Angles 2.1. Exercise Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example

Size: px
Start display at page:

Download "Angles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example"

Transcription

1 2.1 Angles Reognise lternte n orresponing ngles Key wors prllel lternte orresponing vertilly opposite Rememer, prllel lines re stright lines whih never meet or ross. The rrows show tht the lines re prllel Alternte ngles on prllel lines re equl. In this igrm, the lternte ngles re on lternte sies of the re line. Corresponing ngles on prllel lines re equl. In this igrm, the orresponing ngles re on the sme sie of the re line. Rememer, vertilly opposite ngles re lso equl. mple Clulte the size of the lettere ngles in this igrm. Give resons for your nswers euse euse it is the orresponing ngle to 50 euse it is the lternte ngle to 50 erise Fin the size of the lettere ngles. Give resons for your nswers. ) ) ) Angles on stright line up to ) e) f) 34 e f 14 Mths Connet 2

2 Clulte the size of the lettere ngles. Give resons for your nswers. ) ) ) g f e ) s e) f) q n p r l g) h) i) 47 v k j 82 m t u i h w Clulte the size of the lettere ngles. Give resons for your nswers n write own eh step of your working. ) ) ) ) i f e 78 j k 150 It my e helpful to sketh the igrm n work out some of the other ngles first. Investigtion ) Copy this igrm. ) Mrk ll the ngles tht re equl to. ) Mrk ll the ngles tht re equl to. ) Wht o you notie? e) Wht o you notie out the pttern of ngles in prllelogrm? Angles 15

3 2.2 Clulting ngles Know n use the ft tht the sum of the ngles in tringle is 180 Know n use the ft tht the sum of the ngles in qurilterl is 360 Unerstn tht the eterior ngle of tringle is equl to the sum of the two interior opposite ngles Key wors proof interior eterior p q 180 s r p q r s 360 The sum of the ngles in ny tringle The sum of the ngles in ny is 180. qurilterl is 360. An interior ngle is insie the shpe. An eterior ngle is outsie the shpe. It is me y etening one of the lines. e e 180 (ngles on stright line) 180 (ngles in tringle) so e The eterior ngle of tringle is equl to the sum of the two interior opposite ngles. mple Clulte the size of the lettere ngles, giving resons for your nswers. ) ) ) ) The sum of ngles in tringle is 180. The eterior ngle of tringle is equl to the sum of the two interior opposite ngles. 16 Mths Connet 2

4 erise Clulte the size of the lettere ngles, giving resons for your nswers. ) ) ) 122 f ) g e) f) g) h) i j h k Clulte the size of the lettere ngles, giving resons for your nswers. ) ) e Look t the lue tringle first n fin. Look for the two lrge tringles. 34 f 65 Work out the vlue of in eh of the following: ) ) ) Investigtion rw two stright lines A n C. A C A C Mrk point O etween them. O O rw line through O tht rosses the two lines. Mrk the two ngles, unerneth the line, s n y. y Mesure the size of n y with protrtor. rw some more lines through O. h time mesure the ngles unerneth the line n reor your results in tle. Wht o you notie? Cn you eplin why this hppens? ten A n C so they meet. Clulting ngles 17

5 2.3 Qurilterls Reognise n lssify qurilterls y their geometri properties Key wors prllelogrm rhomus isoseles trpezium kite rrowhe or elt A qurilterl is four-sie shpe. There re mny ifferent qurilterls. Some hve speil nmes euse they hve prtiulr geometri properties. Retngle Opposite sies re equl n prllel igonls iset eh other Rottion symmetry orer 2 Squre A retngle with ll four sies the sme length igonls iset eh other t right ngles igonls re lines of symmetry rllelogrm Opposite sies re equl n prllel igonls iset eh other Rottion symmetry orer 2 Rhomus A prllelogrm with ll four sies the sme length igonls iset eh other t right ngles oth igonls re lines of symmetry Isoseles trpezium A trpezium with two opposite non-prllel sies One line of symmetry oth igonls re the sme length Kite Two pirs of jent sies tht re the sme length No interior ngle is lrger thn 180 One line of symmetry igonls ross t right ngles Arrowhe or elt Two pirs of jent sies tht re the sme length One interior ngle is lrger thn 180 One line of symmetry igonls ross t right ngles outsie the shpe mple Julie hs to rw sketh of n isoseles trpezium. She writes the mesurements on the igrm. plin why this sketh nnot show n isoseles trpezium. 4 m m It is not n isoseles trpezium euse it oes not hve ny equl ngles. This mens there is no line of symmetry. 18 Mths Connet 2

