Three squares with sides 3, 4, and 5 units are used to form the right triangle shown. In a right triangle, the sides have special names.


 Cathleen Fox
 1 years ago
 Views:
Transcription
1 1 The Pythgoren Theorem MAIN IDEA Find length using the Pythgoren Theorem. New Voulry leg hypotenuse Pythgoren Theorem Mth Online glenoe.om Extr Exmples Personl Tutor SelfChek Quiz Three squres with sides 3, 4, nd 5 units re used to form the right tringle shown. 3 units 1. Find the re of eh squre.. How re the squres of the sides relted to the res of the squres? 3. Find the sum of the res of the two smller squres. How does the sum ompre to the re of the lrger squre? 4. Use grid pper to ut out three squres 4 units with sides 5, 1, nd 13 units. Form right tringle with these squres. Compre the sum of the res of the two smller squres with the re of the lrger squre. 5 units In right tringle, the sides hve speil nmes. The two sides tht form the right ngle re the legs. The side opposite the right ngle is the hypotenuse. It is the longest side of the tringle. The Pythgoren Theorem desries the reltionship etween the length of the hypotenuse nd the lengths of the legs. Pythgoren Theorem Key Conept Words In right tringle, the squre of the length of the hypotenuse equls the sum of the squres of the lengths of the legs. Model Symols = + When using the Pythgoren Theorem, you will enounter equtions tht involve squre roots. Every positive numer hs oth positive nd negtive squre root. By the definition of squre roots, if n =, then n = ±. The nottion ± indites oth the positive nd negtive squre root of numer. You n use this reltionship to solve equtions tht involve squres. 640 Chpter 1 Geometry nd Mesurement
2 Find the Length of the Hypotenuse 1 Find the length of the hypotenuse of the tringle. = + =8 +4 Pythgoren Theorem Reple with 8 nd with 4. = Evlute 8 nd 4. = 80 Add. = ± 80 Definition of squre root ± 8.9 Simplify. 8 ft ft 4 ft The length of the hypotenuse is out 8.9 feet.. Find the length of the hypotenuse of right tringle with legs 5 yrds nd 7 yrds. Round to the nerest tenth. SCUBA DIVING A su diver dove 14 feet elow the surfe. Then, he swm 16 feet towrd orl formtion. How fr is the diver from his ot? The diver s distne from the ot is the hypotenuse of right tringle. Write nd solve n eqution for x. = + x = Pythgoren Theorem Reple with x, with 14, nd with 16. x = Evlute 14 nd 16. x = 45 Add. x = ± 45 Definition of squre root x ±1.3 Simplify. The diver s distne from the ot is out 1.3 feet. nd se 60 ft mesuring 60 feet on eh side. How fr does plyer on seond se throw when she throws from seond se to home? Round to the nerest tenth. ft. SOFTBALL A softll dimond is squre 60 Chek for Resonleness You n eliminte 8.9 s solution euse the length of side of tringle nnot e negtive numer. home Lesson 1 The Pythgoren Theorem 641
3 You n lso use the Pythgoren Theorem to find the mesure of leg if the mesure of the other leg nd the hypotenuse re known. Find the Length of Leg 3 Find the missing mesure of the tringle. 13 m Round to the nerest tenth if neessry. The missing mesure is the length of leg. = + Pythgoren Theorem 13 = 5 + Reple with 5 nd with = 5 + Evlute 13 nd =  5 Sutrt 5 from eh side. 144 = Simplify. ± 144 = Definition of squre root 1 = Simplify. The length of the leg is 1 entimeters. m 5 m. ft 17 ft d. 9. m 4 m m e. = 7 in., = 5 in. 4 in. 8 ft 15 ft 8.3 m Formuls Some formuls will e given to you during the test. It is good ide to fmilirize yourself with the formuls efore the test. 4 Mr. Thomson reted mosi tile in the shpe of squre to ple in his kithen. 9 in. 9 in. Whih is losest to the length of the digonl of the tile? A 10 in. C 15 in. B 13 in. D 17 in. Red the Item You need to use the Pythgoren Theorem to find the length of the digonl. 64 Chpter 1 Geometry nd Mesurement
