4.3 The Sine Law and the Cosine Law

Size: px
Start display at page:

Download "4.3 The Sine Law and the Cosine Law"

Transcription

1 4.3 Te Sine Lw nd te osine Lw Te ee Tower is te tllest prt of nd s rliment uildings. ronze mst, wi flies te ndin flg, stnds on top of te ee Tower. From point 25 m from te foot of te tower, te ngle of elevtion of te top of te tower is From te sme point, te ngle of elevtion of te top of te mst is INVESTIGTE & INQUIRE To find te eigt of te mst, use te digrm sown. Te given informtion is mrked on te digrm. is te eigt of te mst, nd is te eigt of te tower Find te eigt of te mst, to te nerest tent of metre, using only rigt tringles m 2. ) List mesurements you would use to find te eigt of te mst using te osine lw in. ) Find tese mesurements. ) Use te osine lw to find te eigt of te mst, to te nerest tent of metre. 3. ) List mesurements you would use to find te eigt of te mst using te sine lw in. ) Find tese mesurements. ) Use te sine lw to find te eigt of te mst, to te nerest tent of metre. 4. ompre your nswers from questions 1, 2, nd 3. Wi metod did you prefer? Eplin. 4.3 Te Sine Lw nd te osine Lw MHR 283

2 You ve previously pplied te sine lw nd te osine lw to ute tringles. You ve seen tt te sine lw nd te osine lw lso pply to otuse tringles. Te sine lw for ute nd otuse tringles n e developed s follows. In, drw perpendiulr to, or to etended. is te ltitude or eigt,, of. ute Tringle Otuse Tringle (180 ) In, = sin In, = sin(180 ) Rell tt sin (180 θ) = sin θ. = sin = sin = sin In, = sin In, = sin = sin For ot te ute nd te otuse tringles, sin = sin sin sin ivide ot sides y : = = sin sin sin Simplify: = y drwing te ltitude from, we n similrly sow tt sin sin = omining te results gives te following forms of te sine lw. sin sin sin = = = = sin sin sin 284 MHR pter 4

3 EXMLE 1 Te Sine Lw, Given Two ngles nd Side In RST, S = 40, T = 21, nd r = 46 m. Find t, to te nerest entimetre. SOLUTION rw digrm. Find te mesure of R. R = = 119 Use te sine lw to find t. t r = sin T sin R S 40 R 46 m 21 T t sin 21 = 46 sin sin 21 t = sin 119 t = 19 So, t = 19 m, to te nerest entimetre. EXMLE 2 Te Sine Lw, Given Two Sides nd te ngle Opposite One of Tem In QR, = 105.2, p = 23.2 m, nd r = 18.5 m. Solve te tringle, rounding te side lengt to te nerest tent of entimetre nd te ngles to te nerest tent of degree, if neessry. SOLUTION rw digrm. Use te sine lw to find te mesure of R m sin R r = sin p Q 23.2 m R sin R sin = sin sin R = 23.2 R = Te Sine Lw nd te osine Lw MHR 285

4 Find te mesure of Q. Q = = 24.5 Use te sine lw to find q. q p = sin Q sin q sin = sin sin 24.5 q = sin = 10.0 In QR, R = 50.3, Q = 24.5, nd q = 10.0 m. Te osine lw for ute nd otuse tringles n e developed s follows. In, drw perpendiulr to, or to etended. is te ltitude or eigt,, of. ute Tringle In, = os = os nd 2 = In, 2 = 2 + ( ) 2 = = 2 + ( ) 2 2 = os 286 MHR pter 4

5 Otuse Tringle (180 ) + In, = os (180 ) = os (180 ) Rell tt os (180 θ) = os θ. = os nd 2 = In, 2 = 2 + ( + ) 2 = = 2 + ( ) + 2 = ( os ) 2 = os Te forms of te osine lw re s follows. 2 = os os = 2 2 = os os = 2 2 = os os = 2 EXMLE 3 Te osine Lw, Given Tree Sides In, = 9.6 m, = 20.6 m, nd = 14.7 m. Solve te tringle. Round e ngle mesure to te nerest tent of degree. 4.3 Te Sine Lw nd te osine Lw MHR 287

6 SOLUTION rw digrm. Use te osine lw to find te mesure of n ngle os = = 2(9.6)(14.7) = m 20.6 m 14.7 m Use te sine lw to find te mesure of. sin sin = sin sin = sin sin = 20.6 = 25.1 Find. = = 40.6 In, = 25.1, = 114.3, nd = EXMLE 4 Te osine Lw, Given Two Sides nd te ontined ngle Find te lengt of, to te nerest tent of metre. E m F 3.9 m 288 MHR pter 4

