ac geometry notes semester 2

8/12/2019 AC Geometry Notes Semester 2

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AC Geometry/Trig.

Alan Grothues Semester 2


2/24

Chapter 7: Ratios and Proportions

Ratio

Comparison between two numbers

ProportionEqual ratios

Ex. 1/2 = 2/4

1 4 Means Can change means or extremes and still have true statement

2 8 Extremes Reciprocal PropertyIf 1/2 = 2/4 then 2/1 = 4/2

Finding Ratio Values

!n a pol"gon# "ou can use a $ormula to $ind the measure o$ each angle i$ the ratios o$ theangles are %nown&

Ax + Bx + Cx + = Sum o all interior angles

Letterrepresents the part o$ the ratio

x represents value each must be multiplied b" to reach true value

Similar Pol!gons'ol"gons are similar i$$ their corresponding sides are proportional and theircorresponding angles are congruent

(hen identi$"ing similar pol"gons# ma%e sure to write out which ones aresimilar and the proportions that go along with that

A

"

)* 8 + 4

B C # F

, -

(hen solving for sides of similar triangles# pair up equal ratios with the un%nownvariable and solve

/C 0 1E2

AB/"# = BC/#F = AC/"F


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A$A$ Similarit! Postulate

If two angles of one triangle are congruent to two angles of another triangle, thentriangles are congruent

Bot% triangles are similar

&S$A$S Similarit! Postulate

If two sides of a triangle are proportional to corresponding sides of another triangle and

the included angles of the triangles are congruent, then triangles are similarS$S$S$ Similarit! Postulate

If three sides of a triangle are proportional to corresponding sides of another triangle,then the triangles are congruent

Side'Splitter (%eorem

If a line parallel to one side of a triangle intersects the other two sides, then it dividesthose sides proportionally

A B

A/C = B/"

C " 3ther combinations wor%# also

(riangle Angle'Bise)tor (%eorem

If a ray bisects an angle of a triangle, then it divides the opposite side into segmentsproportional to the other sides

Parallel *ine Segment Similarit! Corollar!

If three parallel lines intersect two transversals, then they divide the transversalsproportionally

i%e the 5ide5plitter 6heorem


4/24

Chapter 8: Right Triangles and Trigonometry

eometri) ,ean

Equal means or extremesEx. )7 9 74

!n a right triangle with all three altitudes# all three triangles are similar

(ith all o$ these similarities# we are able to see two things&

If you have a right triangle, then the altitude to the hypotenuse is a geometric meanbetween the two parts of the hypotenuse

Part o -!pot$ Altitude to-!pot$

Altitude to -!pot$ Part o -!pot$

If you have a right triangle, then a leg of the triangle is a geometric mean between thewhole hypotenuse and the part closest to the leg

.%ole -!pot$ *eg o Rig%t (riangle

*eg o Rig%t (riangle Part o -!pot$ Closest to *eg

P!t%agorean (%eorem

If a triangle is a right triangle, then the sum of the squares of the legs is equal to thesquare of the hypotenuse

a2+ 2= )2

Can be used to $ind an" missing part o$ a right triangle

+: )9 ;

+ : )44 9 ;

+ ; ),< 9 ;

; 9 )-

)

=


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40o'40 o Rig%t (riangles

In a 45o45oright triangle, the hypotenuse is equal to one of the legs times !2

o'3 o Rig%t (riangles

In a "#o$#oright triangle, the smaller leg is half the hypotenuse and the longer leg isequal to the smaller leg times !"

(rigonometr! ntrodu)tion

!n an" right triangle# there is alwa"s , characteristics o$ each side that can be expressed&

Sine 3pposite

="potenuse

Cosine d>acent

="potenuse

(angent 3pposite

d>acent

Cose)ant ="potenuse

3pposite

Se)ant ="potenuse

d>acent

Cotangent d>acent

3pposite

lso can be remembered b" 53=C=63

6ofind the side of any right triangle# >ust set up an equation using this $ormula?example using sine@&

%!p = leg52 ) = a52

-!p = 2a = a5


6/24

Sin6degree o angle7 = sine ratio o triangle = a)tual sine ratio8888ound in )%art or )al)ulator88

