# ELEMENTARY CONSTRUCTIONS IN THE HYPERBOLIC PLANE

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 PROCEEDINGS 13th INTERNTIONL CONFERENCE ON GEOMETRY ND GRPHICS ugust 4-8, 2008, Dresden (Germny) ISBN: ELEMENTRY CONSTRUCTIONS IN THE HYPERBOLIC PLNE Syille MICK Grz University of Tehnology, ustri BSTRCT: Construtions of regulr n-gons in the Poinré disk model nd in the Beltrmi-Klein model of the hyperoli geometry re presented. The fous is on methods tht n e rried out y hnd using only Euliden ompss, Euliden protrtor nd strightedge (i.e. ruler without mesuring mrks). regulr n-gon n e deomposed into 2n right-ngled tringles. This is why we n redue the onstrution of regulr n-gon to the onstrution of right-ngled tringle. Keywords: Elementry Euliden Geometry, Hyperoli Plne, Beltrmi-Klein Model, Poinré Model, Geometry of Cirles. 1. INTRODUCTION Hyperoli geometry nd espeilly its onstrutive spets hve reently een reonsidered (see [1], [5] nd [9]). One reson my e the ft tht some omputer progrmmes like Cinderell [8] provide onstrutions of tringles, regulr polygons nd even tilings in the hyperoli plne with few mouse liks. Figure 1: Regulr pentgons in the Beltrmi-Klein model nd the Poinré disk In Cinderell we n hoose oth the Beltrmi-Klein model nd the Poinré disk model to visulize the result (Figure 1). NonEulid [12] is nother progrmme to visulize hyperoli geometry in the Poinré disk model nd in the hlf plne model. The im of this work is to demonstrte the possiility to visulize hyperoli geometry with onstrutions mde y hnd. s n exmple we onstrut regulr n-gons in the Poinré disk model nd in the Beltrmi-Klein model pplying well-known onstrutions of elementry Euliden geometry. In setion 2 short summries out few properties of the Poinré disk nd the Beltrmi-Klein model re given whih will filitte our onstrutions. In setion 3 properties of regulr n-gons re olleted. Even though we only need to know how to onstrut right-ngled tringle given y two ute ngles with α + β < π / 2, we will onstrut in setion 4 tringle with three ritrry ngles with α + β + γ < π / 2 in the Poinré disk. In setion 5 we onstrut n n-gon with given ngle etween djent edges in the Poinré disk nd in setion 6 n n-gon with given side length in the Beltrmi-Klein model. Setion 7 is short onlusion. 2. MODELS OF THE HYPERBOLIC PLNE s there re lot of textooks out hyperoli geometry, e.g. [3], [4], [6] nd [7], only some si notions nd properties re listed here.

2 presented y MICK, Syille 2.1 Poinré disk model Let m e Euliden irle. The Poinré disk model of hyperoli geometry is the open disk in the Euliden plne with oundry m. Points of the model re inner points of m. Hyperoli lines in this model re open rs on irles orthogonl to m. Hyperoli reflexions re reflexions ross dimeters of m or inversions with respet to irles orthogonl to m. (These irles re lso hrterized s elements of penils of Euliden irles with si points B nd B*, inverse with respet to m.) The inidene is indued from the underlying Euliden plne. The ngle mesurement is identil to the Euliden ngle mesurement, i.e. the Poinré disk is onforml model of the hyperoli plne. 2.2 Beltrmi-Klein model The Beltrmi-Klein model of hyperoli geometry is the open disk in the Euliden plne with oundry m. Points of the model re inner points of m. Hyperoli lines re open ords of m. The inidene is indued from the underlying Euliden plne. The oundry m of the disk is the so-lled solute irle. The hyperoli orthogonlity is determined y the polrity of m. hyperoli right ngle is Euliden right ngle iff one leg is dimeter of m. utomorphi ollinetions of m re hyperoli displements. Espeilly, hrmoni utomorphi ollinetions re hyperoli reflexions. Therefore, enter nd xis of hyperoli reflexion re pole nd polr of the solute irle m. For ngle mesurement we will pply tht the Euliden metri nd the hyperoli metri in the enter of the solute irle m re identil nd we n mesure hyperoli ngles with Euliden protrtor. 2.3 Remrks on mesurement The ongruene of segments is equivlent to the sttement tht the segments hve identil length. similr remrk pplies to ngles. Hene, we n speify the hyperoli h length l = T1T 2 y two points T 1, T 2 in the Poinré disk nd in the Beltrmi-Klein model. The hyperoli mesure of n ngle α h = t 1 t 2 in the Beltrmi-Klein is determined y two lines t 1, t PROPERTIES OF REGULR N-GONS IN THE HYPERBOLIC GEOMETRY The Euliden geometry nd the hyperoli geometry re losely relted to eh other. Therefore, regulr n-gons shre ouple of properties, e.g.: 1...n is regulr if ll its ngles re ongruent nd ll its edges hve the sme length with respet to the hosen metri. ll the verties of 1...n lie on ommon irle nd its enter C is the enter of the regulr n-gon, too. Simple regulr n-gons re lwys onvex. The symmetry group of regulr n-gon is the dihedrl group D n of order 2n. It onsists of the n rottions with enter C nd ngle 2 π / n, together with n reflexions with xes through the enter. If n is even, then hlf of these xes pss through the midpoints of opposite edges. If n is odd, then ll xes pss through vertex nd the midpoint of the opposite edge. No mtter, if n is even or odd, two xes with ngle π / n determine two ongruent right-ngled tringles. The verties of one tringle re one vertex of the n-gon, the midpoint of n djent edge nd the enter of the regulr n-gon. If we strt with one suh tringle (see Figure 4 nd Figure 5), the orit under the dihedrl group is the regulr n-gon. Therefore, it is ler tht the onstrution of right-ngled tringle is the essentil prt of the onstrution of regulr n-gon. But there re no similrities in the hyperoli geometry! Two regulr n-gons 1...n nd B 1...Bn with different edge length hve different ngles etween two djent edges. Therefore, regulr n-gon in the hyperoli geometry n e given either y the ngle σ <π(n-2)/n etween two djent edges (se I) or y the length s of n edge (se II). 2