7 SOLUTION Use EF nd te osine lw to find te lengt of E. E 2 = F 2 + EF 2 2(F)(EF)os F = (3.9)(2.5)os 97.4 E = 4.9 Use E to find te mesure of. = = 75.1 Use E nd te sine lw to find te lengt of. 4.9 = sin 56.7 sin sin 56.7 = sin 75.1 = 4.2 = 4.2 m, to te nerest tent of metre. Key onepts Te forms of te sine lw re = = sin sin sin sin sin sin = = Te sine lw n e used to solve ny tringle wen given ) te mesures of two ngles nd ny side ) te mesures of two sides nd te ngle opposite one of tese sides Te forms of te osine lw re 2 = os os = 2 2 = os os = 2 2 = os os = 2 Te osine lw n e used to solve ny tringle wen given ) te mesures of two sides nd te ontined ngle ) te mesures of tree sides 4.3 Te Sine Lw nd te osine Lw MHR 289

8 ommunite Your Understnding 1. esrie ow you would solve e of te following tringles. Justify your osen metod. ) ) 24 7 m 8 m 18 m E m ) U d) X 10 m m 33 m Y 114 T 18 m S F Z 2. Eplin wy you nnot strt wit te sine lw to solve XYZ. 31 m X m Y Z 3. Eplin wy you nnot strt wit te osine lw to solve KLM. K 3.5 m L M rtise 1. Find te lengt of te indited side, to te nerest tent. ) R ) E 31.2 s 15.2 g 82 G 63 T S F ) d) E F f 290 MHR pter 4

9 e) f) M K k 41.1 L Q R 101 r 8.3 d) In UVW, W = 123.9, V = 22.2, v = 27.5 km. e) In XYZ, X = 92.3, y = 3.1 m, z = 2.8 m. f) In FGH, f = 12.6 m, g = 8.5 m, = 6.3 m. 2. Find te mesure of te indited ngle, to te nerest tent of degree. ) ) 27.3 K J L Find te lengt of te indited side, to te nerest tent. ) 13 m ) X d) L ) 16 mm 17.5 Y Z N 16.8 M 21 mm mm e) R f) Q Solve e tringle. Round nswers to te nerest tent, if neessry. ) In, = 84, = 40, = 5.6 m. ) In QR, R = 28.5, p = 10.4 m, r = 6.3 m. ) In LMN, M = 62, l = 16.9 m, n = 15.1 m. G F E ) d) 10 m m Q 51.8 R S 4.3 Te Sine Lw nd te osine Lw MHR 291

10 e) E 9.5 m F f) m 50.1 G H 5. Find te mesure of te indited ngle, to te nerest tent of degree. ) ) 4.3 m 2.7 m θ m 12.3 m θ m 9.8 m 7.3 m Q 6.5 m 84.3 S 91.7 ) d) 25 θ 10 m 20 m 100 m 104 θ m m R pply, Solve, ommunite 6. Solve. Round nswers to te nerest tent. 7. Mesurement n isoseles tringle s two 5.5-m sides nd two 32.4 ngles. Find ) te perimeter of te tringle, to te nerest tent of entimetre ) te re of te tringle, to te nerest tent of squre entimetre 8. Inquiry/rolem Solving irport X is 150 km est of irport Y. n irrft is 240 km from irport Y, nd 23 nort of due west from irport Y. How fr is te irrft from irport X, to te nerest kilometre? 292 MHR pter m

11 9. pplition To determine te eigt of te ee Tower on rliment Hill in Ottw, mesurements were tken from seline. It ws found tt = 50 m, XY = 42.6, X = 60, nd X = lulte te eigt of te ee Tower, to te nerest metre. Y We onnetion To lern more out te istory nd onstrution of te rliment uildings, visit te ove we site. Go to Mt Resoures, ten to MTHEMTIS 11, to find out were to go net. Write rief report. X 10. Sip nvigtion Two sips left ort Hope on Lke Ontrio t te sme time. One trvelled t 12 km/ on ourse of 235. Te oter trvelled t 15 km/ on ourse of 105. How fr prt were te sips fter four ours, to te nerest kilometre? 270 N Mesurement Find te re of XYZ, to te nerest tent of squre metre m 180 X 6.8 m 12. ommunition ) Use te osine lw to find, to te nerest tent. ) Use te ytgoren teorem to find, to te nerest tent. ) Eplin wy te two metods give te sme results in rigt tringle. Z m 5.2 Y 13. Sine lw in rigt tringles Rigt is sown. Write e of te rtios,, nd in terms of,, or, nd sin sin sin verify tt = = for rigt tringle. sin sin sin 4.3 Te Sine Lw nd te osine Lw MHR 293