7 x

36o

5

#x sin3 = 9/x = $0:9 9 = $0:9x x = 12

6ofind the measure of an angle in any right triangle# >ust set up an equation using this$ormula ?example using sine@&

sin'16sin;x7 = sin'16sine ratio o triangle7

3 5

xo

4

#x sin'16sin;x7 = sin'16/07 x = sin'16/07 x = 9o

mportant (rig Ratios

sin A 7

cos A 7

tan )


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40o

o

sin Bcos A- 7

tan A- 7 -

3o

sin A- 7

cos B

tan A-

Angle o #le


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)2= a2+ 2'26a767)osC

*a o Sines

sed when "ou have AAS ?to $ind a missing side and then the rest o$ the triangle@ orSSA?to $ind the missing angle then the rest o$ the triangle@

sinA = sinB = sinC

a )

Angles o Rotation

3n a coordinate graph# angles are measured $rom two ra"s&

Initial Ray the beginning ra" that starts at -&** and is alwa"s on the xaxisTerminal Ray the ending ra" o$ "our angle# goes wherever needed

!$ ra"s have the same terminal ra"s# the" are said to be conterminal

6his means that a certain angle can be named in more than one wa"

Positive ?initial ra" to terminal ra" in countercloc%wise@

Negative?initial ra" to terminal ra" in cloc%wise@

>360?initial ra" to terminal ra" in countercloc%wise with multiple loops

around Dadd -,* $or each rotation @ !360?initial ra" to terminal ra" in cloc%wise with multiple loops aroundDsubtract -,* $or each rotation @

)+o

-+o


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(rig Fun)tions in rap%s

(hen dealing with graphs# "ou can $ind the trig $unctions o$ a point on a graphrenaming the same parts o$ trig $unctions with names corresponding to the point location

opposite 9x

ad>acent 9 ! r = 5x2+ !2

h"potenuse 9 r

6o $ind the trig values o$ a given angle# "ou can alwa"s $ind it b" $inding the referenceangleF the angle $ormed $rom the terminal ra" to the closest xaxis

6rig ratios o$ the re$erence angle are alwa"s going to be the same as the originalangle measurement

Ge$erence angle +*o )-*o

>uadrant (rig Values!n each quadrant o$ a graph# the ?sine# cosine# or tangent@ ratio ma" be positive ornegative. 6o tell# use this tric% to tell i$ the" are positive&

S A All Students (a%e Calculus

V ( C


10/24

Chapter 9: Cirles

Cir)le

ll points that are equidistant $rom a central point and coplanar

Radius

6he distance $rom the center to a point on the circle

segment whose endpoints are the center and a point on the circle

C%ord

segment whose endpoints are on the circle

"iameter


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chord that contains the center

*ine (!pes

"ecant Line line that goes through two points on a circle

Tangent Line line that goes through one point on a circle

'oint is calledpoint of tangency

Congruent Cir)les

Circles with equal radii

Con)entri) Cir)les

Circles with the same center

ns)ried


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(angent Corollar!

&angents to a circle from a point are congruent

AB = AC /

C

Ar)s

6he degree o$ the angle associated with the arc is the measure o$ that arc

%inor &rc an" arc smaller than )8*o

%a'or &rc an" arc larger than )8*o

"emicircle an arc whose measure is )8*o

Ar) (%eorems and Postulates

Ar) Addition Postulate

&he measure of the arc formed by two ad'acent arcs is the sum of the measuresof these two arcs

Congruent Ar) (%eorem

In the same circle or in congruent circles(

1) *ongruent arcs have congruent chords

2) *ongruent chords have congruent arcs

?C%ord'Center (%eorem

In the same circle or in congruent circles(

1) *hords equally distant from the center are congruent

2) *ongruent chords are equally distant from the center


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,ain (!pes o Angles in Cir)les

$entral &ngle angle whose vertex is the center

Inscri#ed &ngle angle whose vertex is on the circle

Interior "ecant &ngle angle whose vertex is inside the circle

(xterior "ecant &ngle angle whose vertex is outside the circle

ns)ried Angle (%eorems and Postulates

ns)ried Ar) (%eorem

&he measure of an inscribed angle is equal to half the measure of its intercepted

arc

b a m; = @ 6a7

Congruent ns)ried Angle Corollar!

