Line The set of points extending in two directions without end uniquely determined by two points. The set of points on a line between two points

Size: px
Start display at page:

Download "Line The set of points extending in two directions without end uniquely determined by two points. The set of points on a line between two points"

Transcription

1 Lines Line Line segment Perpendiulr Lines Prllel Lines Opposite Angles The set of points extending in two diretions without end uniquely determined by two points. The set of points on line between two points Two lines tht interset t 90 ngle. Two lines tht do not interset or two lines tht re n equl distne prt. The ngles opposite eh other in two interseting lines re equl. Angles = 90 Right < 90 Aute ngle. A tringle n hve three ute ngles. ( ) > 90 Obtuse ngle. A tringle n hve only one obtuse ngle + b = 90 Complementry ngles All the ngles in tringle dd to 180 nd ll the ngles formed on stright line ngle dd to 180. If point is drwn on line nd multiple line segments re drwn from one side of the line in + b + = 180 plne, the sum of ll the ngles formed by these new lines nd the originl line must sum to 180. These ngles re lled supplementry ngles. If only two ngles re under onsidertion nd + b = 180, then + b = 180 the two ngles re lled reiprol ngles beuse they re mirror imges or refletions of one nother. Tringles Slene Isoseles Equilterl All sides different lengths Sum of ll ngles = 180 Two sides the sme length Angles opposite equl sides re equl All sides the sme length All ngles equl 60 Longest side is opposite lrgest ngle Shortest side is opposite smllest ngle sum of shorter sides: < + b greter thn longest side next longest side > mx(+b) mx ngle min ngle longest side < shortest side > shortest side > Aute: < + b All ngles re less thn 90 Obtuse: > + b One ngle is greter thn 90 Right: = + b One ngle = 90 Side lengths governed by Pythgoren Theorem: = + b Hypotenuse The side opposite the right ngle (90 ) Perimeter Are +b+ bh/ Geometry.do, 5/16/04, 6:19 PM 1 of 10

2 Are right Are right isoseles Congruent Similr Angle bisetors Inenter Side bisetors Cirumenter Medins Centriod b/ (one hlf the imginry retngle / (one hlf the imginry squre) = /4 Two tringles re ongruent if ll orresponding ngles nd sides re equl. Two tringles re similr if ngles re ongruent nd sides re proportionl. Tringles re similr when every two pirs of ngles re ongruent. The bisetors of ngles of tringle interset t point tht is equidistnt from sides of the tringle. The perpendiulr bisetors of the sides of tringle interset t point tht is equidistnt from the three verties of the tringle. The medins of tringle interset t point tht is /3 of the distne from eh vertex to the midpoint of the opposite side. The medin is line drwn from the midpoint of one side to the opposite vertex. If the length of this line is x, then the point were the other medin lines interset this line is x/3 distne from the line nd x/3 distne from the vertex. Geometry.do, 5/16/04, 6:19 PM of 10

3 Right Tringles nd the Pythgoren Theorem The most importnt property of right tringles is tht they stisfy the Pythgoren Theorem. The Pythgoren Theorem (PT) sttes tht if the two sides of right tringle re nd b, nd the hypotenuse is, then the following reltionship is true: = + b A set of ommon right tringles is illustrted below: = = =. 36 Two of the bove tringles re speil beuse they hve speil ngles: Rtio of Sides Angles 1 :1: : : 1 : 3 : : 3 : The Pythgoren Theorem is esily proved by reting four identil right tringles of ny size nd rrnging them s illustrted below: b x b y b x y b Geometry.do, 5/16/04, 6:19 PM 3 of 10

4 We know tht the insribed retngle is squre beuse ll four sides re equl nd ll orners re 90. We know tht the orners re 90 beuse the ngles x + y = 90. The re of the insribed squre is equl to the re of the lrge squre minus the re of the four right tringles. The re of the lrge squre is (+b) nd the re of the four tringles is 4b/ = b. This gives whih is the Pythgoren Theorem. = (+b) b = + b + b b = + b Pythgoren Triples A Pythgoren triple (, b, ) is set of three numbers whih stisfy the PT. Note tht if (, b, ) is triple then (k, kb, k) is lso triple. The most ommon triple is the (3, 4, 5) triple nd its ssoited integer multiples (k3, k4, k5). These re summrized below: n Multiples of (3, 4, 5) 1 (3, 4, 5) (6, 8, 10) 3 (9,1,15) 4 (1, 16, 0) 5 (15, 0, 5) 6 (18, 4, 30) The Greek philosopher Plto is sid to hve developed the following formul for generting triples: (n, n -1, n +1) n (n, n -1, n +1) (4, 3, 5) 3 (6, 8, 10) 4 (8, 15, 17) 5 (10, 4, 6) 6 (1, 35, 37) A more generl formul for generting triples for m>n is (mn, m n, m + n ) These triples were generted in EXCEL nd re in the tble on the following pge. The tble immeditely below is summry of triples rrnged in numeril order by the shortest leg. Triples (3, 4, 5) (1, 35, 37) (5, 1, 13) (13, 84, 85) Geometry.do, 5/16/04, 6:19 PM 4 of 10

5 (6, 8, 10) (14, 48, 50) (7, 4, 5) (15, 0, 5) (8, 15, 17) (15, 11, 113) (9,1,15) (16, 30, 34) (9, 40, 41) (16, 63, 65) (10, 4, 6) (17, 144, 145) (11, 60, 61) (18, 4, 30) (1, 16, 0) Triples (mn, m n, m + n ), m>n m/n Another triple formul is (n+1, n +n, n +n+1). The three triple formuls re summrized below. 3n 4n 5n n+1 n +n n +n+1 mn m n m + n Geometry.do, 5/16/04, 6:19 PM 5 of 10

6 Applitions Missing Lengths The most ommon pplition of the Pythgoren Theorem (PT) is to find n unknown side of tringle given two of the three omponents:, b, or. In ddition, the PT n be pplied twie to find the digonl of solid retngle. When solving length problems lwys look for known triples so you do not hve to do the rithmeti. Consider the following isoseles right tringle where the opposite ngles must equl 45. h 1 h h 3 h 4 Wht is the vlue of h 4 in terms of? You ould solve for eh h, but quik inspetion revels tht if i is even, then h i =, but using the Pythgoren Theorem, the formul i n be shown to be true for ll i, both even nd odd. Another interesting exmple of finding n unknown sides follows: It is redily seen tht the unlbeled sides re equl to, 3, 4, 5, Ares For right tringles, the re n be Trigonometry The Pythgoren Theorem is useful for proving the trigonometri identity Geometry.do, 5/16/04, 6:19 PM 6 of 10

