!ans o$)e $#!an(e.

T)e incenter !s $)e &o!n$ o!n$e#se%$!on o $)e $)#ee an(e+!se%$o#s o $)e $#!an(e.

T)e circumcenter !s $)e &o!n$ o!n$e#se%$!on o $)e e# end!%"a#

TYPES O4 TRIANGLE ENTERS


ALTITUDE MEDIAN AND

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ALTITUDE, MEDIAN, AND

ANGLE BISECTOR

FORMULAS ons!de# an a#+!$#a#, $#!an(e !$)s!des a5 b5 and c5 and an(es A5 B5and C. Le$ hc5 mc5 and5 lc +e $)e

en($)s o $)e a$!$"de5 ed!an and+!se%$o# o#!(!na$!n( #o 7e#$e.


ALTITUDE MEDIAN AND

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ALTITUDE, MEDIAN, AND

ANGLE BISECTOR

FORMULAS


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4ORMULAS 4OR THE AREA O4 ATRIANGLE

Gene#a 4o#"a/AC bh

SAS S!de-An(e-S!de0 4o#"a/ACabs!n B

He#ons 4o#"a o# SSS T)#ee S!des0

ase/5


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E?AMPLES

J5 &1/ G!7en a $#!an(eABC

!n )!%)$)e s!des a#eABC 39 !n5ACC 9 !n5 andBCC 9 !n. On $)e s!de AB !s a &o!n$ D$)#o"() )!%) a !ne DE !s d#an and

%onne%$ed $)#o"() a &o!n$ E on s!de ACso $)a$ $)e an(e AED !s e6"a $o an(eABC. I $)e &e#!e$e# o $)e $#!an(eADE!s e6"a $o K !n5 'nd $)e s" o $)een($)s o $)e !ne se(en$s BDand CE.

ANS/ 8 !n


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E?AMPLES


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1.2 E?ERISESJ35 &29/ 4!nd $)e a$!$"de and $)e a#ea o an

e6"!a$e#a $#!an(e $)e s!de o )!%) !s 8 %. ANS/%5 %2

J5 &29/ One s!de o an !sos%ees $#!an(e !s 19 "n!$s

and $)e &e#!e$e# !s 2 "n!$s. 4!nd $)e a#ea o $)e$#!an(e.

JK5 &29/ 4!nd $)e a#ea o an e6"!a$e#a $#!an(e $)e

a$!$"de o )!%) !s K %. ANS/ 1.3 %2

J5 &21/ T)e +ase o an !sos%ees $#!an(e and $)ea$!$"de d#o&&ed on one o $)e %on(#"en$ s!des a#ee6"a $o 18 % and 1K % #es&e%$!7e,. 4!nd $)es!des o $)e $#!an(e. ANS/ 1.28 %


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J85 &21/ To a$!$"des o an !sos%ees $#!an(ea#e e6"a $o 29 % and 39 %. De$e#!ne$)e +ase an(es o $)e $#!an(e.

J125 &21/ 4!nd $)e a#ea o a $#!an(e !$) $os!des $)a$ eas"#e !n and ; !n5 and $)e+!se%$o# o $)e an(e +e$een $)e !s 3 !n.

J135 &21/ In an a%"$e $#!an(eABC5 $)e a$!$"de

AD!s d#an. 4!nd $)e a#ea o $#!an(eABC !ABC 1K !n5ACC 18 !n5 and BDC 19 !n. ANS/13.8 !n2

1.2 E?ERISES


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HOMEWOR 1.2

1.2 E?ERISES/ Js ;5 115 1K5 15 21&&. 21-22


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1.3 QUADRILATERALS

A quadrilateral5 aso knon as$e$#a(on o# 6"ad#an(e5 !s a (ene#a$e# o# a o"#-s!ded &o,(on.


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QUADRILATERALS


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QUADRILATERALS


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QUADRILATERALS


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PARALLELOGRAM

Aparallelogram!s a 6"ad#!a$e#a !n)!%) $)e o&&os!$e s!des a#e &a#ae.


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Pa#aeo(#as )a7e $)e oo!n(!&o#$an$ o&e#$!es/

O&&os!$e s!des a#e e6"a.

O&&os!$e !n$e#!o# an(es a#e%on(#"en$

Ad*a%en$ an(es a#e s"&&een$a#,.

A d!a(ona d!7!des $)e &a#aeo(#a!n$o $o %on(#"en$ $#!an(es.