If two inscribed angles intercept the same arc, then the angles are congruent

ns)ried >uadrilateral Corollar!

If a quadrilateral is inscribed in a circle, then its opposite angles aresupplementary

(angent ns)ried Angle (%eorem

&he measure of an angle formed by a chord and a tangent is equal to half themeasure of the intercepted arc

nterior Se)ant Angle (%eorem

&he measure of an angle formed by two chords that intersect inside a circle +interiorsecant) is equal to half the sum of the measures of the intercepted arcs +average)


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b c

a

m; = @ 6a+)7

#xterior Se)ant Angle (%eorem

&he measure of an angle formed by two chords that intersect outside a circle +eteriorsecant) is equal to half the difference of the larger arc and smaller arc

a b c m;a = @ 6)'7

C%ord *engt% (%eorems

nterse)ting C%ord (%eorem

-hen two chords intersect inside a circle, the product of the segments of onechord equals the product of the segments of another chord

a b

c d a6d7 = 6)7

#xterior Se)ant C%ord (%eorem

-hen two secant segments are drawn to a circle from an eternal point, theproduct of one secant segment and its eternal segment equals the product of theother secant segment and its eternal segment


15/24

/ 6a+7a = 6)+d7)

a

c d

Se)ant'(angent C%ord (%eorem

-hen a secant segment and a tangent segment are drawn to a circle form aneternal point, the product of a secant segment and its eternal segment is equalto the square of the tangent segment

a

b 6a+7 = )2

c

Chapter !!: Area

Area Postulates

&rea ")uare Postulate

&he area of a square is the square of the length of a side

$ongruent *igure Postulate

!$ two $igures are congruent# then the" have equal areas

Base -eig%t

+ase length o$ an" side o$ a parallelogram# the longest side o$ a triangle# or the twoparallel sides o$ a trapeHoid

,eight length o$ perpendicular segment $rom base to opposite side

Parallelogram Area (%eorem


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&he area of a parallelogram equals the product of its base and height

% A = ? %

(riangle Area (%eorem

&he area of a parallelogram equals the half of the product of the base and height

% A = @ 6 ? %7

R%omus Area (%eorem

&he area of a rhombus equals half of the product of the two diagonals

d1 A = @ 6d1? d27

d2

(rapeoid Area (%eorem

&he area of a trape.oid equals the average of the bases times the height

1

% A = @ 61+ 276%7

2


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Regular Pol!gon Area (%eorem

&he area of a regular polygon equals the number of sides times the radius +distance

from center to verte) times the sin+1#/n) times cos+1#/n)

r A = n ? r26sin1:/n76)os1:/n7

&he area of a regular polygon is equal to half of the product of the apothem

+perpendicular segment from center to side) and the perimeter

s

a A = @ 6a ? p7

Cir)le (%eorems

$ircle &rea Theorem

&he area of a circle is pi times the radius squared

r A = r2

$ircumference Theorem

&he circumference of a circle is the diameter times pi

d C = d


18/24

Length of &rc Theorem

&he length of an arc is the quotient of the measure of the arc by "$# times twotimes pi times the radius

x

leng%tARC = 6mARC/3762r7

&rea of "ector Theorem

&he area of a sector is the quotient of the measure of the arc by "$#times pi times the radius squared

x Ase)tor= 6mARC/376r27

Area Ratios o Similar Pol!gons

!$ pol"gons are similar# then all corresponding parts# lengths# etc are proportional

/ut# the areas are di$$erent&

A1/ A2= 6Corresponding Part1/Corresponding Part272

Proailit!