4 Solve the Item = + Pythgoren Theorem = Reple with 9 nd with 9. = Evlute 9 nd 9. = 16 Add. = ± 16 Definition of squre root ±1.7 Simplify. The length is out 1.7 inhes. The nswer hoie losest to 1.7 inhes is 13 inhes. So, the nswer is B. f. A pinter lens ldder ginst the side of uilding. How fr from the ottom of the uilding is the top of the ldder? F F 38. ft H 1.8 ft G 8.0 ft J 0.0 ft 40 ft 1 ft indites multistep prolem Exmples 1, 3 (pp ) Find the missing mesure of eh tringle. Round to the nerest tenth if neessry. 1. mm 6 mm 4 mm 10 mm. 19 in. 4.5 in. 31 in. in. 3. = 1 m, = 8 m 4. = 11 yd, = 1 yd m yd Exmple (p. 641) 5. ARCHITECTURE Wht is the width of the the fene gte shown t the right? Round to the nerest tenth. 4.0 ft.5 ft 4.7 ft Exmple 4 (pp ) 6. MULTIPLE CHOICE A ompny designed puli ply re in the shpe of squre. The ply re will inlude pthwy, s shown. Whih is losest to the length of the pthwy? C 100 yd A 100 yd B 15 yd C 140 yd D 175 yd 100 yd Lesson 1 The Pythgoren Theorem 643
5 HOMEWORK For Exerises 7 8, 11 1, , HELP See Exmples Find the missing mesure of eh tringle. Round to the nerest tenth if neessry in. 8. m 8m 9. 5m 1 in. 8 in. m 14 m m 4 15 m 1. m 4.6 ft ft mm 8.9 mm 11.5 m 8.9 mm.8 ft 13. =.4 yd, = 3.7 yd 14. = 8.5 m, = 10.4 m 15. = 7 in., = 4 in. 16. = 13.5 mm, = 18 mm MEASUREMENT For Exerises 17 nd 18, find eh distne to the nerest tenth mi shool nk 4.6 mi x mi 14.5 ft x ft 1.8 ft store SPORTS For Exerises 19 nd 0, find the length or width of eh piee of sports equipment. Round to the nerest tenth x in in. in in. in. x in. 1. MEASUREMENT A rn door is 10 feet wide nd 15 feet tll. A squre plnk 16 feet on eh side must e tken through the doorwy. Cn the plnk fit through the doorwy? Justify your nswer. 15 ft. MEASUREMENT On weekend trip round Cliforni, EXTRA PRACTICE See pges 70, Sydney left her home in Modesto nd drove 75 miles est to Yosemite Ntionl Prk, then 70 miles south to Fresno, nd finlly 110 miles west to Monterey By. Aout how fr is she from her strting point? Justify your nswer with drwing. Chpter 1 Geometry nd Mesurement 10 ft
6 H.O.T. Prolems 3. CHALLENGE Wht is the length of the digonl shown in the ue t the right? x in. 6 in. 4. FIND THE ERROR Mrus nd Aish re writing n eqution to find the missing mesure of the tringle t the right. Who is orret? Explin. 8 m 1 m x m 1 = 8 + x x = WRITING IN Mrus Aish 5. MATH Write prolem out relworld sitution in whih you would use the Pythgoren Theorem. 6. Whih tringle hs sides,, nd so tht the reltionship + = is true? A B C D 7. An isoseles right tringle hs legs tht re eh 8 inhes long. Aout how long is the hypotenuse? F 1.8 inhes G 11.3 inhes H 8 inhes J 4 inhes 8. ESTIMATION Whih is loser to 55 : 7 or 8? (Lesson 11) 9. MEASUREMENT A ylindershped poporn tin hs height of 1.5 feet nd dimeter of 10 inhes. Find the volume to the nerest ui inh. (Lesson 1110) Write eh perent s deiml. (Lesson 47) % 31. 8% 3. 14% % 34. PREREQUISITE SKILL The verge person tkes out 15 reths per minute. At this rte, how mny reths does the verge person tke in one week? Use the solve simpler prolem strtegy. (Lesson 115) Lesson 1 The Pythgoren Theorem 645