7 sin A + os A = 1 Consider the following tringle A b sin A = b/ nd os A = /. sin A = b / nd os A = / Adding these two expressions gives + b sin A + os A = = 1 Anlyti Geometry The Pythgoren Theorem is used to ompute the distne between two points (x 1,y 1 ) nd (x,y ) in the (x,y) plne. (x,y ) (x 1,y 1 ) The distne between the two points is given by Dist = ( x x ) + ( y ) 1 y1 Crpentry The Pythgoren Theorem is used in rpentry to ly out squre orner in room. The ommon triple to use would be (,b,) = (1,16,0). Twelve feet would be mesured from orner point. Sixteen feet would be mesured in nother diretion. The two end points would be djusted until the hypotenuse ws extly 0 feet. At this point, you ould be ertin the orner ws extly squre. Of ourse, it n lso be used to test for squreness of wll lredy built. Geometry.do, 5/16/04, 6:19 PM 7 of 10

8 Polygons A polygon is plne figure enlosed by three or more line segments. The number of line segments determines the nme of the polygon. The ngles formed where the sides interset re lled verties. Verties must be less thn or equl to 180. Polygons where ll sides re equl re lled regulr polygons or n-gons. Regulr polygons stisfy the following properties: Vertex Perimeter ns Are ½ r (ns) Vertex Angle The interior ngle of polygon. The perimeter of regulr polygon is ns, where n is the number of sides nd s is the length of side. The re of regulr polygon is given s ½ r (ns), where r is the perpendiulr distne from side to the enter of the polygon. Notie tht s the number of sides of polygon gets very lrge, the polygon pprohes irle. The perimeter ns beomes πr nd the re beomes ½ r πr = πr. ( n ) = 180, whih implies tht s the number of sides n n n gets very lrge, the vertex ngle equls 180, nd the polygon pprohes irle. 180 is the number of degrees on line tngent to the irle. Nmed Polygons (ngles if regulr) Sides n Shpe Angles Sides n Shpe Angles 3 Equilterl 60 Otgon Tringle 4 Squre 90 9 Nongon Pentgon Degon Hexgon Undegon 147 3/11 7 Heptgon 18 4/7 1 Dodegon 150 Qudrilterls Qudrilterl Chrteristis Perimeter Are Qudrilterl Any four sided polygon +b++d no formul Trpezoid Two sides prllel +b++d ½(+b)h Prllelogrm Opposite sides prllel (+b) bh Opposite pirs of equl length Rhombus Opposite sides prllel 4s s All sides of equl length A prllelogrm with ll sides equl Retngle Opposite sides equl length (+b) b All ngles = 90 Squre All sides of equl length All ngles 90 4s s Geometry.do, 5/16/04, 6:19 PM 8 of 10

9 Cirles Cirle A set of points in plne ll equl distne from fixed point lled the enter of the irle. Chord Any interior line with end points on the perimeter of the irle. Dimeter A hord pssing through the enter. Twie the rdius Rdius The distne from the enter to the perimeter of the irle. Equl to one-hlf the dimeter. d = r Dimeter = rdius Pi is equl to the irumferene of irle divided by its dimeter. It is pproximtely equl to /7 = Pi to five ples of ury is π Pi long with is one of the most ommon irrtionl numbers. Cirumferene The perimeter of the irle. The distne round the irle. Equl to πr πr. Ar Any pth long the irumferene or perimeter of the irle. The r is θ πr 360 sid to subtend the ngle formed by onneting the end points of the r to the enter of the irle. Think of r s frtion of the irumferene of irle. Insribed The mesure of n insribed ngle is equl to hlf the mesure of its interepted r or the r is twie the insribed ngle. Rdins The number of degrees subtended by n r of length r (the rdius) is θ r = πr 360 lled one rdin. Notie tht if S is the entire irle, then π θ = 360, where θ is expressed in rdins, whih gives 1 rdin = 57.3 or 1 = rdins. It is often onvenient to express ngles in units of 360 rdins to simplify the rithmeti. θ = π Are = πr θ The re of setor is πr, where θ is the ngle of the setor. 360 Tngent Point The single point in ommon with the perimeter of irle nd line or the ommon point between two irles whose outer edges touh t single point. A line tngent to irle is perpendiulr to the dimeter nd the rdius. Insribed A polygon is insribed in irle if the vertex ngles touh the Polygon perimeter of the irle nd re ompletely inside the irle. Cirumsribed A polygon is irumsribed by irle if the sides of the polygon re Polygon tngent to the perimeter of the irle, nd the vertex ngles re outside the perimeter of the irle. Geometry.do, 5/16/04, 6:19 PM 9 of 10

10 Solids Retngulr Solid Cylinder Sphere A three dimensionl figure formed by six retngulr surfes with opposite sides being prllel nd equl in dimension. Eh surfe is lled fe. Eh orner is lled vertex. Surfe re = sum of re of ll surfes Volume = length width height = re of bse height A three dimensionl objet formed by two prllel irles where the line onneting the two irle enters is perpendiulr to the plnes of the irles. The length of this line is lled the ltitude or height (h) of the ylinder. Surfe re = (πr ) + πrh = two irles + urved side Volume = πr h = re of irle height 4 3 Volume = π r 3 Coni Setions Cone Coni Setion Cirle Ellipse Prbol Hyperbol A one is formed by drwing irle in plne nd seleting some point V diretly bove the enter of the irle so tht line from the point is perpendiulr to the plne of the irle. Then, one is formed by the set of ll points in three dimensions from the vertex to the perimeter of the irle. The point V is lled the vertex of the one. The line from the vertex to the enter of the irle is lled the xis of the one. The set of urved points formed when plne intersets one. The set of points formed by plne prllel to the originl plne used to onstrut the one. The set of points formed by plne interseting both sides of one. The set of points formed by plne interseting the one whih is prllel to one side of the one. The set of points formed by pln interseting the one whih is prllel to the xis of the one. Geometry.do, 5/16/04, 6:19 PM 10 of 10

a c = A C AD DB = BD

a c = A C AD DB = BD 1.) SIMILR TRINGLES.) Some possile proportions: Geometry Review- M.. Sntilli = = = = =.) For right tringle ut y its ltitude = = =.) Or for ll possiilities, split into 3 similr tringles: ll orresponding

More information

Final Exam Review F 06 M 236 Be sure to look over all of your tests, as well as over the activities you did in the activity book

Final Exam Review F 06 M 236 Be sure to look over all of your tests, as well as over the activities you did in the activity book inl xm Review 06 M 236 e sure to loo over ll of your tests, s well s over the tivities you did in the tivity oo 1 1. ind the mesures of the numered ngles nd justify your wor. Line j is prllel to line.