12 14. Stikine nyon Te Stikine nyon in entrl ritis olumi is often referred to s nd s Grnd nyon. Two points X nd Y re sigted from seline of lengt 30 m on te opposite side of te nyon. Te ngle mesurements reorded from positions nd were XY = 31.3, XY = 18.5, X = 25.6, nd Y = Find te distne from X to Y, to te nerest metre. X Y 15. Geometry Use te osine lw to sow tt opposite ngles in prllelogrm re ongruent. 16. Mesurement In RST, RS = 4.9 m, ST = 3.7 m, nd RT = 8.1 m. Find te re of RST, to te nerest tent of squre metre. 17. Mesurement In, = 46 m, = 42.2, nd = Find te re of, to te nerest tent of squre metre. 18. Mesurement Find te volume of te rigt prism, to te nerest ui entimetre. 19. Mesurement Find te volume of te rigt prism, to te nerest ui metre. 8.8 m m 15 m m 9.4 m 16.2 m 20. nlyti geometry QR s verties (1, 5), Q(6, 7), nd R( 2, 1). Find te ngle mesures, to te nerest tent of degree. HIEVEMENT ek Knowledge/Understnding Tinking/Inquiry/rolem Solving ommunition pplition n equilterl tringle s een resed nd folded so tt its verte now rests on t, su tt = 1 nd = 2. Find te lengt of ) ) Q ) Q Q MHR pter 4

13 REER ONNETION Surveying Surveying is te sientifi mesurement of nturl or rtifiil fetures on te Ert s surfe. Surveyors re involved in wide vriety of tsks tt require very urte knowledge of lotions. Te distnes nd ngles determined y surveyors re used in mny wys, inluding drwing mps, positioning uildings nd oter strutures orretly, nd defining te property lines tt seprte one piee of lnd from noter. euse nd is te world s seond-lrgest ountry, surveying nd s een n enormous tsk. For emple, it took lmost 60 yers to omplete survey of te nd-u.s order, prt of wi runs troug four of te Gret Lkes. s result of over 150 yers of surveying work, detiled mps now eist for ll prts of nd. 1. From point, te distne to one end of pond is 450 m nd te distne to te oter end is 520 m. Te ngle formed y te lines of sigt is 115. Find te lengt of te pond, to te nerest ten metres. 450 m m 2. Reser Use your reser skills to investigte te following. ) te edution nd trining required to eome surveyor, nd te orgniztions tt employ surveyors ) te use of different types of surveying equipment, inluding mnully ontrolled, eletroni, nd potogrpi instruments, nd te use of stellite tenology ) te work of te Geologil Survey of nd in eploring nd mpping te ountry TTERN ower ) opy nd omplete te pttern = = = ) esrie te pttern in words. ) Eplin wy te pttern works. d) Write te net 2 lines of te pttern. e) Use te pttern to find Te Sine Lw nd te osine Lw MHR 295

Pythagorean Theorem and Trigonometry

Pythagorean Theorem and Trigonometry Ptgoren Teorem nd Trigonometr Te Ptgoren Teorem is nient, well-known, nd importnt. It s lrge numer of different proofs, inluding one disovered merin President Jmes. Grfield. Te we site ttp://www.ut-te-knot.org/ptgors/inde.stml

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Nme Dte hpter 9 Mintining Mthemtil Profiieny Simplify the epression. 1. 500. 189 3. 5 4. 4 3 5. 11 5 6. 8 Solve the proportion. 9 3 14 7. = 8. = 9. 1 7 5 4 = 4 10. 0 6 = 11. 7 4 10 = 1. 5 9 15 3 = 5 +

More information

Section 2.1 Special Right Triangles

Section 2.1 Special Right Triangles Se..1 Speil Rigt Tringles 49 Te --90 Tringle Setion.1 Speil Rigt Tringles Te --90 tringle (or just 0-60-90) is so nme euse of its ngle mesures. Te lengts of te sies, toug, ve very speifi pttern to tem