The area of the larger square is: IF it s a right triangle, THEN + =
8.1 Pythgoren Theorem nd 2D Applitions The Pythgoren Theorem sttes tht IF tringle is right tringle, THEN the sum of the squres of the lengths of the legs equls the squre of the hypotenuse lengths. Tht
More informationThe 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 information4.5 The Converse of the
Pge 1 of. The onverse of the Pythgoren Theorem Gol Use the onverse of Pythgoren Theorem. Use side lengths to lssify tringles. Key Words onverse p. 13 grdener n use the onverse of the Pythgoren Theorem
More information11. PYTHAGORAS THEOREM
11. PYTHAGORAS THEOREM 111 Along the Nile 2 112 Proofs of Pythgors theorem 3 113 Finding sides nd ngles 5 114 Semiirles 7 115 Surds 8 116 Chlking hndll ourt 9 117 Pythgors prolems 10 118 Designing
More information81. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary
81 The Pythgoren Theorem nd Its Converse Voulry Review 1. Write the squre nd the positive squre root of eh numer. Numer Squre Positive Squre Root 9 81 3 1 4 1 16 1 2 Voulry Builder leg (noun) leg Relted
More informationEssential Question What are the Law of Sines and the Law of Cosines?
9.7 TEXS ESSENTIL KNOWLEDGE ND SKILLS G.6.D Lw of Sines nd Lw of osines Essentil Question Wht re the Lw of Sines nd the Lw of osines? Disovering the Lw of Sines Work with prtner.. opy nd omplete the tle
More informationKnow the sum of angles at a point, on a straight line and in a triangle
2.1 ngle sums Know the sum of ngles t point, on stright line n in tringle Key wors ngle egree ngle sum n ngle is mesure of turn. ngles re usully mesure in egrees, or for short. ngles tht meet t point mke
More informationThank you for participating in Teach It First!
Thnk you for prtiipting in Teh It First! This Teh It First Kit ontins Common Core Coh, Mthemtis teher lesson followed y the orresponding student lesson. We re onfident tht using this lesson will help you
More informationLesson 18.2: Right Triangle Trigonometry
Lesson 8.: Right Tringle Trigonometry lthough Trigonometry is used to solve mny prolems, historilly it ws first pplied to prolems tht involve right tringle. This n e extended to nonright tringles (hpter
More informationSECTION 72 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 informationLesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle
: Using Trigonometry to Find Side Lengths of n Aute Tringle Clsswork Opening Exerise. Find the lengths of d nd e.. Find the lengths of x nd y. How is this different from prt ()? Exmple 1 A surveyor needs
More informationGeometry 71 Geometric Mean and the Pythagorean Theorem
Geometry 71 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 informationSect 8.3 Triangles and Hexagons
13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed twodimensionl geometric figure consisting of t lest three line segments for its
More informationThe Pythagorean Theorem Tile Set
The Pythgoren Theorem Tile Set Guide & Ativities Creted y Drin Beigie Didx Edution 395 Min Street Rowley, MA 01969 www.didx.om DIDAX 201 #211503 1. Introdution The Pythgoren Theorem sttes tht in right
More informationSOLVING EQUATIONS BY FACTORING
316 (560) 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 informationFinal Exam covers: Homework 0 9, Activities 1 20 and GSP 1 6 with an emphasis on the material covered after the midterm exam.