More information

UNCORRECTED SAMPLE PAGES. Angle relationships and properties of 6geometrical figures 1. Online resources. What you will learn

UNCORRECTED SAMPLE PAGES. Angle relationships and properties of 6geometrical figures 1. Online resources. What you will learn Online resoures uto-mrked hpter pre-test Video demonstrtions of ll worked exmples Intertive widgets Intertive wlkthroughs Downlodle HOTsheets ess to ll HOTmths ustrlin urriulum ourses ess to the HOTmths

More information

Review Packet #3 Notes

Review Packet #3 Notes SCIE 40, Fll 05 Miller Review Pket # Notes Prllel Lines If two prllel lines re onneted y third line (lled the trnsversl), the resulting ngles re either ongruent or supplementry. Angle pirs re nmed s follows:

More information

Review Packet #3 Notes

Review Packet #3 Notes SCIE 40, Spring 0 Miller Review Pket # Notes Mpping Nottion We use mpping nottion to note how oordinte hnges. For exmple, if the point ( ) trnsformed under mpping nottion of ( x, y) ( x, y), then it eomes

More information

Measurement and geometry

Measurement and geometry Mesurement nd geometry 4 Geometry Geometry is everywhere. Angles, prllel lines, tringles nd qudrilterls n e found ll round us, in our homes, on trnsport, in onstrution, rt nd nture. This sene from Munih

More information

Geometry/Trig 2 Unit 3 Review Packet Answer Key

Geometry/Trig 2 Unit 3 Review Packet Answer Key Unit 3 Review Pcket nswer Key Section I Nme the five wys to prove tht prllel lines exist. 1. If two lines re cut y trnsversl nd corresponding ngles re congruent, then the lines re prllel.. If two lines

More information

MATHS LECTURE # 09. Plane Geometry. Angles

MATHS LECTURE # 09. Plane Geometry. Angles Mthemtics is not specttor sport! Strt prcticing. MTHS LTUR # 09 lne eometry oint, line nd plne There re three sic concepts in geometry. These concepts re the point, line nd plne. oint fine dot, mde y shrp

More information

Chapter 2. Chapter 2 5. Section segments: AB, BC, BD, BE. 32. N 53 E GEOMETRY INVESTIGATION Answers will vary. 34. (a) N. sunset.

Chapter 2. Chapter 2 5. Section segments: AB, BC, BD, BE. 32. N 53 E GEOMETRY INVESTIGATION Answers will vary. 34. (a) N. sunset. Chpter 2 5 Chpter 2 32. N 53 E GEOMETRY INVESTIGATION Answers will vry. 34. () N Setion 2.1 2. 4 segments: AB, BC, BD, BE sunset sunrise 4. 2 rys: CD (or CE ), CB (or CA ) 6. ED, EC, EB W Oslo, Norwy E

More information

9.1 PYTHAGOREAN THEOREM (right triangles)

9.1 PYTHAGOREAN THEOREM (right triangles) Simplifying Rdicls: ) 1 b) 60 c) 11 d) 3 e) 7 Solve: ) x 4 9 b) 16 80 c) 9 16 9.1 PYTHAGOREAN THEOREM (right tringles) c If tringle is right tringle then b, b re the legs * c is clled the hypotenuse (side

More information

TRIANGLE. The sides of a triangle (any type of triangle) are proportional to the sines of the angle opposite to them in triangle.

TRIANGLE. The sides of a triangle (any type of triangle) are proportional to the sines of the angle opposite to them in triangle. 19. SOLUTIONS OF TRINGLE 1. INTRODUTION In ny tringle, the side, opposite to the ngle is denoted by ; the side nd, opposite to the ngles nd respetively re denoted by b nd respetively. The semi-perimeter

More information

Angle Properties in Polygons. Part 1 Interior Angles

Angle Properties in Polygons. Part 1 Interior Angles 2.4 Angle Properties in Polygons YOU WILL NEED dynmic geometry softwre OR protrctor nd ruler EXPLORE A pentgon hs three right ngles nd four sides of equl length, s shown. Wht is the sum of the mesures

More information

Doubts about how to use azimuth values from a Coordinate Object. Juan Antonio Breña Moral

Doubts about how to use azimuth values from a Coordinate Object. Juan Antonio Breña Moral Douts out how to use zimuth vlues from Coordinte Ojet Jun Antonio Breñ Morl # Definition An Azimuth is the ngle from referene vetor in referene plne to seond vetor in the sme plne, pointing towrd, (ut

More information

50 AMC LECTURES Lecture 2 Analytic Geometry Distance and Lines. can be calculated by the following formula:

50 AMC LECTURES Lecture 2 Analytic Geometry Distance and Lines. can be calculated by the following formula: 5 AMC LECTURES Lecture Anlytic Geometry Distnce nd Lines BASIC KNOWLEDGE. Distnce formul The distnce (d) between two points P ( x, y) nd P ( x, y) cn be clculted by the following formul: d ( x y () x )

More information

Pythagoras theorem and trigonometry (2)

Pythagoras theorem and trigonometry (2) HPTR 10 Pythgors theorem nd trigonometry (2) 31 HPTR Liner equtions In hpter 19, Pythgors theorem nd trigonometry were used to find the lengths of sides nd the sizes of ngles in right-ngled tringles. These

More information

Geometrical reasoning 1

Geometrical reasoning 1 MODULE 5 Geometril resoning 1 OBJECTIVES This module is for study y n individul teher or group of tehers. It: looks t pprohes to developing pupils visulistion nd geometril resoning skills; onsiders progression

More information

4-1 NAME DATE PERIOD. Study Guide. Parallel Lines and Planes P Q, O Q. Sample answers: A J, A F, and D E

4-1 NAME DATE PERIOD. Study Guide. Parallel Lines and Planes P Q, O Q. Sample answers: A J, A F, and D E 4-1 NAME DATE PERIOD Pges 142 147 Prllel Lines nd Plnes When plnes do not intersect, they re sid to e prllel. Also, when lines in the sme plne do not intersect, they re prllel. But when lines re not in

More information

Fig.1. Let a source of monochromatic light be incident on a slit of finite width a, as shown in Fig. 1.