6 erise Nme eh shpe, using its geometri properties to eplin your nswers. You will nee to mesure lines n ngles. ) ) ) The igonls re shown s she lines. Jules rws some skethes of qurilterls, mrks the size of their sies n ngles n nmes them. Some of the nmes re inorret. Fin whih skethes hve inorret nmes n eplin why they re inorret. ) 3 m ) 3 m 2 m ) 3 m 3 m 6 m ISOSCLS TRAZIUM ARALLLOGRAM ) e) 4 m 50 4 m 50 3 m RHOMUS 3 m m 50 4 m KIT LTA ) rw long oth eges of ruler. Tke the ruler wy n then ple it ross your lines t n ngle. rw long oth eges of the ruler gin. She the shpe enlose y the lines. Wht is the nme of the shpe? plin how you know. ) It is possile to mke nother qurilterl using the sme metho. Wht is the nme of the qurilterl? plin how you know. Investigtion You nee 3 y 3 pin or, or Resoure sheet 61, n ruer n. Mke s mny ifferent qurilterls s you n. rw n lel eh shpe. Cut out your shpes n rrnge them into groups tht hve similr properties. Stik the groups into your workook n eplin the properties tht link the shpes in eh group. Qurilterls 19

7 2.4 Solving geometril prolems Solve geometril prolems using sie n ngle properties of speil tringles n qurilterls Key wors lternte orresponing vertilly opposite Angles in tringle up to 180. Angles on stright line up to 180. Alternte ngles re equl. Corresponing ngles re equl. Vertilly opposite ngles re equl. We n use this knowlege to fin the size of missing ngles. Sine we re lulting the sizes of the ngles y following hin of resoning n not mesuring them, our igrms o not hve to e urte. mple 1 Clulte the size of ngles,,, n e, giving resons for your nswers. 53 e Angle is 53 euse tringle AC is isoseles n the se ngles of n isoseles tringle re equl. Angle is 53 euse it is n lternte ngle to ngle. Angle is 74 euse ngles on stright line up to 180 n Angle is 53 euse it is orresponing ngle to ngle. Angle e is 53 euse it is n lternte ngle to ngle. A 53 C We n see from the igrm tht AC is n isoseles tringle n tht n C re prllel. C n re lso prllel. mple 2 Clulte the size of the missing ngles giving resons for your nswers. 3 y 2 z euse ngles in tringle up to so 3 54 n 2 36 y euse ngles in stright line up to 180 so y 126 z 36 euse it is vertilly opposite to the ngle Mths Connet 2

8 erise Clulte the size of the lettere ngles, giving resons for your nswers. ) 76 ) A C F ) 58 A C Clulte the size of the lettere ngles, giving resons for your nswers. ) A ) ) A C y C 33 Clulte the size of eh ngle, giving resons for your nswers. ) ) Investigtion The ngle t the verte of regulr pentgon is 108. Two igonls re rwn to the sme verte to mke three tringles. ) Clulte the sizes of the ngles in eh tringle. ) The mile tringle n one of the other tringles re ple together like this: 108 plin why the tringles fit together to mke new tringle. Wht re its ngles? ) Investigte the other shpes you n mke y putting the three tringles together. Solving geometril prolems 21

9 2.5 Constrution Construt isetor of n ngle, using ruler n ompsses Construt the mi-point n perpeniulr isetor of line segment, using ruler n ompsses Key wors isetor ompsses equiistnt perpeniulr isetor mi-point equiistnt The isetor of n ngle is line tht ivies the ngle into two equl prts. You n onstrut the isetor of n ngle using ompsses. In this igrm, is the isetor of the ngle AC. very point on the line is equiistnt from the lines A n C. A C The perpeniulr isetor of line segment ivies the line segment into two equl prts t right ngles. In this igrm, is the perpeniulr isetor of AC. It rossses AC t the mi-point (M) of the line. very point on the line is equiistnt from oth A n C. If you join AC, rhomus is forme. A M C mple 1 F is 100. Construt the isetor of F. G An r is prt of irle. You n rw rs with ompsses. F F F 1) Open the ompsses n put the point on. rw n r tht intersets with n F. mple 2 1) 2.5 m 22 Mths Connet 2 2) o not hnge the opening of the ompsses. ut the point first on the intersetion of the r with n then with F. rw new rs to interset t G. Join G. The line segment Q is 2.5 m long. Construt the perpeniulr isetor of Q. Q 2) Q 3) Q 1) rw line Q of 2.5 m. Open up the ompsses to over hlf the length of Q. 2) le the point t n rw n r. Keep the opening of the ompsses the sme n repet t Q. Join the points where the rs interset.