3 presented y MICK, Syille 4. CONSTRUCTION OF TRINGLE WITH THREE SPECIFIED NGLES IN THE POINCRÉ DISK 4.1 Euliden irles interseting two lines under speified ngles Let nd e two interseting lines with = α. To onstrut Euliden irle * with * = γ nd * = β we hoose some ritrry points on nd nd drw the nglesγ nd β, respetively (Figure 2). * = γ nd * = β. In order to find irle m* with enter nd orthogonl to *, we drw the polr of with respet to *. The points of intersetion with * determine the rdius of m*. The diltion with enter tht mps m* on m lso mps the irle * on irle orthogonl to m (Figure 3). m* m * r* r* g B C g Figure 2: Euliden irles interset two interseting lines under speified ngles n ritrry distne r* is mrked on lines orthogonl to the seond legs of γ nd β, respetively. Prllel lines to nd through the new points interset in the enter of irle * with the desired properties. 4.2 Tringles with three speified ngles in the Poinré disk Let m e the solute irle of the Poinré disk, α, β nd γ e ngles with α + β + γ < π. Then tringles BC with ngles α, β nd γ exist (they re ongruent). Let us onstrut one. Without loss of generlity we put the vertex y hyperoli displement in the enter of m. The edges nd through re segments on dimeters of m. The missing edge of the tringle is n r on Euliden irle orthogonl to m, interseting nd under = γ nd = β. Due to susetion 4.1, we n drw irle * with 3 Figure 3: Tringle BC with three speified ngles nd with vertex in the enter of the solute irle of the Poinré disk The r BC on the irle is the third edge of the tringle. If we put γ = π / 2 the onstrution is simplified. This will e demonstrted in Figure 4. Remrk: In Euliden geometry there exist up to similrities extly one right-ngled isoseles tringle with α = π / 2, β = γ = π / 4 nd extly one equilterl tringle with α = β = γ = π / 3. The onstrution in Figure 3 shows tht in hyperoli geometry every speifition α = π / 2, 0 < β = γ < π / 4 yields right-ngled isoseles tringle nd every ngle α < π / 3 yields n equilterl tringle.