Fig.1. Let a source of monochromatic light be incident on a slit of finite width a, as shown in Fig. 1. Answer on Question #5692, Physics, Optics Stte slient fetures of single slit Frunhofer diffrction pttern. The slit is verticl nd illuminted by point source. Also, obtin n expression for intensity distribution

More information

PROBLEM OF APOLLONIUS

PROBLEM OF APOLLONIUS PROBLEM OF APOLLONIUS In the Jnury 010 issue of Amerin Sientist D. Mkenzie isusses the Apollonin Gsket whih involves fining the rius of the lrgest irle whih just fits into the spe etween three tngent irles

More information

Section 10.4 Hyperbolas

Section 10.4 Hyperbolas 66 Section 10.4 Hyperbols Objective : Definition of hyperbol & hyperbols centered t (0, 0). The third type of conic we will study is the hyperbol. It is defined in the sme mnner tht we defined the prbol

More information

Chapter44. Polygons and solids. Contents: A Polygons B Triangles C Quadrilaterals D Solids E Constructing solids

Chapter44. Polygons and solids. Contents: A Polygons B Triangles C Quadrilaterals D Solids E Constructing solids Chpter44 Polygons nd solids Contents: A Polygons B Tringles C Qudrilterls D Solids E Constructing solids 74 POLYGONS AND SOLIDS (Chpter 4) Opening prolem Things to think out: c Wht different shpes cn you

More information

Can Pythagoras Swim?

Can Pythagoras Swim? Overview Ativity ID: 8939 Mth Conepts Mterils Students will investigte reltionships etween sides of right tringles to understnd the Pythgoren theorem nd then use it to solve prolems. Students will simplify

More information

6.3 Volumes. Just as area is always positive, so is volume and our attitudes towards finding it.

6.3 Volumes. Just as area is always positive, so is volume and our attitudes towards finding it. 6.3 Volumes Just s re is lwys positive, so is volume nd our ttitudes towrds finding it. Let s review how to find the volume of regulr geometric prism, tht is, 3-dimensionl oject with two regulr fces seprted

More information

3 4. Answers may vary. Sample: Reteaching Vertical s are.

3 4. Answers may vary. Sample: Reteaching Vertical s are. Chpter 7 Answers Alterntive Activities 7-2 1 2. Check students work. 3. The imge hs length tht is 2 3 tht of the originl segment nd is prllel to the originl segment. 4. The segments pss through the endpoints

More information

5 ANGLES AND POLYGONS

5 ANGLES AND POLYGONS 5 GLES POLYGOS urling rige looks like onventionl rige when it is extene. However, it urls up to form n otgon to llow ots through. This Rolling rige is in Pington sin in Lonon, n urls up every Friy t miy.

More information

B. Definition: The volume of a solid of known integrable cross-section area A(x) from x = a

B. Definition: The volume of a solid of known integrable cross-section area A(x) from x = a Mth 176 Clculus Sec. 6.: Volume I. Volume By Slicing A. Introduction We will e trying to find the volume of solid shped using the sum of cross section res times width. We will e driving towrd developing

More information

Angle properties of lines and polygons

Angle properties of lines and polygons chievement Stndrd 91031 pply geometric resoning in solving problems Copy correctly Up to 3% of workbook Copying or scnning from ES workbooks is subject to the NZ Copyright ct which limits copying to 3%

More information

POLYGON NAME UNIT # ASSIGN # 2.) STATE WHETHER THE POLYGON IS EQUILATERAL, REGULAR OR EQUIANGULAR

POLYGON NAME UNIT # ASSIGN # 2.) STATE WHETHER THE POLYGON IS EQUILATERAL, REGULAR OR EQUIANGULAR POLYGONS POLYGON CLOSED plane figure that is formed by three or more segments called sides. 2.) STTE WHETHER THE POLYGON IS EQUILTERL, REGULR OR EQUINGULR a.) b.) c.) VERTEXThe endpoint of each side of

More information

Calculus Differentiation

Calculus Differentiation //007 Clulus Differentition Jeffrey Seguritn person in rowot miles from the nerest point on strit shoreline wishes to reh house 6 miles frther down the shore. The person n row t rte of mi/hr nd wlk t rte

More information

ANALYTICAL GEOMETRY. The curves obtained by slicing the cone with a plane not passing through the vertex are called conics.

ANALYTICAL GEOMETRY. The curves obtained by slicing the cone with a plane not passing through the vertex are called conics. ANALYTICAL GEOMETRY Definition of Conic: The curves obtined by slicing the cone with plne not pssing through the vertex re clled conics. A Conic is the locus directrix of point which moves in plne, so

More information

Stained Glass Design. Teaching Goals:

Stained Glass Design. Teaching Goals: Stined Glss Design Time required 45-90 minutes Teching Gols: 1. Students pply grphic methods to design vrious shpes on the plne.. Students pply geometric trnsformtions of grphs of functions in order to

More information

Answer Key Lesson 6: Workshop: Angles and Lines

Answer Key Lesson 6: Workshop: Angles and Lines nswer Key esson 6: tudent Guide ngles nd ines Questions 1 3 (G p. 406) 1. 120 ; 360 2. hey re the sme. 3. 360 Here re four different ptterns tht re used to mke quilts. Work with your group. se your Power

More information

Grade 7/8 Math Circles Geometric Arithmetic October 31, 2012

Grade 7/8 Math Circles Geometric Arithmetic October 31, 2012 Fculty of Mthemtics Wterloo, Ontrio N2L 3G1 Grde 7/8 Mth Circles Geometric Arithmetic Octoer 31, 2012 Centre for Eduction in Mthemtics nd Computing Ancient Greece hs given irth to some of the most importnt