10 erise rw ny ute ngle. Construt the ngle isetor using only ruler n ompsses. rw line 10 m in length. Construt the perpeniulr isetor of the line using only ruler n ompsses. Mrk the mi-point (M) of the line. rw ny otuse ngle. Construt the ngle isetor using only ruler n ompsses. Chek it is orret y mesuring the two ngles n mking sure they re the sme. A new wter hnnel is to e uilt t the zoo. It is to e ple etween two niml houses so tht the houses re equiistnt from the hnnel. Copy this igrm, n onstrut re line to show the position of the wter hnnnel. Animl house 1 Animl house 2 A new fene is to e uilt in the prk. The fene is to e ple etween two lrge trees so tht the trees re equiistnt from the fene. The he grener eies to pint in line to mrk the position of the fene. Copy this igrm n onstrut lue line to show the position of the fene. Tree 1 Tree 2 Investigtion ) rw ny tringle with sies longer thn 4 m ut shorter thn 8 m. Construt the ngle isetor for eh of the three ngles. Mke sure the ngle isetors re long enough to ross eh other. Wht o you notie? ) rw nother tringle pproimtely the sme size. Construt the perpeniulr isetor for eh of the three sies. Mke sure the perpeniulr isetors re long enough to ross eh other. Wht o you notie? Constrution 23

11 2.6 erpeniulrs Use stright ege n ompsses to onstrut the perpeniulr from point to line n from point on line Key wors onstrut perpeniulr r We n use ruler n ompsses to: onstrut the perpeniulr from point to line N onstrut the perpeniulr from point on line An r is prt of irle. You n rw rs with your ompsses. mple 1 Mke opy of the igrm. Using only ruler n ompsses rw perpeniulr from the point on the line. 1) Q 2) Q 1) Using ompsses, rw two rs from to mke two intersetions on the line. From the two intersetions, rw two rs tht interset n lel the intersetion Q. 2) le stright ege from to Q n join them with stright line. mple 2 Mke opy of the igrm. Using only ruler n ompsses rw perpeniulr from the point A to the line. A 1) A 2) A 1) Using ompsses, rw two rs from the point A tht interset with the line. From the two rs rw two more rs tht interset n lel the intersetion. 2) le stright ege from A to n join A to the stright line. 24 Mths Connet 2

12 erise Mke opy of the igrm. Using ruler n ompsses, onstrut perpeniulr from the point on the line. 2 m 3.5 m Mke opy of the igrm. Using ruler n ompsses, onstrut perpeniulr from the point to the line. Jmin is wlking her og in the prk when it strts to rin. She wnts to tke the shortest route k to the pth. Copy the igrm n onstrut the shortest route to the pth, using ruler n ompsses. Jmin th ) rw n equilterl tringle, using ruler n protrtor, with sies 5 m in length, s shown here. Construt the perpeniulrs from eh verte to the opposite sie. Wht o you notie? ) rw n isoseles tringle with se 5 m n se ngles of 50, s shown here. Construt the perpeniulrs from eh verte to the opposite sie. Wht o you notie? 5 m m 5 m ) rw slene tringle with se 5 m n se ngles of 70 n 30, s shown here. Construt the perpeniulrs from eh verte to the opposite sie. Wht o you notie? m m erpeniulrs 25

Interior and exterior angles add up to 180. Level 5 exterior angle

Interior and exterior angles add up to 180. Level 5 exterior angle 22 ngles n proof Ientify interior n exterior ngles in tringles n qurilterls lulte interior n exterior ngles of tringles n qurilterls Unerstn the ie of proof Reognise the ifferene etween onventions, efinitions