MTH 494.594 / FINL EXM REVIEW Finl Exm overs: Homework 0 9, tivities 1 0 nd GSP 1 6 with n emphsis on the mteril overed fter the midterm exm. You my use oth sides of one 3 5 rd of notes on the exm onepts
More informationTHE PYTHAGOREAN THEOREM
THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most wellknown nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this
More informationSquare 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
More informationPYTHAGORAS THEOREM. Answers. Edexcel GCSE Mathematics (Linear) 1MA0
Edexel GSE Mthemtis (Liner) 1M0 nswers PYTHGORS THEOREM Mterils required for exmintion Ruler grduted in entimetres nd millimetres, protrtor, ompsses, pen, H penil, erser. Tring pper my e used. Items inluded
More informationProving the Pythagorean Theorem
Proving the Pythgoren Theorem Proposition 47 of Book I of Eulid s Elements is the most fmous of ll Eulid s propositions. Disovered long efore Eulid, the Pythgoren Theorem is known y every high shool geometry
More informationFor the Final Exam, you will need to be able to:
Mth B Elementry Algebr Spring 0 Finl Em Study Guide The em is on Wednesdy, My 0 th from 7:00pm 9:0pm. You re lloed scientific clcultor nd " by 6" inde crd for notes. On your inde crd be sure to rite ny
More informationEnd 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 informationState 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
More informationRight Triangle Trigonometry 8.7
304470_Bello_h08_se7_we 11/8/06 7:08 PM Pge R1 8.7 Right Tringle Trigonometry R1 8.7 Right Tringle Trigonometry T E G T I N G S T R T E D The origins of trigonometry, from the Greek trigonon (ngle) nd
More informationAngles 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
More informationAngles 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 informationD e c i m a l s DECIMALS.
D e i m l s DECIMALS www.mthletis.om.u Deimls DECIMALS A deiml numer is sed on ple vlue. 214.84 hs 2 hundreds, 1 ten, 4 units, 8 tenths nd 4 hundredths. Sometimes different 'levels' of ple vlue re needed
More informationSOLVING 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 informationRight Triangle Trigonometry
CONDENSED LESSON 1.1 Right Tringle Trigonometr In this lesson ou will lern out the trigonometri rtios ssoited with right tringle use trigonometri rtios to find unknown side lengths in right tringle use
More informationMATH 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 information9.1 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 informationThe Parallelogram Law. Objective: To take students through the process of discovery, making a conjecture, further exploration, and finally proof.
The Prllelogrm Lw Objective: To tke students through the process of discovery, mking conjecture, further explortion, nd finlly proof. I. Introduction: Use one of the following Geometer s Sketchpd demonstrtion
More informationChapter. 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 informationBasic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }
ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All
More informationSimple Electric Circuits
Simple Eletri Ciruits Gol: To uild nd oserve the opertion of simple eletri iruits nd to lern mesurement methods for eletri urrent nd voltge using mmeters nd voltmeters. L Preprtion Eletri hrges move through
More information8.2 Trigonometric Ratios
8.2 Trigonometri Rtios Ojetives: G.SRT.6: Understnd tht y similrity, side rtios in right tringles re properties of the ngles in the tringle, leding to definitions of trigonometri rtios for ute ngles. For
More informationChess and Mathematics
Chess nd Mthemtis in UK Seondry Shools Dr Neill Cooper Hed of Further Mthemtis t Wilson s Shool Mnger of Shool Chess for the English Chess Federtion Mths in UK Shools KS (up to 7 yers) Numers: 5 + 7; x
More informationPythagoras theorem is one of the most popular theorems. Paper Folding And The Theorem of Pythagoras. Visual Connect in Teaching.
in the lssroom Visul Connet in Tehing Pper Folding And The Theorem of Pythgors Cn unfolding pper ot revel proof of Pythgors theorem? Does mking squre within squre e nything more thn n exerise in geometry
More informationTwo special Righttriangles 1. The
Mth Right Tringle Trigonometry Hndout B (length of )  c  (length of side ) (Length of side to ) Pythgoren s Theorem: for tringles with right ngle ( side + side = ) + = c Two specil Righttringles. The