More information

Trigonometry. cosθ. sinθ tanθ. Mathletics Instant Workbooks. Copyright

Trigonometry. cosθ. sinθ tanθ. Mathletics Instant Workbooks. Copyright Student Book - Series K- sinθ tnθ osθ Mtletis Instnt Workooks Copyrigt Student Book - Series K Contents Topis Topi - Nming te sides of rigt-ngled tringle Topi 2 - Te trigonometri rtios Topi 3 - Using lultor

More information

Chapter 5 Worked Solutions to the Problems

Chapter 5 Worked Solutions to the Problems Mtemtis for Queenslnd, Yer Mtemtis, Grpis lultor ppro 00. Kiddy olger, Rex oggs, Rond Frger, Jon elwrd pter 5 Worked Solutions to te Problems Hints. Strt by writing formul for te re of tringle. Note tt

More information

Precalculus Notes: Unit 6 Law of Sines & Cosines, Vectors, & Complex Numbers. A can be rewritten as

Precalculus Notes: Unit 6 Law of Sines & Cosines, Vectors, & Complex Numbers. A can be rewritten as Dte: 6.1 Lw of Sines Syllus Ojetie: 3.5 Te student will sole pplition prolems inoling tringles (Lw of Sines). Deriing te Lw of Sines: Consider te two tringles. C C In te ute tringle, sin In te otuse tringle,

More information

CHENG Chun Chor Litwin The Hong Kong Institute of Education

CHENG Chun Chor Litwin The Hong Kong Institute of Education PE-hing Mi terntionl onferene IV: novtion of Mthemtis Tehing nd Lerning through Lesson Study- onnetion etween ssessment nd Sujet Mtter HENG hun hor Litwin The Hong Kong stitute of Edution Report on using

More information

Activities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions

Activities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd

More information

GM1 Consolidation Worksheet

GM1 Consolidation Worksheet Cmridge Essentils Mthemtis Core 8 GM1 Consolidtion Worksheet GM1 Consolidtion Worksheet 1 Clulte the size of eh ngle mrked y letter. Give resons for your nswers. or exmple, ngles on stright line dd up

More information

PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:

PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL: PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for right-ngled tringles

More information

PYTHAGORAS THEOREM,TRIGONOMETRY,BEARINGS AND THREE DIMENSIONAL PROBLEMS

PYTHAGORAS THEOREM,TRIGONOMETRY,BEARINGS AND THREE DIMENSIONAL PROBLEMS PYTHGORS THEOREM,TRIGONOMETRY,ERINGS ND THREE DIMENSIONL PROLEMS 1.1 PYTHGORS THEOREM: 1. The Pythgors Theorem sttes tht the squre of the hypotenuse is equl to the sum of the squres of the other two sides

More information

Date Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( )

Date Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( ) UNIT 5 TRIGONOMETRI RTIOS Dte Lesson Text TOPI Homework pr. 4 5.1 (48) Trigonometry Review WS 5.1 # 3 5, 9 11, (1, 13)doso pr. 6 5. (49) Relted ngles omplete lesson shell & WS 5. pr. 30 5.3 (50) 5.3 5.4

More information

Math Lesson 4-5 The Law of Cosines

Math Lesson 4-5 The Law of Cosines Mth-1060 Lesson 4-5 The Lw of osines Solve using Lw of Sines. 1 17 11 5 15 13 SS SSS Every pir of loops will hve unknowns. Every pir of loops will hve unknowns. We need nother eqution. h Drop nd ltitude

More information

Geometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272.

Geometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272. Geometry of the irle - hords nd ngles Geometry of the irle hord nd ngles urriulum Redy MMG: 272 www.mthletis.om hords nd ngles HRS N NGLES The irle is si shpe nd so it n e found lmost nywhere. This setion

More information

Something found at a salad bar

Something found at a salad bar Nme PP Something found t sld r 4.7 Notes RIGHT TRINGLE hs extly one right ngle. To solve right tringle, you n use things like SOH-H-TO nd the Pythgoren Theorem. n OLIQUE TRINGLE hs no right ngles. To solve

More information

Section 1.3 Triangles

Section 1.3 Triangles Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior

More information

Non Right Angled Triangles

Non Right Angled Triangles Non Right ngled Tringles Non Right ngled Tringles urriulum Redy www.mthletis.om Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in non-right ngled tringles. This unit

More information

Trigonometry and Constructive Geometry

Trigonometry and Constructive Geometry Trigonometry nd Construtive Geometry Trining prolems for M2 2018 term 1 Ted Szylowie tedszy@gmil.om 1 Leling geometril figures 1. Prtie writing Greek letters. αβγδɛθλµπψ 2. Lel the sides, ngles nd verties

More information

*GMT62* *20GMT6201* Mathematics. Unit T6 Paper 2 (With calculator) Higher Tier [GMT62] MONDAY 11 JUNE 3.00 pm 4.15 pm. 1 hour 15 minutes.