More information

Class-XI Mathematics Conic Sections Chapter-11 Chapter Notes Key Concepts

Class-XI Mathematics Conic Sections Chapter-11 Chapter Notes Key Concepts Clss-XI Mthemtics Conic Sections Chpter-11 Chpter Notes Key Concepts 1. Let be fixed verticl line nd m be nother line intersecting it t fixed point V nd inclined to it t nd ngle On rotting the line m round

More information

N-Level Math (4045) Formula List. *Formulas highlighted in yellow are found in the formula list of the exam paper. 1km 2 =1000m 1000m

N-Level Math (4045) Formula List. *Formulas highlighted in yellow are found in the formula list of the exam paper. 1km 2 =1000m 1000m *Formul highlighted in yellow re found in the formul lit of the em pper. Unit Converion Are m =cm cm km =m m = m = cm Volume m =cm cm cm 6 = cm km/h m/ itre =cm (ince mg=cm ) 6 Finncil Mth Percentge Incree

More information

CS 241 Week 4 Tutorial Solutions

CS 241 Week 4 Tutorial Solutions CS 4 Week 4 Tutoril Solutions Writing n Assemler, Prt & Regulr Lnguges Prt Winter 8 Assemling instrutions utomtilly. slt $d, $s, $t. Solution: $d, $s, nd $t ll fit in -it signed integers sine they re 5-it

More information

6.2 Volumes of Revolution: The Disk Method

6.2 Volumes of Revolution: The Disk Method mth ppliction: volumes by disks: volume prt ii 6 6 Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem 6) nd the ccumultion process is to determine so-clled volumes

More information

10.2 Graph Terminology and Special Types of Graphs

10.2 Graph Terminology and Special Types of Graphs 10.2 Grph Terminology n Speil Types of Grphs Definition 1. Two verties u n v in n unirete grph G re lle jent (or neighors) in G iff u n v re enpoints of n ege e of G. Suh n ege e is lle inient with the

More information

Lesson 4.4. Euler Circuits and Paths. Explore This

Lesson 4.4. Euler Circuits and Paths. Explore This Lesson 4.4 Euler Ciruits nd Pths Now tht you re fmilir with some of the onepts of grphs nd the wy grphs onvey onnetions nd reltionships, it s time to egin exploring how they n e used to model mny different

More information

Triangles. Learning Objectives. Pre-Activity

Triangles. Learning Objectives. Pre-Activity Setion 3.2 Pre-tivity Preparation Triangles Geena needs to make sure that the dek she is building is perfetly square to the brae holding the dek in plae. How an she use geometry to ensure that the boards

More information

Ray surface intersections

Ray surface intersections Ry surfce intersections Some primitives Finite primitives: polygons spheres, cylinders, cones prts of generl qudrics Infinite primitives: plnes infinite cylinders nd cones generl qudrics A finite primitive

More information

Right Angled Trigonometry. Objective: To know and be able to use trigonometric ratios in rightangled

Right Angled Trigonometry. Objective: To know and be able to use trigonometric ratios in rightangled C2 Right Angled Trigonometry Ojetive: To know nd e le to use trigonometri rtios in rightngled tringles opposite C Definition Trigonometry ws developed s method of mesuring ngles without ngulr units suh

More information

Mathematics Background

Mathematics Background For more roust techer experience, plese visit Techer Plce t mthdshord.com/cmp3 Mthemtics Bckground Extending Understnding of Two-Dimensionl Geometry In Grde 6, re nd perimeter were introduced to develop

More information

Thirty-fourth Annual Columbus State Invitational Mathematics Tournament. Instructions

Thirty-fourth Annual Columbus State Invitational Mathematics Tournament. Instructions Thirty-fourth Annul Columbus Stte Invittionl Mthemtics Tournment Sponsored by Columbus Stte University Deprtment of Mthemtics Februry, 008 ************************* The Mthemtics Deprtment t Columbus Stte

More information

a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1

a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1 Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the

More information

If f(x, y) is a surface that lies above r(t), we can think about the area between the surface and the curve.

If f(x, y) is a surface that lies above r(t), we can think about the area between the surface and the curve. Line Integrls The ide of line integrl is very similr to tht of single integrls. If the function f(x) is bove the x-xis on the intervl [, b], then the integrl of f(x) over [, b] is the re under f over the

More information

Math 4 Review for Quarter 2 Cumulative Test

Math 4 Review for Quarter 2 Cumulative Test Mth 4 Review for Qurter 2 Cumultive Test Nme: I. Right Tringle Trigonometry (3.1-3.3) Key Fcts Pythgoren Theorem - In right tringle, 2 + b 2 = c 2 where c is the hypotenuse s shown below. c b Trigonometric

More information

9.1 apply the distance and midpoint formulas

9.1 apply the distance and midpoint formulas 9.1 pply the distnce nd midpoint formuls DISTANCE FORMULA MIDPOINT FORMULA To find the midpoint between two points x, y nd x y 1 1,, we Exmple 1: Find the distnce between the two points. Then, find the

More information

MA1008. Calculus and Linear Algebra for Engineers. Course Notes for Section B. Stephen Wills. Department of Mathematics. University College Cork

MA1008. Calculus and Linear Algebra for Engineers. Course Notes for Section B. Stephen Wills. Department of Mathematics. University College Cork MA1008 Clculus nd Liner Algebr for Engineers Course Notes for Section B Stephen Wills Deprtment of Mthemtics University College Cork s.wills@ucc.ie http://euclid.ucc.ie/pges/stff/wills/teching/m1008/ma1008.html

More information

MTH 146 Conics Supplement

MTH 146 Conics Supplement 105- Review of Conics MTH 146 Conics Supplement In this section we review conics If ou ne more detils thn re present in the notes, r through section 105 of the ook Definition: A prol is the set of points

More information

Order these angles from smallest to largest by wri ng 1 to 4 under each one. Put a check next to the right angle.

