lesson 2 math13-1

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Lesson 2.1 CIRCLESLesson 2.2 MISCELLANEOUS

PLANES

Week 3 and Week 4

MATH13-1Solid Mens!a"ion

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2.1 CIRCLES

A circle is a se" o# $oin"s% ea&' o#('i&' is e)idis"an" #!o* a +,ed$oin" &alled "'e center.

T'e line oinin "'e &en"e! o# a &i!&le"o an/ $oin"s on "'e &i!&le is kno(nas "'e radius.

An arc is a $o!"ion o# a &i!&le "'a"&on"ains "(o end$oin"s and all "'e$oin"s on "'e &i!&le 0e"(een "'e

end$oin"s.Re#e!en&e Solid Mens!a"ion Unde!s"andin "'e 3- S$a&e 0/

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CIRCLES

/ &'oosin an/ "(o $oin"s on "'e&i!&le% "(o a!&s (ill 0e #o!*ed amajor arc 5"'e lone! a!&6% and aminor arc5"'e s'o!"e! one6.

Re#e!en&e Solid Mens!a"ion Unde!s"andin "'e 3- S$a&e 0/

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CIRCLES

A chord is a line se*en" oinin an/"(o $oin"s on "'e &i!&le. T'e &'o!d"'a" $asses "'!o' "'e &en"e! o# "'e

&i!&le is &alled "'e diameter o# a&i!&le.

A &'o!d di7ides "'e &i!&le in"o "(o

!eions% "'e major segmentand "'eminor segment.


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CIRCLES

A sectoris "'e +!e #o!*ed 0/ "(o!adii and an in&lded a!&. T'ecentral angle is "'e anle in ('i&'

"'e 7e!"e, lies a" "'e &en"e! o# "'e&i!&le and ('i&' sides a!e "'e "(o!adii.

T'e inscribed angle is "'e anle in('i&' "'e 7e!"e, lies on "'e &i!&leand ('i&' "(o sides a!e &'o!ds o#

"'e &i!&le.Re#e!en&e Solid Mens!a"ion Unde!s"andin "'e 3- S$a&e 0/

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CIRCLES


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CIRCLES

I# &i!&les o# di8e!en" !adii 'a7e&o**on &en"e! "'en "'e/ a!e !e#e!!ed "oas concentric circles.

T'e !eion 0onded 0/ an/ "(o&on&en"!i& &i!&les is kno(n as "'e annulus.


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CIRCLES

A line in "'e sa*e $lane as "'e &i!&le is atangent line o# "'e &i!&le i# i" in"e!se&"s "'e&i!&le a" e,a&"l/ one $oin" on "'e &i!&le.

A line is &alled a secant line i# i" in"e!se&"s

"'e &i!&le a" "(o $oin"s on "'e &i!&le.


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C9CLIC :UARILATERAL

A cyclic quadrilateral is a #o!-sided +!e ins&!i0ed in a &i!&le% (i"'ea&' 7e!"e, 5&o!ne!6 o# "'e

)ad!ila"e!al "o&'in "'e&i!&*#e!en&e o# "'e &i!&le.

T'e s* o# "'e o$$osi"e anles o#

s&' a )ad!ila"e!al is 1;

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C9CLIC :UARILATERAL

x> w? 1;z? 1;

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THEOREMS ON CIRCLES

1. I# "(o &'o!ds in"e!se&" a" a $oin" inside"'e &i!&le% "'en "'e $!od&" o# "'ese*en"s o# one &'o!d is e)al "o "'e

$!od&" o# "'e se*en"s o# "'e o"'e! &'o!d.


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THEOREMS ON CIRCLES

2. I# "(o se&an" lines o# a &i!&lein"e!se&" a" an e,"e!io! $oin"% "'en "'e$!od&" o# len"'s o# "'e en"i!e se&an"

line and i"s e,"e!nal se*en" is e)al"o "'e $!od&" o# "'e len"'s o# "'eo"'e! se&an" line and i"s e,"e!nal

se*en".


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THEOREMS ON CIRCLES

3. I# a "anen" line and a se&an" line o#a &i!&le in"e!se&" a" a $oin" e,"e!io! "o"'e &i!&le% "'en "'e $!od&" o# "'e

len"'s o# "'e se&an" line and i"se,"e!nal se*en" is e)al "o "'es)a!e o# "'e len"' o# "'e "anen"

line.