More information

Lesson 2.1 Inductive Reasoning

Lesson 2.1 Inductive Reasoning Lesson.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 1, 16,,. 400, 00, 100, 0,,,. 1 8, 7, 1, 4,, 4.,,, 1, 1, 0,,. 60, 180, 10,

More information

H SERIES. Area and Perimeter. Curriculum Ready. www.mathletics.com

H SERIES. Area and Perimeter. Curriculum Ready. www.mathletics.com Are n Perimeter Curriulum Rey www.mthletis.om Copyright 00 3P Lerning. All rights reserve. First eition printe 00 in Austrli. A tlogue reor for this ook is ville from 3P Lerning Lt. ISBN 78--86-30-7 Ownership

More information

Maximum area of polygon

Maximum area of polygon Mimum re of polygon Suppose I give you n stiks. They might e of ifferent lengths, or the sme length, or some the sme s others, et. Now there re lots of polygons you n form with those stiks. Your jo is

More information

The remaining two sides of the right triangle are called the legs of the right triangle.

The remaining two sides of the right triangle are called the legs of the right triangle. 10 MODULE 6. RADICAL EXPRESSIONS 6 Pythgoren Theorem The Pythgoren Theorem An ngle tht mesures 90 degrees is lled right ngle. If one of the ngles of tringle is right ngle, then the tringle is lled right

More information

Lesson 4.1 Triangle Sum Conjecture

Lesson 4.1 Triangle Sum Conjecture Lesson 4.1 ringle um onjecture Nme eriod te n ercises 1 9, determine the ngle mesures. 1. p, q 2., y 3., b 31 82 p 98 q 28 53 y 17 79 23 50 b 4. r, s, 5., y 6. y t t s r 100 85 100 y 30 4 7 y 31 7. s 8.

More information

End of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below.

End of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below. End of term: TEST A You will need penil nd ruler. Yer Nme Clss Dte Complete the missing numers in the sequenes elow. 8 30 3 28 2 9 25 00 75 25 2 Put irle round ll of the following shpes whih hve 3 shded.

More information

Chapter. Contents: A Constructing decimal numbers

Chapter. Contents: A Constructing decimal numbers Chpter 9 Deimls Contents: A Construting deiml numers B Representing deiml numers C Deiml urreny D Using numer line E Ordering deimls F Rounding deiml numers G Converting deimls to frtions H Converting

More information

UNCORRECTED SAMPLE PAGES

UNCORRECTED SAMPLE PAGES 6 Chpter Length, re, surfe re n volume Wht you will lern 6A Length n perimeter 6B Cirumferene of irles n perimeter of setors 6C Are of qurilterls n tringles 6D Are of irles 6E Perimeter n re of omposite

More information

Ratio and Proportion

Ratio and Proportion Rtio nd Proportion Rtio: The onept of rtio ours frequently nd in wide vriety of wys For exmple: A newspper reports tht the rtio of Repulins to Demorts on ertin Congressionl ommittee is 3 to The student/fulty

More information

Further applications of area and volume

Further applications of area and volume 2 Further pplitions of re n volume 2A Are of prts of the irle 2B Are of omposite shpes 2C Simpson s rule 2D Surfe re of yliners n spheres 2E Volume of omposite solis 2F Error in mesurement Syllus referene

More information

The art of Paperarchitecture (PA). MANUAL

The art of Paperarchitecture (PA). MANUAL The rt of Pperrhiteture (PA). MANUAL Introution Pperrhiteture (PA) is the rt of reting three-imensionl (3D) ojets out of plin piee of pper or ror. At first, esign is rwn (mnully or printe (using grphil

More information

50 MATHCOUNTS LECTURES (10) RATIOS, RATES, AND PROPORTIONS

50 MATHCOUNTS LECTURES (10) RATIOS, RATES, AND PROPORTIONS 0 MATHCOUNTS LECTURES (0) RATIOS, RATES, AND PROPORTIONS BASIC KNOWLEDGE () RATIOS: Rtios re use to ompre two or more numers For n two numers n ( 0), the rtio is written s : = / Emple : If 4 stuents in

More information

Chapter. Fractions. Contents: A Representing fractions

Chapter. Fractions. Contents: A Representing fractions Chpter Frtions Contents: A Representing rtions B Frtions o regulr shpes C Equl rtions D Simpliying rtions E Frtions o quntities F Compring rtion sizes G Improper rtions nd mixed numers 08 FRACTIONS (Chpter