Order these angles from smallest to largest by wri ng 1 to 4 under each one. Put a check next to the right angle. Lines nd ngles Connect ech set of lines to the correct nme: prllel perpendiculr Order these ngles from smllest to lrgest y wri ng to 4 under ech one. Put check next to the right ngle. Complete this tle

More information

Moments and products of inertia and radii of gyration about central axes. I x ¼ I y ¼ Ix 0 ¼ a4. r x ¼ r y ¼ r 0 x ¼ 0:2887a.

Moments and products of inertia and radii of gyration about central axes. I x ¼ I y ¼ Ix 0 ¼ a4. r x ¼ r y ¼ r 0 x ¼ 0:2887a. TBLE.1 Properties of sections NOTTION: ¼ re ðlengthþ 2 ; y ¼ distnce to extreme fiber (length); I ¼ moment of inerti ðlength Þ; r ¼ rdius of gyrtion (length); Z ¼ plstic section modulus ðlength 3 Þ;SF¼

More information

Adjacency. Adjacency Two vertices u and v are adjacent if there is an edge connecting them. This is sometimes written as u v.

Adjacency. Adjacency Two vertices u and v are adjacent if there is an edge connecting them. This is sometimes written as u v. Terminology Adjeny Adjeny Two verties u nd v re djent if there is n edge onneting them. This is sometimes written s u v. v v is djent to nd ut not to. 2 / 27 Neighourhood Neighourhood The open neighourhood

More information

MENSURATION-IV

MENSURATION-IV MENSURATION-IV Theory: A solid is figure bounded by one or more surfce. Hence solid hs length, bredth nd height. The plne surfces tht bind solid re clled its fces. The fundmentl difference between plne

More information

Solids. Solids. Curriculum Ready.

Solids. Solids. Curriculum Ready. Curriulum Rey www.mthletis.om This ooklet is ll out ientifying, rwing n mesuring solis n prisms. SOM CUES The Som Cue ws invente y Dnish sientist who went y the nme of Piet Hein. It is simple 3 # 3 #

More information

Area & Volume. Chapter 6.1 & 6.2 September 25, y = 1! x 2. Back to Area:

Area & Volume. Chapter 6.1 & 6.2 September 25, y = 1! x 2. Back to Area: Bck to Are: Are & Volume Chpter 6. & 6. Septemer 5, 6 We cn clculte the re etween the x-xis nd continuous function f on the intervl [,] using the definite integrl:! f x = lim$ f x * i )%x n i= Where fx

More information

Introduction to Algebra

Introduction to Algebra INTRODUCTORY ALGEBRA Mini-Leture 1.1 Introdution to Alger Evlute lgeri expressions y sustitution. Trnslte phrses to lgeri expressions. 1. Evlute the expressions when =, =, nd = 6. ) d) 5 10. Trnslte eh

More information

Paradigm 5. Data Structure. Suffix trees. What is a suffix tree? Suffix tree. Simple applications. Simple applications. Algorithms

Paradigm 5. Data Structure. Suffix trees. What is a suffix tree? Suffix tree. Simple applications. Simple applications. Algorithms Prdigm. Dt Struture Known exmples: link tble, hep, Our leture: suffix tree Will involve mortize method tht will be stressed shortly in this ourse Suffix trees Wht is suffix tree? Simple pplitions History

More information

Geo 9 Ch 11 1 AREAS OF POLYGONS SQUARE EQUILATERAL TRIANGLE

Geo 9 Ch 11 1 AREAS OF POLYGONS SQUARE EQUILATERAL TRIANGLE Geo 9 h 11 1 RES OF POLYGONS SQURE RETNGLE PRLLELOGRM TRINGLE EQUILTERL TRINGLE RHOMUS TRPEZOI REGULR POLY IRLE R LENGTH SETOR SLIVER RTIO OF RES SME SE SME HEIGHT Geo 9 h 11 2 11.1 reas of Polygons Postulate

More information

Here is an example where angles with a common arm and vertex overlap. Name all the obtuse angles adjacent to

Here is an example where angles with a common arm and vertex overlap. Name all the obtuse angles adjacent to djcent tht do not overlp shre n rm from the sme vertex point re clled djcent ngles. me the djcent cute ngles in this digrm rm is shred y + + me vertex point for + + + is djcent to + djcent simply mens

More information

45 Wyner Math Academy I Spring 2016

45 Wyner Math Academy I Spring 2016 45 Wyner Math cademy I Spring 2016 HPTER FIVE: TRINGLES Review January 13 Test January 21 Other than circles, triangles are the most fundamental shape. Many aspects of advanced abstract mathematics and

More information

Tight triangulations: a link between combinatorics and topology

Tight triangulations: a link between combinatorics and topology Tight tringultions: link between ombintoris nd topology Jonthn Spreer Melbourne, August 15, 2016 Topologil mnifolds (Geometri) Topology is study of mnifolds (surfes) up to ontinuous deformtion Complited

More information

Width and Bounding Box of Imprecise Points

Width and Bounding Box of Imprecise Points Width nd Bounding Box of Impreise Points Vhideh Keikh Mrten Löffler Ali Mohdes Zhed Rhmti Astrt In this pper we study the following prolem: we re given set L = {l 1,..., l n } of prllel line segments,

More information

Naming 3D objects. 1 Name the 3D objects labelled in these models. Use the word bank to help you.

Naming 3D objects. 1 Name the 3D objects labelled in these models. Use the word bank to help you. Nming 3D ojects 1 Nme the 3D ojects lelled in these models. Use the word nk to help you. Word nk cue prism sphere cone cylinder pyrmid D A C F A B C D cone cylinder cue cylinder E B E prism F cue G G pyrmid

More information

COMP108 Algorithmic Foundations

COMP108 Algorithmic Foundations Grph Theory Prudene Wong http://www.s.liv..uk/~pwong/tehing/omp108/201617 How to Mesure 4L? 3L 5L 3L ontiner & 5L ontiner (without mrk) infinite supply of wter You n pour wter from one ontiner to nother

More information

OPTICS. (b) 3 3. (d) (c) , A small piece

OPTICS. (b) 3 3. (d) (c) , A small piece AQB-07-P-106 641. If the refrctive indices of crown glss for red, yellow nd violet colours re 1.5140, 1.5170 nd 1.518 respectively nd for flint glss re 1.644, 1.6499 nd 1.685 respectively, then the dispersive

More information

Honors Geometry CHAPTER 7. Study Guide Final Exam: Ch Name: Hour: Try to fill in as many as possible without looking at your book or notes.