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THEOREMS ON CIRCLES

4. E7e!/ "anen" line o# a &i!&le is$e!$endi&la! "o "'e !adis o# "'e &i!&led!a(n "'!o' "'e $oin" o# "anen&/.


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THEOREMS ON CIRCLES

@. T'e dia*e"e! "'a" is $e!$endi&la! "o a&'o!d 0ise&"s "'e &'o!d and i"s "(o a!&s.

Con7e!sel/% "'e dia*e"e! "'a" 0ise&"s a&'o!d is $e!$endi&la! "o "'e &'o!d.

Conse)en"l/% "'e $e!$endi&la! 0ise&"o! o#a &'o!d is "'e dia*e"e! o# "'e &i!&le ('i&'*s" $ass "'!o' "'e &en"e! o# "'e &i!&le.


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THEOREMS ON CIRCLES

. Cen"!al anles o# "'e sa*e o! e)al&i!&les 'a7e "'e sa*e !a"io as "'ei!in"e!&e$"ed a!&s.

B. Anles #o!*ed #!o* "(o $oin"s on"'e &i!&le a!e e)al "o "'e o"'e!anles% in "'e sa*e a!&% #o!*ed #!o*

"'ose "(o $oin"s.


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THEOREMS ON CIRCLES

;. T'e line o# &en"e!s o# "(o "anen" &i!&les$asses "'!o' "'e $oin" o# "anen&/.

. T'e *eas!e o# a &en"!al anle is "'e anla!

*eas!e o# i"s in"e!&e$"ed a!&. T'e in"e!&e$"eda!& is *eas!ed 0/ i"s &en"!al anle.


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THEOREMS ON CIRCLES

1

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THEOREMS ON CIRCLES

12. T'e s* o# "'e $!od&"s o# o$$osi"esides o# a &/&li& )ad!ila"e!al is e)al "o"'e $!od&" o# "'e diaonals.


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THEOREMS ON CIRCLES

13. T'e len"'s o# "(o "anen"s #!o*"'e $oin"s o# "anen&/ on "'e &i!&le "o"'ei! $oin" o# in"e!se&"ion a!e "'e

sa*e.


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THEOREMS ON CIRCLES

14. T'e *eas!e o# "'e anle #o!*ed 0/"(o se&an" lines in"e!se&"in a" an in"e!io!$oin" is one-'al# "'e s* o# "'e anla!

*eas!e*en"s o# "'e a!&s in"e!&e$"ed 0/"'e anle and i"s 7e!"i&al anle.


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THEOREMS ON CIRCLES

I# "(o se&an"s in"e!se&" a" an e,"e!io!$oin" o# a &i!&le% "'e *eas!e o# "'eanle #o!*ed 0/ "'e se&an"s is one-'al#

"'e di8e!en&e 0e"(een "'e anla!*eas!e*en"s o# "'e in"e!&e$"ed a!&s.

Re#e!en&e Solid Mens!a"ion Unde!s"andin"'e 3- S$a&e 0/

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DORMULAS ON CIRCLES

The Perimeters

Circumference:C? 2r

Arc length:s? rFPerimeter of a sector:P? r52 > F6 ?rF > 2r

Perimeter of a segment:P? s> LLength of chord:


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DORMULAS ON CIRCLES

Areas of Circle, Sector, Segment,and Annulus

Area:A? r2? d24

Area of a Semicircle:A? r2

Area of a Quarter Circle:A? Jr2

Area of a Sector:A? r2

F ? rsArea of Segment:A? r25F K sin F6


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DORMULAS ON CIRCLES

Area of Annulus region:

Apolygon is inscribed in a circlei# "'e

7e!"i&es o# "'e $ol/on lie on "'e &i!&le. I# "'e$ol/on is !ela!% "'en "'e *eas!e o# i"sside &an 0e de"e!*ined 0/ "'e &osine la(.

r? !adis% s? side o# $ol/on% F ? &en"!alanle


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DORMULAS ON CIRCLES

Polygon Inscribed in a Circle


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CIRCLES

T'e polygon is circumscribedabout the circle i# ea&' side o# "'e$ol/on is "anen" "o "'e &i!&le.

I# "'e $ol/on is !ela!% "'en "'eradius of the circle is equal tothe apothem ao# "'e $ol/on.


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CIRCLES

Polygon Circumscribed About theCircle


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DORMULAS ON CIRCLES

adius of the CircleCircumscribing a Triangle

('e!eAis "'e a!ea o# "'e "!ianle and.