More information

SECTION 7-2 Law of Cosines

SECTION 7-2 Law of Cosines 516 7 Additionl Topis in Trigonometry h d sin s () tn h h d 50. Surveying. The lyout in the figure t right is used to determine n inessile height h when seline d in plne perpendiulr to h n e estlished

More information

Quick Guide to Lisp Implementation

Quick Guide to Lisp Implementation isp Implementtion Hndout Pge 1 o 10 Quik Guide to isp Implementtion Representtion o si dt strutures isp dt strutures re lled S-epressions. The representtion o n S-epression n e roken into two piees, the

More information

Geometry 7-1 Geometric Mean and the Pythagorean Theorem

Geometry 7-1 Geometric Mean and the Pythagorean Theorem Geometry 7-1 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the

More information

Volumes by Cylindrical Shells: the Shell Method

Volumes by Cylindrical Shells: the Shell Method olumes Clinril Shells: the Shell Metho Another metho of fin the volumes of solis of revolution is the shell metho. It n usull fin volumes tht re otherwise iffiult to evlute using the Dis / Wsher metho.

More information

1 Fractions from an advanced point of view

1 Fractions from an advanced point of view 1 Frtions from n vne point of view We re going to stuy frtions from the viewpoint of moern lger, or strt lger. Our gol is to evelop eeper unerstning of wht n men. One onsequene of our eeper unerstning

More information

Angles and Triangles

Angles and Triangles nges nd Tringes n nge is formed when two rys hve ommon strting point or vertex. The mesure of n nge is given in degrees, with ompete revoution representing 360 degrees. Some fmiir nges inude nother fmiir

More information

MATH PLACEMENT REVIEW GUIDE

MATH PLACEMENT REVIEW GUIDE MATH PLACEMENT REVIEW GUIDE This guie is intene s fous for your review efore tking the plement test. The questions presente here my not e on the plement test. Although si skills lultor is provie for your

More information

84 cm 30 cm. 12 in. 7 in. Proof. Proof of Theorem 7-4. Given: #QXY with 6 Prove: * RS * XY

84 cm 30 cm. 12 in. 7 in. Proof. Proof of Theorem 7-4. Given: #QXY with 6 Prove: * RS * XY -. Pln Ojetives o use the ie-plitter heorem o use the ringle-ngle- isetor heorem Emples Using the ie-plitter heorem el-worl onnetion Using the ringle-ngle- isetor heorem Mth kgroun - Wht ou ll Lern o use

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1 PROBLEMS - APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.

More information

1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5.

1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5. . Definition, Bsi onepts, Types. Addition nd Sutrtion of Mtries. Slr Multiplition. Assignment nd nswer key. Mtrix Multiplition. Assignment nd nswer key. Determinnt x x (digonl, minors, properties) summry

More information

SOLVING EQUATIONS BY FACTORING

SOLVING EQUATIONS BY FACTORING 316 (5-60) Chpter 5 Exponents nd Polynomils 5.9 SOLVING EQUATIONS BY FACTORING In this setion The Zero Ftor Property Applitions helpful hint Note tht the zero ftor property is our seond exmple of getting

More information

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

More information

Words Symbols Diagram. abcde. a + b + c + d + e

Words Symbols Diagram. abcde. a + b + c + d + e Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To

More information

If two triangles are perspective from a point, then they are also perspective from a line.

If two triangles are perspective from a point, then they are also perspective from a line. Mth 487 hter 4 Prtie Prolem Solutions 1. Give the definition of eh of the following terms: () omlete qudrngle omlete qudrngle is set of four oints, no three of whih re olliner, nd the six lines inident

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions. Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

More information

WHAT HAPPENS WHEN YOU MIX COMPLEX NUMBERS WITH PRIME NUMBERS?

WHAT HAPPENS WHEN YOU MIX COMPLEX NUMBERS WITH PRIME NUMBERS? WHAT HAPPES WHE YOU MIX COMPLEX UMBERS WITH PRIME UMBERS? There s n ol syng, you n t pples n ornges. Mthemtns hte n t; they love to throw pples n ornges nto foo proessor n see wht hppens. Sometmes they

More information

AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.