More informationThe 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,
More informationUse 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 information1. 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 informationRatio 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 informationThe theorem of. Pythagoras. Opening problem
The theorem of 8 Pythgors ontents: Pythgors theorem [4.6] The onverse of Pythgors theorem [4.6] Prolem solving [4.6] D irle prolems [4.6, 4.7] E Threedimensionl prolems [4.6] Opening prolem The Louvre
More informationexcenters and excircles
21 onurrene IIi 2 lesson 21 exenters nd exirles In the first lesson on onurrene, we sw tht the isetors of the interior ngles of tringle onur t the inenter. If you did the exerise in the lst lesson deling
More informationIntroduction. Law of Cosines. a 2 b2 c 2 2bc cos A. b2 a 2 c 2 2ac cos B. c 2 a 2 b2 2ab cos C. Example 1
3330_060.qxd 1/5/05 10:41 M Pge 439 Setion 6. 6. Lw of osines 439 Lw of osines Wht you should lern Use the Lw of osines to solve olique tringles (SSS or SS). Use the Lw of osines to model nd solve rellife
More informationWords 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 information10.3 Systems of Linear Equations: Determinants
758 CHAPTER 10 Systems of Equtions nd Inequlities 10.3 Systems of Liner Equtions: Determinnts OBJECTIVES 1 Evlute 2 y 2 Determinnts 2 Use Crmer s Rule to Solve System of Two Equtions Contining Two Vriles
More informationSection 55 Solving Right Triangles*
55 Solving Right Tringles 379 79. Geometry. The re of retngulr nsided 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
More informationNapoleon and Pythagoras with Geometry Expressions
Npoleon nd Pythgors with eometry xpressions NPOLON N PYTORS WIT OMTRY XPRSSIONS... 1 INTROUTION... xmple 1: Npoleon s Theorem... 3 xmple : n unexpeted tringle from Pythgorslike digrm... 5 xmple 3: Penequilterl
More information5.6 The Law of Cosines
44 HPTER 5 nlyti Trigonometry 5.6 The Lw of osines Wht you ll lern out Deriving the Lw of osines Solving Tringles (SS, SSS) Tringle re nd Heron s Formul pplitions... nd why The Lw of osines is n importnt
More informationThe Quadratic Formula and the Discriminant
99 The Qudrtic Formul nd the Discriminnt Objectives Solve qudrtic equtions by using the Qudrtic Formul. Determine the number of solutions of qudrtic eqution by using the discriminnt. Vocbulry discriminnt
More informationLet us recall some facts you have learnt in previous grades under the topic Area.
6 Are By studying this lesson you will be ble to find the res of sectors of circles, solve problems relted to the res of compound plne figures contining sectors of circles. Ares of plne figures Let us
More information4 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
More informationContent Objectives: After completing the activity, students will gain experience of informally proving Pythagoras Theorem
Pythgors Theorem S Topic 1 Level: Key Stge 3 Dimension: Mesures, Shpe nd Spce Module: Lerning Geometry through Deductive Approch Unit: Pythgors Theorem Student ility: Averge Content Ojectives: After completing
More informationCHAPTER 4: POLYGONS AND SOLIDS. 3 Which of the following are regular polygons? 4 Draw a pentagon with equal sides but with unequal angles.
Mthemtis for Austrli Yer 6  Homework POLYGONS AND SOLIDS (Chpter 4) CHAPTER 4: POLYGONS AND SOLIDS 4A POLYGONS 3 Whih of the following re regulr polygons? A polygon is lose figure whih hs only stright
More informationThe 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 informationSection 54 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 informationGRADE 4. Fractions WORKSHEETS
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
More informationEXAMPLE EXAMPLE. Quick Check EXAMPLE EXAMPLE. Quick Check. EXAMPLE RealWorld Connection EXAMPLE
 Wht You ll Lern To use the Pthgoren Theorem To use the onverse of the Pthgoren Theorem... nd Wh To find the distne etween two doks on lke, s in Emple The Pthgoren Theorem nd Its onverse hek Skills You
More information2 If a branch is prime, no other factors
Chpter 2 Multiples, nd primes 59 Find the prime of 50 by drwing fctor tree. b Write 50 s product of its prime. 1 Find fctor pir of the given 50 number nd begin the fctor tree (50 = 5 10). 5 10 2 If brnch
More informationQuadratic Equations  1
Alger Module A60 Qudrtic Equtions  1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions  1 Sttement of Prerequisite