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DORMULAS ON CIRCLES

Circle Circumscribing a Triangle !Triangle Inscribed in a Circle


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DORMULAS ON CIRCLES

adius of the Circle Inscribed in aTriangle

r = As


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DORMULAS ON CIRCLES

Circle Inscribed in a Triangle !Triangle Circumscribing a Circle


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AREA OD A C9CLIC :UARILATERAL

W'e!e % se*i-$e!i*e"e! o# "'e)ad!ila"e!al.


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EAMPLES

1% $@4 W'a" is "'e a!ea o# a &i!&le(i"' a &i!&*#e!en&e o# 1; &* [email protected] &*2

3% $@@ A &i!&le ('i&' 'as an a!ea o#144 &*2 is &" in"o "(o se*en"s 0/ a

&'o!d "'a" is &* #!o* "'e &en"e! o#"'e &i!&le. W'a" is "'e a!ea o# "'es*alle! se*en" ANS ;;.4@ &*2


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EAMPLES

4% $@@ T'e "anen"ABand "'e se&an" BCDa!e d!a(n "o a &i!&le #!o* "'e sa*e e,"e!io!$oin" B. I# "'e len"' o# "'e "anen" is ;in&'es and "'e e,"e!nal se*en" o# "'e

se&an" is 4 in&'es% "'en ('a" is "'e len"' o#"'e se&an"

ANS 1 in


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EAMPLES

% $@ An e)ila"e!al "!ianle isins&!i0ed in a &i!&le (i"' an a!ea e)al"o ;1 s)a!e ni"s. Dind "'e a!ea o#

"'e "!ianle. ANS 1

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2.1 EERCISES

% $

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2.1 EERCISES1% $

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2.1 EERCISES

2B% $2 A &i!&le 'as an a!ea e)al "o 2@&*2. I"s dia*e"e!AB&oin&ides (i"' one o# "'esides o# "!ianleACBin ('i&' "'e 7e!"e, Clies

on "'e &i!&le. I# "'e "!ianle 'as an a!ea e)al"o 11 &*2% +nd i"s $e!i*e"e!. ANS 22 &*

31% $3 Dind "'e a!ea o# a &/&li&)ad!ila"e!al (i"' "(o sides "'a" *eas!e 4

and @ ni"s and one diaonal &oin&ides (i"' adia*e"e! o# "'e &i!&le% "'e !adis o# ('i&' is23 ni"s. ANS 23.3 s. .


2 2 SC O S

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2.2 MISCELLANEOUSPLANES

A star is a s"a!-like +!e ('i&'ene!all/ &onsis"s o# a $ol/on (i"'"!ianles on i"s sides.

I" is a regular star i# "'e $ol/onin7ol7ed is a !ela! $ol/on

T'epentagram% also kno(n as Germano! witch star, is a +7e-$oin"ed !ela!

s"a!. T'e hexagram ('i&' is alsokno(n as Davids staro! Solomons sealis a si,-$oin"ed !ela! s"a!.


2 2 MISCELLANEOUS

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2.2 MISCELLANEOUSPLANES


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ELLIPTICAL SECTION

Area A ? ab

a is "'e len"' o# *ao! se*en"5*ao! !adis6

b is "'e len"' o# "'e *ino! se*en"5*ino! !adis6

Perimeter:

Res"!i&"ion

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ELLIPTICAL SECTION

a b


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ELLIPTICAL SECTION

No"e "'a" "'is #o!*la &on#o!*s (i"'"'e #o!*la o# "'e &i!&*#e!en&e o# "'e&i!&le ('e!e r? a? b.


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PARAOLIC SECTION

Area of Parabola: A? bh

b? 0ase% h? al"i"de

Area of Spandrel:A? bh

Perimeter of Parabola:


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PARAOLIC SECTION


IRREULAR SHAPE

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IRREULAR SHAPEDIURES

Simpson"s uleI#y

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2.2 EERCISES

2% $B

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2.2 EERCISES

12% $B1 Usin Si*$sonYs Rle andan in"e!7al o# d?

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2.2 EERCISES

13% $B2 T(o se*i&i!&les a!eins&!i0ed in a s)a!e (i"' side ; &* ass'o(n. Dind "'e a!ea o# "'e s'aded

!eion. ANS 22.;B &*2


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HOMEWORZ 2

2.1 EERCISES Ys @% B% 1@% 23% 2% [33 $$. @-3

2.2 EERCISES Ys ;% 1

lesson 2 math13-1

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