Honors Geometry CHAPTER 7. Study Guide Final Exam: Ch Name: Hour: Try to fill in as many as possible without looking at your book or notes. Honors Geometry Study Guide Final Exam: h 7 12 Name: Hour: Try to fill in as many as possible without looking at your book or notes HPTER 7 1 Pythagorean Theorem: Pythagorean Triple: 2 n cute Triangle

More information

Chapter 4 Fuzzy Graph and Relation

Chapter 4 Fuzzy Graph and Relation Chpter 4 Fuzzy Grph nd Reltion Grph nd Fuzzy Grph! Grph n G = (V, E) n V : Set of verties(node or element) n E : Set of edges An edge is pir (x, y) of verties in V.! Fuzzy Grph ~ n ( ~ G = V, E) n V :

More information

CONSTRUCTING CONGRUENT LINE SEGMENTS

CONSTRUCTING CONGRUENT LINE SEGMENTS NME: 1. Given: Task: Construct a segment congruent to. CONSTRUCTING CONGRUENT LINE SEGMENTS B a) Draw a new, longer segment with your straightedge. b) Place an endpoint on the left side of the new segment

More information

Fall 2018 Midterm 1 October 11, ˆ You may not ask questions about the exam except for language clarifications.

Fall 2018 Midterm 1 October 11, ˆ You may not ask questions about the exam except for language clarifications. 15-112 Fll 2018 Midterm 1 October 11, 2018 Nme: Andrew ID: Recittion Section: ˆ You my not use ny books, notes, extr pper, or electronic devices during this exm. There should be nothing on your desk or

More information

Iterated Integrals. f (x; y) dy dx. p(x) To evaluate a type I integral, we rst evaluate the inner integral Z q(x) f (x; y) dy.

Iterated Integrals. f (x; y) dy dx. p(x) To evaluate a type I integral, we rst evaluate the inner integral Z q(x) f (x; y) dy. Iterted Integrls Type I Integrls In this section, we begin the study of integrls over regions in the plne. To do so, however, requires tht we exmine the importnt ide of iterted integrls, in which inde

More information

MATH STUDENT BOOK. 12th Grade Unit 3

MATH STUDENT BOOK. 12th Grade Unit 3 MTH STUDENT OOK 12th Grade Unit 3 MTH 1203 RIGHT TRINGLE TRIGONOMETRY INTRODUTION 3 1. SOLVING RIGHT TRINGLE LENGTHS OF SIDES NGLE MESURES 13 INDIRET MESURE 18 SELF TEST 1: SOLVING RIGHT TRINGLE 23 2.

More information

Right Triangle Trigonometry

Right Triangle Trigonometry Rigt Tringle Trigonometry Trigonometry comes from te Greek trigon (tringle) nd metron (mesure) nd is te study of te reltion between side lengts nd ngles of tringles. Angles A ry is strigt lf line tt stretces

More information

There are three ways to classify triangles based on sides

There are three ways to classify triangles based on sides Unit 4 Notes: Triangles 4-1 Triangle ngle-sum Theorem ngle review, label each angle with the correct classification: Triangle a polygon with three sides. There are two ways to classify triangles: by angles

More information

Year 11 GCSE Revision - Re-visit work

Year 11 GCSE Revision - Re-visit work Week beginning 6 th 13 th 20 th HALF TERM 27th Topis for revision Fators, multiples and primes Indies Frations, Perentages, Deimals Rounding 6 th Marh Ratio Year 11 GCSE Revision - Re-visit work Understand

More information

Math 227 Problem Set V Solutions. f ds =

Math 227 Problem Set V Solutions. f ds = Mth 7 Problem Set V Solutions If is urve with prmetriztion r(t), t b, then we define the line integrl f ds b f ( r(t) ) dr dt (t) dt. Evlute the line integrl f(x,y,z)ds for () f(x,y,z) xosz, the urve with

More information

Duality in linear interval equations

Duality in linear interval equations Aville online t http://ijim.sriu..ir Int. J. Industril Mthemtis Vol. 1, No. 1 (2009) 41-45 Dulity in liner intervl equtions M. Movhedin, S. Slhshour, S. Hji Ghsemi, S. Khezerloo, M. Khezerloo, S. M. Khorsny

More information

1 Drawing 3D Objects in Adobe Illustrator

1 Drawing 3D Objects in Adobe Illustrator Drwing 3D Objects in Adobe Illustrtor 1 1 Drwing 3D Objects in Adobe Illustrtor This Tutoril will show you how to drw simple objects with three-dimensionl ppernce. At first we will drw rrows indicting

More information

CS311H: Discrete Mathematics. Graph Theory IV. A Non-planar Graph. Regions of a Planar Graph. Euler s Formula. Instructor: Işıl Dillig

CS311H: Discrete Mathematics. Graph Theory IV. A Non-planar Graph. Regions of a Planar Graph. Euler s Formula. Instructor: Işıl Dillig CS311H: Discrete Mthemtics Grph Theory IV Instructor: Işıl Dillig Instructor: Işıl Dillig, CS311H: Discrete Mthemtics Grph Theory IV 1/25 A Non-plnr Grph Regions of Plnr Grph The plnr representtion of

More information

such that the S i cover S, or equivalently S

such that the S i cover S, or equivalently S MATH 55 Triple Integrls Fll 16 1. Definition Given solid in spce, prtition of consists of finite set of solis = { 1,, n } such tht the i cover, or equivlently n i. Furthermore, for ech i, intersects i

More information

Area and Volume. Introduction

Area and Volume. Introduction CHAPTER 3 Are nd Volume Introduction Mn needs mesurement for mny tsks. Erly records indicte tht mn used ody prts such s his hnd nd forerm nd his nturl surroundings s mesuring instruments. Lter, the imperil

More information

AML710 CAD LECTURE 16 SURFACES. 1. Analytical Surfaces 2. Synthetic Surfaces

AML710 CAD LECTURE 16 SURFACES. 1. Analytical Surfaces 2. Synthetic Surfaces AML7 CAD LECTURE 6 SURFACES. Anlticl Surfces. Snthetic Surfces Surfce Representtion From CAD/CAM point of view surfces re s importnt s curves nd solids. We need to hve n ide of curves for surfce cretion.