More information

10 AREA AND VOLUME 1. Before you start. Objectives

10 AREA AND VOLUME 1. Before you start. Objectives 10 AREA AND VOLUME 1 The Tower of Pis is circulr bell tower. Construction begn in the 1170s, nd the tower strted lening lmost immeditely becuse of poor foundtion nd loose soil. It is 56.7 metres tll, with

More information

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2 7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6

More information

Orthopoles and the Pappus Theorem

Orthopoles and the Pappus Theorem Forum Geometriorum Volume 4 (2004) 53 59. FORUM GEOM ISSN 1534-1178 Orthopoles n the Pppus Theorem tul Dixit n Drij Grinerg strt. If the verties of tringle re projete onto given line, the perpeniulrs from

More information

One Minute To Learn Programming: Finite Automata

One Minute To Learn Programming: Finite Automata Gret Theoreticl Ides In Computer Science Steven Rudich CS 15-251 Spring 2005 Lecture 9 Fe 8 2005 Crnegie Mellon University One Minute To Lern Progrmming: Finite Automt Let me tech you progrmming lnguge

More information

Unit 6: Exponents and Radicals

Unit 6: Exponents and Radicals Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -

More information

Vectors Summary. Projection vector AC = ( Shortest distance from B to line A C D [OR = where m1. and m

Vectors Summary. Projection vector AC = ( Shortest distance from B to line A C D [OR = where m1. and m . Slr prout (ot prout): = osθ Vetors Summry Lws of ot prout: (i) = (ii) ( ) = = (iii) = (ngle etween two ientil vetors is egrees) (iv) = n re perpeniulr Applitions: (i) Projetion vetor: B Length of projetion

More information

How To Find The Re Of Tringle

How To Find The Re Of Tringle Heron s Formul for Tringulr Are y Christy Willims, Crystl Holom, nd Kyl Gifford Heron of Alexndri Physiist, mthemtiin, nd engineer Tught t the museum in Alexndri Interests were more prtil (mehnis, engineering,

More information

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324 A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................

More information

LISTENING COMPREHENSION

LISTENING COMPREHENSION PORG, přijímí zkoušky 2015 Angličtin B Reg. číslo: Inluded prts: Points (per prt) Points (totl) 1) Listening omprehension 2) Reding 3) Use of English 4) Writing 1 5) Writing 2 There re no extr nswersheets

More information

c b 5.00 10 5 N/m 2 (0.120 m 3 0.200 m 3 ), = 4.00 10 4 J. W total = W a b + W b c 2.00

c b 5.00 10 5 N/m 2 (0.120 m 3 0.200 m 3 ), = 4.00 10 4 J. W total = W a b + W b c 2.00 Chter 19, exmle rolems: (19.06) A gs undergoes two roesses. First: onstnt volume @ 0.200 m 3, isohori. Pressure inreses from 2.00 10 5 P to 5.00 10 5 P. Seond: Constnt ressure @ 5.00 10 5 P, isori. olume

More information

Rotating DC Motors Part II

Rotating DC Motors Part II Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors

More information

The Cat in the Hat. by Dr. Seuss. A a. B b. A a. Rich Vocabulary. Learning Ab Rhyming

The Cat in the Hat. by Dr. Seuss. A a. B b. A a. Rich Vocabulary. Learning Ab Rhyming MINI-LESSON IN TION The t in the Ht y Dr. Seuss Rih Voulry tme dj. esy to hndle (not wild) LERNING Lerning Rhyming OUT Words I know it is wet nd the sun is not sunny. ut we n hve Lots of good fun tht is

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

More information

Pure C4. Revision Notes

Pure C4. Revision Notes Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd

More information

Introduction. Teacher s lesson notes The notes and examples are useful for new teachers and can form the basis of lesson plans.

Introduction. Teacher s lesson notes The notes and examples are useful for new teachers and can form the basis of lesson plans. Introduction Introduction The Key Stge 3 Mthemtics series covers the new Ntionl Curriculum for Mthemtics (SCAA: The Ntionl Curriculum Orders, DFE, Jnury 1995, 0 11 270894 3). Detiled curriculum references

More information

0.1 Basic Set Theory and Interval Notation

0.1 Basic Set Theory and Interval Notation 0.1 Bsic Set Theory nd Intervl Nottion 3 0.1 Bsic Set Theory nd Intervl Nottion 0.1.1 Some Bsic Set Theory Notions Like ll good Mth ooks, we egin with definition. Definition 0.1. A set is well-defined

More information

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right.