4 presented y MICK, Syille 5. CONSTRUCTION OF REGULR N-GON WITH GIVEN NGLE BETWEEN TWO DJCENT EDGES IN THE POINCRÉ DISK We onstrut regulr n-gon with given ngle σ etween djent edges in the Poinré disk model. We ssume tht the enter of the regulr n-gon oinides with the enter of the Poinré disk. We drw tringle (see setion 4) with ngles α = π / n, β = σ / 2, γ = π / 2 ; n=5 nd σ = 60. The onstrution of BC simplifies euse the enter of irles * nd oinide with (Figure 4). B C m* * Figure 4: Right-ngled tringle BC with α = π / 5 = 36, β = 30, γ = π / 2 nd vertex in the enter nd regulr pentgon in the Poinré disk The imges from BC under the reflexion ross nd the rottions out through 2 k π /5, k = 1,...,4 uild up the regulr pentgon. Remrk: In Euliden geometry only the regulr hexgon is deomposle in six equilterl tringles. The sitution in hyperoli geometry is different. If regulr n-gon is deomposle in n equilterl tringles, then it is deomposle in 2n m 4 right-ngled tringles with α = π / n nd β = 2π / n with α + β < π / 2. From this it follows tht for every n > 6 up to hyperoli displements regulr n-gon exists deomposle in n equilterl tringles. The onstrution is the sme s in Figure 4 with speifitions α = π / n, β = 2 π / n, γ = π / 2 with n CONSTRUCTION OF REGULR N-GON WITH GIVEN SIDE LENGTH IN THE BELTRMI-KLEIN MODEL 6.1 Equidistnt urves in the Beltrmi-Klein model In the hyperoli geometry n equidistnt urve is irle with ultr-prllel dimeters, ll orthogonl to its xis o, while its enter is n ultr-idel point O. The equidistnt urve intersets ll its dimeters in segments of the sme length. In the Beltrmi-Klein model xis o nd enter O re pole nd polr with respet to the solute irle m. If o is dimeter of m, then O is point t infinity in the underlying Euliden plne. 6.2 Some right-ngled tringles in the Beltrmi Klein model We onsider right-ngled tringles in the hyperoli geometry given y n ute ngle α nd length of the opposite side nd we ssume tht the vertex is the enter of m (Figure 5). Beuse n ngle with vertex in the enter of the solute irle hs the sme mgnitude in Euliden nd in hyperoli geometry we n use Euliden protrtor to drw n ngleα. Then one leg of α is the hypotenuse nd the other one thetus of the tringle on dimeter o. The seond endpoint B of the hypotenuse hs the distne of o. Hene, B is point of n equidistnt urve with xis o. If the hyperoli length is given y segment then hyperoli reflexion exists tht mps this segment on B on the dimeter orthogonl to o. The point B determines together with o s xis the

5 presented y MICK, Syille equidistnt urve in the hyperoli plne n ellipse in Euliden sense. The point of intersetion of the urve nd the seond leg of α is the seond point B on the hypotenuse. Finlly, the Euliden right-ngled tringle BC is the requested hyperoli right-ngled tringle, too. m o Figure 5 Right-ngled tringle BC with vertex in the enter of the solute irle nd regulr hexgon in the Beltrmi-Klein model 7. CONSTRUCTION OF REGULR N-GON WITH GIVEN SIDE LENGTH IN THE BELTRMI-KLEIN MODEL lso in Figure 5 we onstrut regulr n-gon with given side length s in the Beltrmi-Klein model. Its enter oinides with the enter of the Beltrmi-Klein model. We speify α = π / n, s / 2 = nd n = 6 nd onstrut the right-ngled tringle BC. The imges from BC under the reflexion ross o nd the rottions out through 2 k π / 6, k = 1,...,5, form the regulr hexgon. 8. CONCLUSIONS Visuliztion of hyperoli geometry in different models, i.e. the Poinré disk model nd the Beltrmi-Klein model, mkes the xiomti development of hyperoli geometry more lively. Computer progrmmes re ville, ut onstrutions y hnd re stright ess to visulize geometry. s B' B C exmples we onstruted right-ngled tringles nd regulr n-gons using elementry Euliden onstrutions. REFERENCES [1] Bić, I., Sliepčević,., Regulr Polygons in the Projetively Extended Hyperoli Plne, KoG 11 (2007), [2] Benz, W., Clssil Geometries in Modern Contexts, Birkhäuser Verlg, Bsel, [3] Coolidge, J., The Elements of Non-Euliden Geometry, Oxford, Clrendon Press, 1 st ed (reprinted 1927). [4] Coxeter, H.M.S., Non-Euliden Geometry, 5 th ed. (reprinted), Univ. of Toronto Press, [5] Goodmn-Struss, C., Compss nd Strightedge in the Poinré Disk, m. Mth. Mon. 108 (2001), No.1, [6] Greenerg, M.J., Euliden nd Non-Euliden Geometries, 3 rd ed., New York, [7] Klein, F., Vorlesungen üer Niht-Euklidishe Geometrie, Springer, Berlin [8] Rihter-Geert, J., Kortenkmp, U. H., User Mnul for the Intertive Geometry Softwre Cinderell, Springer, [9] Shwiger, J., Gronu, D., The Poinré model of hyperoli geometry in n ritrry rel inner produt spe nd n elementry onstrution of hyperoli tringles with presried ngles, J. of Geometry, epted for pulition, [10] Stillwell, J., The four Pillrs of Geometry, Springer, [11] Stillwell, J., Soures of hyperoli geometry, m. Mth. So., Providene,