*GMT62* *20GMT6201* Mathematics. Unit T6 Paper 2 (With calculator) Higher Tier [GMT62] MONDAY 11 JUNE 3.00 pm 4.15 pm. 1 hour 15 minutes. entre Numer ndidte Numer Mtemtis Generl ertifite Seondry Edution 0 Unit T6 Pper (Wit lultor) Higer Tier [GMT6] MONDAY JUNE 3.00 pm4.5 pm *GMT6* *GMT6* TIME our 5 minutes. INSTRUTIONS TO ANDIDATES Write

More information

Similar Right Triangles

Similar Right Triangles Geometry V1.noteook Ferury 09, 2012 Similr Right Tringles Cn I identify similr tringles in right tringle with the ltitude? Cn I identify the proportions in right tringles? Cn I use the geometri mens theorems

More information

LESSON 11: TRIANGLE FORMULAE

LESSON 11: TRIANGLE FORMULAE . THE SEMIPERIMETER OF TRINGLE LESSON : TRINGLE FORMULE In wht follows, will hve sides, nd, nd these will e opposite ngles, nd respetively. y the tringle inequlity, nd..() So ll of, & re positive rel numers.

More information

Alg 3 Ch 7.2, 8 1. C 2) If A = 30, and C = 45, a = 1 find b and c A

Alg 3 Ch 7.2, 8 1. C 2) If A = 30, and C = 45, a = 1 find b and c A lg 3 h 7.2, 8 1 7.2 Right Tringle Trig ) Use of clcultor sin 10 = sin x =.4741 c ) rete right tringles π 1) If = nd = 25, find 6 c 2) If = 30, nd = 45, = 1 find nd c 3) If in right, with right ngle t,

More information

THE PYTHAGOREAN THEOREM

THE PYTHAGOREAN THEOREM THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this

More information

12.4 Similarity in Right Triangles

12.4 Similarity in Right Triangles Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right

More information

Lesson 4.1 Triangle Sum Conjecture

Lesson 4.1 Triangle Sum Conjecture Lesson 4.1 ringle um onjeture me eriod te n erises 1 9, determine the ngle mesures. 1. p, q 2., 3., 31 82 p 98 q 28 53 17 79 23 50 4. r, s, 5., 6. t t s r 100 85 100 30 4 7 31 7. s 8. m 9. m s 76 35 m

More information

Mathematics 10 Page 1 of 5 Properties of Triangle s and Quadrilaterals. Isosceles Triangle. - 2 sides and 2 corresponding.

Mathematics 10 Page 1 of 5 Properties of Triangle s and Quadrilaterals. Isosceles Triangle. - 2 sides and 2 corresponding. Mthemtis 10 Pge 1 of 5 Properties of s Pthgoren Theorem 2 2 2 used to find the length of sides of right tringle Tpe of s nd Some s Theorems ngles s Slene Isoseles Equilterl ute - ll ngles re less thn 90

More information

Trigonometry Revision Sheet Q5 of Paper 2

Trigonometry Revision Sheet Q5 of Paper 2 Trigonometry Revision Sheet Q of Pper The Bsis - The Trigonometry setion is ll out tringles. We will normlly e given some of the sides or ngles of tringle nd we use formule nd rules to find the others.

More information

ONLINE PAGE PROOFS. Trigonometry Kick off with CAS 12.2 Trigonometry 12.3 Pythagorean triads

ONLINE PAGE PROOFS. Trigonometry Kick off with CAS 12.2 Trigonometry 12.3 Pythagorean triads 12 12.1 Kik off with S 12.2 Trigonometry 12.3 Pythgoren trids Trigonometry 12.4 Three-dimensionl Pythgors theorem 12.5 Trigonometri rtios 12.6 The sine rule 12.7 miguous se of the sine rule 12.8 The osine

More information

A Study on the Properties of Rational Triangles

A Study on the Properties of Rational Triangles Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn

More information

Naming the sides of a right-angled triangle

Naming the sides of a right-angled triangle 6.2 Wht is trigonometry? The word trigonometry is derived from the Greek words trigonon (tringle) nd metron (mesurement). Thus, it literlly mens to mesure tringle. Trigonometry dels with the reltionship