More informationReasoning 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 informationActivity I: Proving the Pythagorean Theorem (Grade Levels: 69)
tivity I: Proving the Pythgoren Theorem (Grde Levels: 69) Stndrds: Stndrd 7: Resoning nd Proof Ojetives: The Pythgoren theorem n e proven using severl different si figures. This tivity introdues student
More informationAddition and subtraction of rational expressions
Lecture 5. Addition nd subtrction of rtionl expressions Two rtionl expressions in generl hve different denomintors, therefore if you wnt to dd or subtrct them you need to equte the denomintors first. The
More informationQuadrilaterals Here are some examples using quadrilaterals
Qudrilterls Here re some exmples using qudrilterls Exmple 30: igonls of rhomus rhomus hs sides length nd one digonl length, wht is the length of the other digonl? 4  Exmple 31: igonls of prllelogrm Given
More information2.1 ANGLES AND THEIR MEASURE. y I
.1 ANGLES AND THEIR MEASURE Given two interseting lines or line segments, the mount of rottion out the point of intersetion (the vertex) required to ring one into orrespondene with the other is lled the
More informationRightangled triangles
13 13A Pythgors theorem 13B Clulting trigonometri rtios 13C Finding n unknown side 13D Finding ngles 13E Angles of elevtion nd depression Rightngled tringles Syllus referene Mesurement 4 Rightngled tringles
More informationLines and angles. Name. Use a ruler and pencil to draw: a 2 parallel lines. c 2 perpendicular lines. b 2 intersecting lines. Complete the following:
Lines nd s 1 Use ruler nd pencil to drw: 2 prllel lines 2 intersecting lines c 2 perpendiculr lines 2 Complete the following: drw in the digonls on this shpe mrk the interior s on this shpe c mrk equl
More informationa 2 + b 2 = c 2. There are many proofs of this theorem. An elegant one only requires that we know that the area of a square of side L is L 2
Pythgors Pythgors A right tringle, suh s shown in the figure elow, hs one 90 ngle. The long side of length is the hypotenuse. The short leg (or thetus) hs length, nd the long leg hs length. The theorem
More informationRight Triangle Trigonometry for College Algebra
Right Tringle Trigonometry for ollege Alger B A sin os A = = djent A = = tn A = = djent sin B = = djent os B = = tn B = = djent ontents I. Bkground nd Definitions (exerises on pges 34) II. The Trigonometri
More informationThis unit will help you to calculate perimeters and areas of circles and sectors, and to find the radius given the circumference or area.
Get strte 1 Cirles This unit will help you to lulte perimeters n res of irles n setors, n to fin the rius given the irumferene or re. AO1 Flueny hek 1 Roun 4.635 to 2 eiml ples (.p.) 2 Roun 5.849 to 1.p.
More informationChapter15 SAMPLE. Simultaneous equations. Contents: A B C D. Graphical solution Solution by substitution Solution by elimination Problem solving
Chpter15 Simultneous equtions Contents: A B C D Grphil solution Solution y sustitution Solution y elimintion Prolem solving 308 SIMULTANEOUS EQUATIONS (Chpter 15) Opening prolem Ewen wnts to uy pie, ut
More informationUnit 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 informationInterior 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 informationLesson 18.3: Triangle Trigonometry ( ) : OBTUSE ANGLES
Lesson 1.3: Tringle Trigonometry We now extend te teory of rigt tringle trigonometry to nonrigt or olique tringles. Of te six omponents wi form tringle, tree sides nd tree ngles, te possiilities for omintion
More informationIt may be helpful to review some right triangle trigonometry. Given the right triangle: C = 90º
Ryn Lenet Pge 1 Chemistry 511 Experiment: The Hydrogen Emission Spetrum Introdution When we view white light through diffrtion grting, we n see ll of the omponents of the visible spetr. (ROYGBIV) The diffrtion
More informationProblem Set 2 Solutions
University of Cliforni, Berkeley Spring 2012 EE 42/100 Prof. A. Niknej Prolem Set 2 Solutions Plese note tht these re merely suggeste solutions. Mny of these prolems n e pprohe in ifferent wys. 1. In prolems
More informationPYTHAGORAS THEOREM 8YEARS. A guide for teachers  Years 8 9. The Improving Mathematics Education in Schools (TIMES) Project
The Improving Mthemtis Edution in Shools (TIMES) Projet PYTHGORS THEOREM guide for tehers  Yers 8 9 MESUREMENT ND GEOMETRY Module 15 June 2011 8YERS 9 Pythgors theorem (Mesurement nd Geometry: Module