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right. Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction

More information

Factoring Polynomials

Factoring Polynomials Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

More information

1.2 The Integers and Rational Numbers

1.2 The Integers and Rational Numbers .2. THE INTEGERS AND RATIONAL NUMBERS.2 The Integers n Rtionl Numers The elements of the set of integers: consist of three types of numers: Z {..., 5, 4, 3, 2,, 0,, 2, 3, 4, 5,...} I. The (positive) nturl

More information

The Pythagorean Theorem

The Pythagorean Theorem The Pythgoren Theorem Pythgors ws Greek mthemtiin nd philosopher, orn on the islnd of Smos (. 58 BC). He founded numer of shools, one in prtiulr in town in southern Itly lled Crotone, whose memers eventully

More information

6.2 Volumes of Revolution: The Disk Method

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

More information

OxCORT v4 Quick Guide Revision Class Reports

OxCORT v4 Quick Guide Revision Class Reports OxCORT v4 Quik Guie Revision Clss Reports This quik guie is suitble for the following roles: Tutor This quik guie reltes to the following menu options: Crete Revision Clss Reports pg 1 Crete Revision Clss

More information

Experiment 6: Friction

Experiment 6: Friction Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

More information

In order to master the techniques explained here it is vital that you undertake the practice exercises provided.

In order to master the techniques explained here it is vital that you undertake the practice exercises provided. Tringle formule m-ty-tringleformule-009-1 ommonmthemtilprolemistofindthenglesorlengthsofthesidesoftringlewhen some,utnotllofthesequntitiesreknown.itislsousefultoeletolultethere of tringle from some of

More information

Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chapter 40 Key Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100 hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

More information

Binary Representation of Numbers Autar Kaw

Binary Representation of Numbers Autar Kaw Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

More information

Answer, Key Homework 10 David McIntyre 1

Answer, Key Homework 10 David McIntyre 1 Answer, Key Homework 10 Dvid McIntyre 1 This print-out should hve 22 questions, check tht it is complete. Multiple-choice questions my continue on the next column or pge: find ll choices efore mking your

More information

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1. Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose

More information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

More information

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001 CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic

More information

10.6 Applications of Quadratic Equations

10.6 Applications of Quadratic Equations 10.6 Applictions of Qudrtic Equtions In this section we wnt to look t the pplictions tht qudrtic equtions nd functions hve in the rel world. There re severl stndrd types: problems where the formul is given,

More information

DATABASDESIGN FÖR INGENJÖRER - 1056F

DATABASDESIGN FÖR INGENJÖRER - 1056F DATABASDESIGN FÖR INGENJÖRER - 06F Sommr 00 En introuktionskurs i tssystem http://user.it.uu.se/~ul/t-sommr0/ lt. http://www.it.uu.se/eu/course/homepge/esign/st0/ Kjell Orsorn (Rusln Fomkin) Uppsl Dtse

More information

ONLINE PAGE PROOFS. Trigonometry. 6.1 Overview. topic 6. Why learn this? What do you know? Learning sequence. measurement and geometry

ONLINE PAGE PROOFS. Trigonometry. 6.1 Overview. topic 6. Why learn this? What do you know? Learning sequence. measurement and geometry mesurement nd geometry topic 6 Trigonometry 6.1 Overview Why lern this? Pythgors ws gret mthemticin nd philosopher who lived in the 6th century BCE. He is est known for the theorem tht ers his nme. It

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

National Firefighter Ability Tests And the National Firefighter Questionnaire

National Firefighter Ability Tests And the National Firefighter Questionnaire Ntionl Firefighter Aility Tests An the Ntionl Firefighter Questionnire PREPARATION AND PRACTICE BOOKLET Setion One: Introution There re three tests n questionnire tht mke up the NFA Tests session, these

More information

CHAPTER 31 CAPACITOR

CHAPTER 31 CAPACITOR . Given tht Numer of eletron HPTER PITOR Net hrge Q.6 9.6 7 The net potentil ifferene L..6 pitne v 7.6 8 F.. r 5 m. m 8.854 5.4 6.95 5 F... Let the rius of the is R re R D mm m 8.85 r r 8.85 4. 5 m.5 m

More information

2 DIODE CLIPPING and CLAMPING CIRCUITS

2 DIODE CLIPPING and CLAMPING CIRCUITS 2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of

More information

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a. Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn 33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

More information

RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS

RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is

More information

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

Vectors 2. 1. Recap of vectors

Vectors 2. 1. Recap of vectors Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

More information

Integration. 148 Chapter 7 Integration

Integration. 148 Chapter 7 Integration 48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

More information

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.