6 presented y MICK, Syille LINKS [12] Cstellnos, J., et l., NonEulid, NonEulid.html BOUT THE UTHOR Syille Mik, Dr., is memer of the Institute of Geometry t the Grz University of Tehnology. Her reserh interests regrd Non-Euliden Geometry nd Grphis Edutions. She n e rehed y e-mil: or y mil: Grz University of Tehnology, Institute of Geometry, Kopernikusgsse 24, Grz, ustri. 6

### State the size of angle x. Sometimes the fact that the angle sum of a triangle is 180 and other angle facts are needed. b y 127

ngles 2 CHTER 2.1 Tringles Drw tringle on pper nd lel its ngles, nd. Ter off its orners. Fit ngles, nd together. They mke stright line. This shows tht the ngles in this tringle dd up to 180 ut it is not

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

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

### 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

### 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

### 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

### 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

### Heron s Formula for Triangular Area

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,

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### How to Graphically Interpret the Complex Roots of a Quadratic Equation

Universit of Nersk - Linoln DigitlCommons@Universit of Nersk - Linoln MAT Em Epositor Ppers Mth in the Middle Institute Prtnership 7-007 How to Grphill Interpret the Comple Roots of Qudrti Eqution Crmen

### 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

### Radius of the Earth - Radii Used in Geodesy James R. Clynch Naval Postgraduate School, 2002

dius of the Erth - dii Used in Geodesy Jmes. Clynh vl Postgrdute Shool, 00 I. Three dii of Erth nd Their Use There re three rdii tht ome into use in geodesy. These re funtion of ltitude in the ellipsoidl

### 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,

### 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.

### 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?

### 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

### 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

### 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

### 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

### 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

### New combinatorial features for knots and virtual knots. Arnaud MORTIER

New omintoril fetures for knots nd virtul knots Arnud MORTIER April, 203 2 Contents Introdution 5. Conventions.................................... 9 2 Virtul knot theories 2. The lssil se.................................

### 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.

### 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.................................................

### Lecture 5. Inner Product

Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right

### 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

### 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

### 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.

### Section 5-5 Solving Right Triangles*

5-5 Solving Right Tringles 379 79. Geometry. The re of retngulr n-sided polygon irumsried out irle of rdius is given y A n tn 80 n (A) Find A for n 8, n 00, n,000, nd n 0,000. Compute eh to five deiml

### PHY 140A: Solid State Physics. Solution to Homework #2

PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.

### 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

### 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

### 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

### 4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A

Geometry: Shpes. Circumference nd re of circle HOMEWORK D C 3 5 6 7 8 9 0 3 U Find the circumference of ech of the following circles, round off your nswers to dp. Dimeter 3 cm Rdius c Rdius 8 m d Dimeter

### Clause Trees: a Tool for Understanding and Implementing Resolution in Automated Reasoning

Cluse Trees: Tool for Understnding nd Implementing Resolution in Automted Resoning J. D. Horton nd Brue Spener University of New Brunswik, Frederiton, New Brunswik, Cnd E3B 5A3 emil : jdh@un. nd spener@un.

### 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

### 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

### 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

### KEY SKILLS INFORMATION TECHNOLOGY Level 3. Question Paper. 29 January 9 February 2001

KEY SKILLS INFORMATION TECHNOLOGY Level 3 Question Pper 29 Jnury 9 Ferury 2001 WHAT YOU NEED This Question Pper An Answer Booklet Aess to omputer, softwre nd printer You my use ilingul ditionry Do NOT

### 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}.

### The Math Learning Center PO Box 12929, Salem, Oregon 97309 0929 Math Learning Center

Resource Overview Quntile Mesure: Skill or Concept: 1010Q Determine perimeter using concrete models, nonstndrd units, nd stndrd units. (QT M 146) Use models to develop formuls for finding res of tringles,

### Lecture 3: orientation. Computer Animation

Leture 3: orienttion Computer Animtion Mop tutoril sessions Next Thursdy (Feb ) Tem distribution: : - :3 - Tems 7, 8, 9 :3 - : - Tems nd : - :3 - Tems 5 nd 6 :3 - : - Tems 3 nd 4 Pper ssignments Pper ssignment