6he ratio o$ the target over the whole

Proailit! = Atarget/ A%ole


19/24

Chapter !2: Area " #olume o$ Regular %& Shapes

Sura)e Area

Lateral "urface &rea?*$S$A@ area o$ the sides o$ the three dimensional shape ?notbases@

Total "urface &rea ?($S$A@ area o$ all o$ the sides o$ a three dimensional shape?including bases@

2ormula is alwa"s *$S$A + Aases

Volume

6hreedimensional area o$ a shape ?measured in unit-@

Prism


20/24

6hreedimensional shape with congruent# parallel pol"gons $or bases

+ases congruent# parallel pol"gons

,eight altitude $rom one base to the other

6wo t"pes o$ prisms&

Right Prism all sides o$ prism are perpendicular

-#li)ue Prism sides are not perpendicular

L.".&. of Prism Theorem

&he lateral surface area of a prism equals the perimeter of the base times the

height

*$S$Aprism= 6Pase76%7

/olume of Prism Theorem

&he volume of a prism equals the area of the base times the height

Vprism= 6B76%7

P!ramid

6hreedimensional shape with a regular pol"gon base that $orms lateral $aces ?triangles@that meet at a vertex

+ase regular pol"gon $rom which the lateral $aces use each side as a base

Lateral face triangles going $rom the base o$ the p"ramid to the vertex

/ertex point that is perpendicular with the center o$ the base and in which alllateral $aces meet

,eight altitude $rom the center o$ the base to the vertex ?perpendicular@

"lant ,eight?l@ the height o$ a lateral $ace o$ the p"ramid


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L.".&. of Pyramid Theorem

&he lateral surface area of a pyramid equals half of the product of the perimetertimes the slant height

*$S$Ap!ramid= @ 6pl7

/olume of Pyramid Theorem

&he volume of a pyramid equals onethird of the area of the base times theheight

Vp!ramid= 1/ 6B76%7

C!linder

6hreedimensional shape with congruent# parallel# circle bases

L.".& of $ylinder Theorem

&he lateral surface area of a cylinder equals the product of the circumference ofthe base times the height

*$S$A)!linder= 6C76%7


22/24

/olume of $ylinder Theorem

&he volume of a cylinder equals the area of the base times the height

V)!linder= 6B76%7

Cone

6hreedimensional shape with a circular base whose lateral $aces meet at a vertex

"lant ,eight?l@ height $rom point on the circle to the vertex

/ertex point that is perpendicular with the center o$ the base and in which alllateral $aces meet

L.".& of $one Theorem

&he lateral surface area of a cone equals the radius of the base times the slant

height times pi*$S$A)one= rl

/olume of $one Theorem

&he volume of a cone equals onethird of the area of the base times the height

V)one= 1/ 6B76%7

Sp%ere

6hreedimensional circle


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L.".& of "phere

&he lateral surface area of a sphere equals four times pi times the radiussquared

*$S$Asp%ere= 4r2

/olume of "phere

&he volume of a sphere equals fourthirds the product of pi times the radiuscubed

Vsp%ere= 4/r

Chapter !%: Coordinate Geometry

"istan)e Formula

1eals with the distance between two points on a coordinate graph

6hree di$$erent $ormulas that amount to the same distance $ormula

Pythagorean Theorem

a2+ 2= )2

()uation of a $ircle

r

2

= 6x'a7

2

+ 6!'7

2

istance *ormula

d = 56!2'!172+ 6x2'x172


24/24

,idpoint Formula

6o $ind the midpoint between two given points# $ind the average o$ the x coordinates toreceive "our midpoint x# and the average o$ the " coordinates to get "ou midpoint "coordinates

, = 6 Dx1+x2E/2 D!1+!2E/2 7

Slope

6o $ind the slope o$ a line# use the change in " coordinates over the change in xcoordinates

slope 6m7= G

H

#Iuation o a *ine

6he equation o$ a line gives "ou the abilit" to determine an" point that is on that line

!t can be set up in a $ew di$$erent $ormats&

"tandard *orm& 0 y = b"lope!Intercept *orm& m+) 0 b = y

6o ma%e sure "ou set it up correctl" ever" time# use the I2in%e 2ormJ

G ! J

H x J a

Parallel and Perpendi)ular *ines

Parallel Lines parallel lines have congruent slopes

Perpendicular Lines perpendicular lines have opposite reciprocal slopes

9

ac geometry notes semester 2

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