More information

Pentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simple-looking set of objects through which some powerful

Pentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simple-looking set of objects through which some powerful Pentominoes Bruce Bguley Cscde Mth Systems, LLC Astrct. Pentominoes nd their reltives the polyominoes, polycues, nd polyhypercues will e used to explore nd pply vrious importnt mthemticl concepts. In this

More information

and thus, they are similar. If k = 3 then the Jordan form of both matrices is

and thus, they are similar. If k = 3 then the Jordan form of both matrices is Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If

More information

Student Access to Virtual Desktops from personally owned Windows computers

Student Access to Virtual Desktops from personally owned Windows computers Student Aess to Virtul Desktops from personlly owned Windows omputers Mdison College is plesed to nnoune the ility for students to ess nd use virtul desktops, vi Mdison College wireless, from personlly

More information

Practice Test 2. a. 12 kn b. 17 kn c. 13 kn d. 5.0 kn e. 49 kn

Practice Test 2. a. 12 kn b. 17 kn c. 13 kn d. 5.0 kn e. 49 kn Prtie Test 2 1. A highwy urve hs rdius of 0.14 km nd is unnked. A r weighing 12 kn goes round the urve t speed of 24 m/s without slipping. Wht is the mgnitude of the horizontl fore of the rod on the r?

More information

Eufic Guide Enfant UK 14/12/04 15:45 Page 1 Healthy Eatin 10 g Play with us! Tips for Kids

Eufic Guide Enfant UK 14/12/04 15:45 Page 1 Healthy Eatin 10 g Play with us! Tips for Kids Eting Kids Helthy Ply with us! Tips for 10 Do you rememer when you lerned to ride ike? The most importnt prt ws getting the lnce right. Once you could lnce esily, the pedls could turn smoothly, to drive

More information

Calculating Principal Strains using a Rectangular Strain Gage Rosette

Calculating Principal Strains using a Rectangular Strain Gage Rosette Clulting Prinipl Strins using Retngulr Strin Gge Rosette Strin gge rosettes re used often in engineering prtie to determine strin sttes t speifi points on struture. Figure illustrtes three ommonly used

More information

Radius of the Earth - Radii Used in Geodesy James R. Clynch February 2006

Radius of the Earth - Radii Used in Geodesy James R. Clynch February 2006 dius of the Erth - dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.

More information

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding 1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

More information

addition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix.

addition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix. APPENDIX A: The ellipse August 15, 1997 Becuse of its importnce in both pproximting the erth s shpe nd describing stellite orbits, n informl discussion of the ellipse is presented in this ppendix. The

More information

MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!

MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent! MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more

More information

5.6 POSITIVE INTEGRAL EXPONENTS

5.6 POSITIVE INTEGRAL EXPONENTS 54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section

More information

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the

More information

Regular Sets and Expressions

Regular Sets and Expressions Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite

More information

Review Problems for the Final of Math 121, Fall 2014

Review Problems for the Final of Math 121, Fall 2014 Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since

More information

SOLVING QUADRATIC EQUATIONS BY FACTORING

SOLVING QUADRATIC EQUATIONS BY FACTORING 6.6 Solving Qudrti Equtions y Ftoring (6 31) 307 In this setion The Zero Ftor Property Applitions 6.6 SOLVING QUADRATIC EQUATIONS BY FACTORING The tehniques of ftoring n e used to solve equtions involving

More information

Exam 1 Study Guide. Differentiation and Anti-differentiation Rules from Calculus I

Exam 1 Study Guide. Differentiation and Anti-differentiation Rules from Calculus I Exm Stuy Guie Mth 2020 - Clculus II, Winter 204 The following is list of importnt concepts from ech section tht will be teste on exm. This is not complete list of the mteril tht you shoul know for the

More information