More informationLesson 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
More informationLesson 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 informationWorksheet 24: Optimization
Worksheet 4: Optimiztion Russell Buehler b.r@berkeley.edu 1. Let P 100I I +I+4. For wht vlues of I is P mximum? P 100I I + I + 4 Tking the derivtive, www.xkcd.com P (I + I + 4)(100) 100I(I + 1) (I + I
More informationThe AVL Tree Rotations Tutorial
The AVL Tree Rottions Tutoril By John Hrgrove Version 1.0.1, Updted Mr222007 Astrt I wrote this doument in n effort to over wht I onsider to e drk re of the AVL Tree onept. When presented with the tsk
More informationGeometry Notes SIMILAR TRIANGLES
Similr Tringles Pge 1 of 6 SIMILAR TRIANGLES Objectives: After completing this section, you shoul be ble to o the following: Clculte the lengths of sies of similr tringles. Solve wor problems involving
More informationAppendix 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 informationAssuming 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;
B26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndomnumer genertor supplied s stndrd with ll computer systems Stn KellyBootle,
More information8. Hyperbolic triangles
8. Hyperoli tringles Note: This yer, I m not doing this mteril, prt from Pythgors theorem, in the letures (nd, s suh, the reminder isn t exminle). I ve left the mteril s Leture 8 so tht (i) nyody interested
More informationExample
6. SOLVING RIGHT TRINGLES In the right tringle B shwn in Figure 6.1, the ngles re dented y α t vertex, β t vertex B, nd t vertex. The lengths f the sides ppsite the ngles α, β, nd re dented y,, nd. Nte
More informationProving the Pythagorean Theorem
CONCEPT DEVELOPMENT Mthemtis Assessment Projet CLASSROOM CHALLENGES A Formtive Assessment Lesson Proving the Pythgoren Theorem Mthemtis Assessment Resoure Servie University of Nottinghm & UC Berkeley For
More informationFractions: Arithmetic Review
Frtions: Arithmeti Review Frtions n e interprete s rtios omprisons of two quntities. For given numer expresse in frtion nottion suh s we ll the numertor n the enomintor n it is helpful to interpret this
More informationTriangles, Altitudes, and Area Instructor: Natalya St. Clair
Tringle, nd ltitudes erkeley Mth ircles 015 Lecture Notes Tringles, ltitudes, nd re Instructor: Ntly St. lir *Note: This M session is inspired from vriety of sources, including wesomemth, reteem Mth Zoom,
More informationSurface Area and Volume
Surfce Are nd Volume Student Book  Series J Mthletics Instnt Workooks Copyright Surfce re nd volume Student Book  Series J Contents Topics Topic  Surfce re of right prism Topic 2  Surfce re of right
More information9 LENGTH, AREA AND VOLUME
E 9E Are of prllelogrm 9B Perimeter 9F Are of tringle M PL 9A Unerstning length 9C Unerstning re 9G Surfe re 9D Are of retngle 9H Volume n pity SA MEASUREMENT AND GEOMETRY 9 LENGTH, AREA AND VOLUME E SS
More informationLecture 15  Curve Fitting Techniques
Lecture 15  Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting  motivtion For root finding, we used given function to identify where it crossed zero where does fx
More informationPrealgebra 7* In your group consider the following problems:
Prelger * Group Activit # Group Memers: In our group consider the following prolems: 1) If ever person in the room, including the techer, were to shke hnds with ever other person ectl one time, how mn
More informationPROJECTILE 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
More informationAREA 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 informationHow 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 7007 How to Grphill Interpret the Comple Roots of Qudrti Eqution Crmen
More informationChapter 10 Geometry: Angles, Triangles and Distance
hpter 10 Geometry: ngles, Tringles nd Distne In setion 1 we egin y gthering together fts out ngles nd tringles tht hve lredy een disussed in previous grdes. This time the ide is to se student understnding
More information15. Let f (x) = 3x Suppose rx 2 + sx + t = 0 where r 0. Then x = 24. Solve 5x 25 < 20 for x. 26. Let y = 7x
Pretest Review The pretest will onsist of 0 problems, eh of whih is similr to one of the following 49 problems If you n do problems like these 49 listed below, you will hve no problem with the pretest
More information