### . 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

### 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

### European Convention on Products Liability in regard to Personal Injury and Death

Europen Trety Series - No. 91 Europen Convention on Produts Liility in regrd to Personl Injury nd Deth Strsourg, 27.I.1977 The memer Sttes of the Counil of Europe, signtory hereto, Considering tht the

### 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

### 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

### Square Roots Teacher Notes

Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this

### Real Analysis HW 10 Solutions

Rel Anlysis HW 10 Solutions Problem 47: Show tht funtion f is bsolutely ontinuous on [, b if nd only if for eh ɛ > 0, there is δ > 0 suh tht for every finite disjoint olletion {( k, b k )} n of open intervls

### Right-angled triangles

13 13A Pythgors theorem 13B Clulting trigonometri rtios 13C Finding n unknown side 13D Finding ngles 13E Angles of elevtion nd depression Right-ngled tringles Syllus referene Mesurement 4 Right-ngled tringles

### PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * challenge questions

PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * hllenge questions e The ll will strike the ground 1.0 s fter it is struk. Then v x = 20 m s 1 nd v y = 0 + (9.8 m s 2 )(1.0 s) = 9.8 m s 1 The speed

### 4.11 Inner Product Spaces

314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

### Lesson 8.1 Areas of Rectangles and Parallelograms

Leon 8.1 Are of Rectngle nd Prllelogrm In Eercie 1 4, find the re of the hded region. 1.. 1 cm 1 cm. 17 cm 4. 9 cm 5 cm 1.5 cm 1 cm cm cm 5. Rectngle ABCD h re 684 m nd width 44 m. Find it length. 6. Drw

### NCERT INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS. Trigonometric Ratios of the angle A in a triangle ABC right angled at B are defined as:

INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS (A) Min Concepts nd Results Trigonometric Rtios of the ngle A in tringle ABC right ngled t B re defined s: side opposite to A BC sine of A = sin A = hypotenuse

### Drawing Diagrams From Labelled Graphs

Drwing Digrms From Lbelled Grphs Jérôme Thièvre 1 INA, 4, venue de l Europe, 94366 BRY SUR MARNE FRANCE Anne Verroust-Blondet 2 INRIA Rocquencourt, B.P. 105, 78153 LE CHESNAY Cedex FRANCE Mrie-Luce Viud

### 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

### - DAY 1 - Website Design and Project Planning

Wesite Design nd Projet Plnning Ojetive This module provides n overview of the onepts of wesite design nd liner workflow for produing wesite. Prtiipnts will outline the sope of wesite projet, inluding

### GENERAL OPERATING PRINCIPLES

KEYSECUREPC USER MANUAL N.B.: PRIOR TO READING THIS MANUAL, YOU ARE ADVISED TO READ THE FOLLOWING MANUAL: GENERAL OPERATING PRINCIPLES Der Customer, KeySeurePC is n innovtive prout tht uses ptente tehnology:

### Lesson 12.1 Trigonometric Ratios

Lesson 12.1 rigonometric Rtios Nme eriod Dte In Eercises 1 6, give ech nswer s frction in terms of p, q, nd r. 1. sin 2. cos 3. tn 4. sin Q 5. cos Q 6. tn Q p In Eercises 7 12, give ech nswer s deciml

### 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

### 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

### 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

### 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

### QUANTITATIVE REASONING

Guide For Exminees Inter-University Psychometric Entrnce Test QUNTITTIVE RESONING The Quntittive Resoning domin tests your bility to use numbers nd mthemticl concepts to solve mthemticl problems, s well

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

### Module 5. Three-phase AC Circuits. Version 2 EE IIT, Kharagpur

Module 5 Three-hse A iruits Version EE IIT, Khrgur esson 8 Three-hse Blned Suly Version EE IIT, Khrgur In the module, ontining six lessons (-7), the study of iruits, onsisting of the liner elements resistne,

### Rational Functions. Rational functions are the ratio of two polynomial functions. Qx bx b x bx b. x x x. ( x) ( ) ( ) ( ) and

Rtionl Functions Rtionl unctions re the rtio o two polynomil unctions. They cn be written in expnded orm s ( ( P x x + x + + x+ Qx bx b x bx b n n 1 n n 1 1 0 m m 1 m + m 1 + + m + 0 Exmples o rtionl unctions