More information

Journal of Combinatorial Theory, Series A

Journal of Combinatorial Theory, Series A Journl of Comintoril Theory, Series A 0 (0) Contents lists ville t SiVerse SieneDiret Journl of Comintoril Theory, Series A www.elsevier.om/lote/jt Spheril tiling y ongruent pentgons Hongho Go, Nn Shi,

More information

UNIT 2 NOTE PACKET. Triangle Proofs

UNIT 2 NOTE PACKET. Triangle Proofs Name GEOMETRY UNIT 2 NOTE PKET Triangle Proofs ate Page Topic Homework 9/19 2-3 Vocabulary Study Vocab 9/20 4 Vocab ont. and No Homework Reflexive/ddition/Subtraction 9/23 5-6 rawing onclusions from Vocab

More information

Lily Yen and Mogens Hansen

Lily Yen and Mogens Hansen SKOLID / SKOLID No. 8 Lily Yen nd Mogens Hnsen Skolid hs joined Mthemticl Myhem which is eing reformtted s stnd-lone mthemtics journl for high school students. Solutions to prolems tht ppered in the lst

More information

TOPIC 10 THREE DIMENSIONAL GEOMETRY

TOPIC 10 THREE DIMENSIONAL GEOMETRY TOPIC THREE DIMENSIONAL GEOMETRY SCHEMATIC DIAGRAM Topi Conept Degree of importne Three Dimensionl Geometr (i Diretion Rtios n Diretion Cosines (iicrtesin n Vetor eqution of line in spe & onversion of

More information

10.5 Graphing Quadratic Functions

10.5 Graphing Quadratic Functions 0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions

More information

Apply the Tangent Ratio. You used congruent or similar triangles for indirect measurement. You will use the tangent ratio for indirect measurement.

Apply the Tangent Ratio. You used congruent or similar triangles for indirect measurement. You will use the tangent ratio for indirect measurement. 7.5 pply the Tangent Ratio efore Now You used congruent or similar triangles for indirect measurement. You will use the tangent ratio for indirect measurement. Why? So you can find the height of a roller

More information

Angle Relationships. Geometry Vocabulary. Parallel Lines November 07, 2013

Angle Relationships. Geometry Vocabulary. Parallel Lines November 07, 2013 Geometr Vocbulr. Point the geometric figure formed t the intersecon of two disnct lines 2. Line the geometric figure formed b two points. A line is the stright pth connecng two points nd etending beond

More information

Inequalities in Triangles Geometry 5-5

Inequalities in Triangles Geometry 5-5 Inequalities in Triangles Geometry 5-5 Name: ate: Period: Theorem 5-10 Theorem 5-11 If two sides of a triangle are not If two angles of a triangle are not congruent, then the larger angle congruent, then

More information

ZZ - Advanced Math Review 2017

ZZ - Advanced Math Review 2017 ZZ - Advnced Mth Review Mtrix Multipliction Given! nd! find the sum of the elements of the product BA First, rewrite the mtrices in the correct order to multiply The product is BA hs order x since B is

More information

Before We Begin. Introduction to Spatial Domain Filtering. Introduction to Digital Image Processing. Overview (1): Administrative Details (1):

Before We Begin. Introduction to Spatial Domain Filtering. Introduction to Digital Image Processing. Overview (1): Administrative Details (1): Overview (): Before We Begin Administrtive detils Review some questions to consider Winter 2006 Imge Enhncement in the Sptil Domin: Bsics of Sptil Filtering, Smoothing Sptil Filters, Order Sttistics Filters

More information

A. 180 B. 108 C. 360 D. 540

A. 180 B. 108 C. 360 D. 540 Part I - Multiple Choice - Circle your answer: REVIEW FOR FINAL EXAM - GEOMETRY 2 1. Find the area of the shaded sector. Q O 8 P A. 2 π B. 4 π C. 8 π D. 16 π 2. An octagon has sides. A. five B. six C.

More information

Graphing Conic Sections

Graphing Conic Sections Grphing Conic Sections Definition of Circle Set of ll points in plne tht re n equl distnce, clled the rdius, from fixed point in tht plne, clled the center. Grphing Circle (x h) 2 + (y k) 2 = r 2 where

More information

Honors Thesis: Investigating the Algebraic Properties of Cayley Digraphs

Honors Thesis: Investigating the Algebraic Properties of Cayley Digraphs Honors Thesis: Investigting the Algebri Properties of Cyley Digrphs Alexis Byers, Wittenberg University Mthemtis Deprtment April 30, 2014 This pper utilizes Grph Theory to gin insight into the lgebri struture

More information

Lecture 12 : Topological Spaces

Lecture 12 : Topological Spaces Leture 12 : Topologil Spes 1 Topologil Spes Topology generlizes notion of distne nd loseness et. Definition 1.1. A topology on set X is olletion T of susets of X hving the following properties. 1. nd X

More information

Angles. Angles. Curriculum Ready.

Angles. Angles. Curriculum Ready. ngles ngles urriculum Redy www.mthletics.com ngles mesure the mount of turn in degrees etween two lines tht meet t point. Mny gmes re sed on interpreting using ngles such s pool, snooker illirds. lck

More information

Points that live on the same line are. Lines that live on the same plane are. Two lines intersect at a.

Points that live on the same line are. Lines that live on the same plane are. Two lines intersect at a. For points through E, plot and label the points on the coordinate plane and then state the quadrant each point is located in. If the point does not live in a quadrant, state where it falls. LOTION (-3,

More information

Geometry Chapter 1 Basics of Geometry

Geometry Chapter 1 Basics of Geometry Geometry Chapter 1 asics of Geometry ssign Section Pages Problems 1 1.1 Patterns and Inductive Reasoning 6-9 13-23o, 25, 34-37, 39, 47, 48 2 ctivity!!! 3 1.2 Points, Lines, and Planes 13-16 9-47odd, 55-59odd

More information

Integration. October 25, 2016

Integration. October 25, 2016 Integrtion October 5, 6 Introduction We hve lerned in previous chpter on how to do the differentition. It is conventionl in mthemtics tht we re supposed to lern bout the integrtion s well. As you my hve

More information