T)e $o d!a(onas +!se%$ ea%) o$)e#.

PARALLELOGRAM


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4ORMULAS

d2C a2 b2 2ab%os B

P

C 2a

2b

A C bh

A = ab s!n B


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RETANGLE

A rectangle !s essen$!a, a&a#aeo(#a !n )!%) $)e !n$e#!o#an(es a#e a #!()$ an(es.


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4ORMULAS

PC 2b 2h

AC bh


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SQUARE

A square !s a s&e%!a $,&e o a#e%$an(e !n )!%) a $)e s!des a#ee6"a.


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4ORMULAS

dC a2

PC a

AC a2


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RHOM@US A rhombus!s a &a#aeo(#a !n )!%) a

s!des a#e e6"a.


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4ORMULAS

B C 2 $an1d1Fd20

PC b

AC

d1d

2

AC bh

AC b2s!n B


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A trape!oid!s a 6"ad#!a$e#a !$) one&a!# o &a#ae s!des.

I $)e non-&a#ae s!des a#e

%on(#"en$5 $)e $#a&eo!d !s %aed anisosceles trape!oid.

TRAPEOID


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A $#a&eo!d )!%) %on$a!ns $o #!()$an(es !s %aed a right trape!oid"

TRAPEOID


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AREA O4 TRAPEOID

AC a b0h


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TRAPEIUM

A trape!ium !s a 6"ad#!a$e#a !$)no $o s!des $)a$ a#e &a#ae.

Aand Ca#e an, $o o&&os!$e !n$e#!o#an(es.

s!s $)e se!-&e#!e$e#.


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E?AMPLES

J85 &31/ T)e d!a(ona o a s6"a#e !s 12"n!$s. W)a$ !s $)e eas"#e o one s!de o$)e s6"a#eV 4!nd !$s a#ea and &e#!e$e#.

ANS/AC 25 PC 22

J195 &32/I ABCD !s a #)o+"s5 ACC 5and ADC !s an e6"!a$e#a $#!an(e5 )a$

!s $)e a#ea o $)e #)o+"sV ANS/ A C13.8 s. ".


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J125 &33/ 4!nd $)e a#ea and $)e&e#!e$e# o $)e #!()$ $#a&eo!ds)on !n $)e '("#e.

ANS/AC ;.5 PC 39.2

E?AMPLES


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1.3 E?ERISESJ15 &38/ T)e d!a(ona o a #e%$an(e !s 2K

e$e#s on( and akes an an(e o 3:!$) one s!de o $)e #e%$an(e. 4!nd $)ea#ea and $)e &e#!e$e# o $)e #e%$an(e.

ANS/ 2;.2 2

5 ;.8 J5 &38/ A #e%$an(e and a s6"a#e )a7e $)e

sae a#ea. I $)e en($) o $)e s!de o $)es6"a#e !s "n!$s and $)e on(es$ s!de o $)e

#e%$an(e !s K o#e $)an $)e eas"#e o$)e s)o#$e# s!de. 4!nd $)e d!ens!ons o $)e#e%$an(e.


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J85 &3;/ T)e a#ea o an !sos%ees $#a&eo!d!s 2 2. I $)e )e!()$ and $)e en($) oone o !$s %on(#"en$ s!des eas"#e

and 19 #es&e%$!7e,5 'nd $)e $o +ases.J195 &3;/ A &!e%e o !#e o en($) K2 !s

%"$ !n$o $o &a#$s. Ea%) &a#$ !s $)en +en$$o o# a s6"a#e. I$ !s o"nd $)a$ $)e

%o+!ned a#ea o $)e $o s6"a#es !s 19;2. 4!nd $)e s!des o $)e $o s6"a#es.

1.3 E?ERISES


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J115 &3;/ A #)o+"s )as d!a(onas o32 and 29 !n%)es. 4!nd $)e a#ea and$)e an(e o&&os!$e $)e on(e#

d!a(ona. ANS/ 329 !n25 11:

J 25 &9/ 4!nd $)e a#ea o a #)o+"s

!n )!%) one s!de eas"#es 19 %and one o $)e d!a(onas eas"#es12 %.

1.3 E?ERISES


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HOMEWOR 1.3

1.3 E?ERISES/ Js 5 ;5 1K5 295 2352K5 285 2; &&. 38-9

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Lesson 1 math 13-1