### European Convention on Social and Medical Assistance

Europen Convention on Soil nd Medil Assistne Pris, 11.XII.1953 Europen Trety Series - No. 14 The governments signtory hereto, eing memers of the Counil of Europe, Considering tht the im of the Counil of

### 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

### Volumes of solids of revolution

Volumes of solids of revolution We sometimes need to clculte the volume of solid which cn be obtined by rotting curve bout the x-xis. There is strightforwrd technique which enbles this to be done, using

### 5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

### Intersection Problems

Intersetion Prolems Determine pirs of interseting ojets? C A B E D Complex shpes forme y oolen opertions: interset, union, iff. Collision etetion in rootis n motion plnning. Visiility, olusion, renering

### S-Parameters for Three and Four Two- Port Networks

the ehnology Interfe/pring 2007 Mus, diku, nd Akujuoi -Prmeters for hree nd Four wo- Port Networks rhn M. Mus, Mtthew N.O. diku, nd Cjetn M. Akujuoi Center of Exellene for Communition ystems ehnology Reserh

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics. W02D3_0 Group Problem: Pulleys and Ropes Constraint Conditions

MSSCHUSES INSIUE OF ECHNOLOGY Deprtment of hysics 8.0 W02D3_0 Group roblem: ulleys nd Ropes Constrint Conditions Consider the rrngement of pulleys nd blocks shown in the figure. he pulleys re ssumed mssless

### 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

### 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.

### CHAPTER 9: Moments of Inertia

HPTER 9: Moments of nerti! Moment of nerti of res! Second Moment, or Moment of nerti, of n re! Prllel-is Theorem! Rdius of Grtion of n re! Determintion of the Moment of nerti of n re ntegrtion! Moments

### 11.2 The Law of Sines

894 Applitions of Trigonometry 11. The Lw of Sines Trigonometry literlly mens mesuring tringles nd with Chpter 10 under our belts, we re more thn prepred to do just tht. The min gol of this setion nd the

### Integration by Substitution

Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is

### 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

### Trowel Notch Sizes for Installation of Floor Coverings, Wood Flooring and Tiles

TKB-Tehnil Briefing Note 6 Trowel Noth s for Instlltion of Floor Coverings, Wood Flooring nd Tiles Version: My 2007 Prepred y the Tehnishe Kommission Buklestoffe (TKB) (Tehnil Commission on Constrution

### Vector differentiation. Chapters 6, 7

Chpter 2 Vectors Courtesy NASA/JPL-Cltech Summry (see exmples in Hw 1, 2, 3) Circ 1900 A.D., J. Willird Gis invented useful comintion of mgnitude nd direction clled vectors nd their higher-dimensionl counterprts

### Geometry Made Easy Handbook Common Core Standards Edition

Geometry Made Easy Handbook ommon ore Standards Edition y: Mary nn asey. S. Mathematics, M. S. Education 2015 Topical Review ook ompany, Inc. ll rights reserved. P. O. ox 328 Onsted, MI. 49265-0328 This

### 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

### 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

### 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

GRADE Frtions WORKSHEETS Types of frtions equivlent frtions This frtion wll shows frtions tht re equivlent. Equivlent frtions re frtions tht re the sme mount. How mny equivlent frtions n you fin? Lel eh

### End-to-end development solutions

TECHNICAL SERVICES Endtoend development solutions Mnged y TFE HOTELS TFE Hotels re the only Austrlin Hotel group with inhouse end to end development solutions. We hve expertise in Arhiteturl nd Interior

### Double Integrals over General Regions

Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing

### Sine and Cosine Ratios. For each triangle, find (a) the length of the leg opposite lb and (b) the length of the leg adjacent to lb.

- Wht You ll ern o use sine nd osine to determine side lengths in tringles... nd Wh o use the sine rtio to estimte stronomil distnes indiretl, s in Emple Sine nd osine tios hek Skills You ll Need for Help

### Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;

B-26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndom-numer genertor supplied s stndrd with ll computer systems Stn Kelly-Bootle,

### Version 001 CIRCUITS holland (1290) 1

Version CRCUTS hollnd (9) This print-out should hve questions Multiple-choice questions my continue on the next column or pge find ll choices efore nswering AP M 99 MC points The power dissipted in wire

### 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.

### Answers to Exercises ABC ABC

nswers to ercises HTR HTR LSSON. HTR HTR. postulte is sttement ccepted s true without proof. theorem is deduced from other theorems or postultes.. Sutrction: quls minus equls re equl. Multipliction: quls

### MODULE 3. 0, y = 0 for all y

Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)