More information

m A 1 1 A ! and AC 6

m A 1 1 A ! and AC 6 REVIEW SET A Using sle of m represents units, sketh vetor to represent: NON-CALCULATOR n eroplne tking off t n ngle of 8 ± to runw with speed of 6 ms displement of m in north-esterl diretion. Simplif:

More information

Trigonometry. Trigonometry. labelling conventions. Evaluation of areas of non-right-angled triangles using the formulas A = 1 ab sin (C )

Trigonometry. Trigonometry. labelling conventions. Evaluation of areas of non-right-angled triangles using the formulas A = 1 ab sin (C ) 8 8 Pythgors theorem 8 Pythgoren trids 8 Three-dimensionl Pythgors theorem 8D Trigonometri rtios 8E The sine rule 8F miguous se of the sine rule 8G The osine rule 8H Speil tringles 8I re of tringles res

More information

PROPERTIES OF TRIANGLES

PROPERTIES OF TRIANGLES PROPERTIES OF TRINGLES. RELTION RETWEEN SIDES ND NGLES OF TRINGLE:. tringle onsists of three sides nd three ngles lled elements of the tringle. In ny tringle,,, denotes the ngles of the tringle t the verties.

More information

Green s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e

Green s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e Green s Theorem. Let be the boundry of the unit squre, y, oriented ounterlokwise, nd let F be the vetor field F, y e y +, 2 y. Find F d r. Solution. Let s write P, y e y + nd Q, y 2 y, so tht F P, Q. Let

More information

3 Angle Geometry. 3.1 Measuring Angles. 1. Using a protractor, measure the marked angles.

3 Angle Geometry. 3.1 Measuring Angles. 1. Using a protractor, measure the marked angles. 3 ngle Geometry MEP Prtie ook S3 3.1 Mesuring ngles 1. Using protrtor, mesure the mrked ngles. () () (d) (e) (f) 2. Drw ngles with the following sizes. () 22 () 75 120 (d) 90 (e) 153 (f) 45 (g) 180 (h)

More information

Applications of trigonometry

Applications of trigonometry 3 3 3 3 3D 3E 3F 3G 3H Review of right-ngled tringles erings Using the sine rule to find side lengths Using the sine rule to find ngles re of tringle Using the osine rule to find side lengths Using the

More information

Use of Trigonometric Functions

Use of Trigonometric Functions Unit 03 Use of Trigonometric Functions 1. Introduction Lerning Ojectives of tis UNIT 1. Lern ow te trigonometric functions re relted to te rtios of sides of rigt ngle tringle. 2. Be le to determine te

More information

2. There are an infinite number of possible triangles, all similar, with three given angles whose sum is 180.

2. There are an infinite number of possible triangles, all similar, with three given angles whose sum is 180. SECTION 8-1 11 CHAPTER 8 Setion 8 1. There re n infinite numer of possile tringles, ll similr, with three given ngles whose sum is 180. 4. If two ngles α nd β of tringle re known, the third ngle n e found

More information

2 Calculate the size of each angle marked by a letter in these triangles.

2 Calculate the size of each angle marked by a letter in these triangles. Cmridge Essentils Mthemtics Support 8 GM1.1 GM1.1 1 Clculte the size of ech ngle mrked y letter. c 2 Clculte the size of ech ngle mrked y letter in these tringles. c d 3 Clculte the size of ech ngle mrked

More information

MATHEMATICS AND STATISTICS 1.6

MATHEMATICS AND STATISTICS 1.6 MTHMTIS N STTISTIS 1.6 pply geometri resoning in solving prolems ternlly ssessed 4 redits S 91031 inding unknown ngles When finding the size of unknown ngles in figure, t lest two steps of resoning will

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

Objective: Use the Pythagorean Theorem and its converse to solve right triangle problems. CA Geometry Standard: 12, 14, 15

Objective: Use the Pythagorean Theorem and its converse to solve right triangle problems. CA Geometry Standard: 12, 14, 15 Geometry CP Lesson 8.2 Pythgoren Theorem nd its Converse Pge 1 of 2 Ojective: Use the Pythgoren Theorem nd its converse to solve right tringle prolems. CA Geometry Stndrd: 12, 14, 15 Historicl Bckground

More information

SECTION A STUDENT MATERIAL. Part 1. What and Why.?

SECTION A STUDENT MATERIAL. Part 1. What and Why.? SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are

More information

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem. 27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we

More information

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then. pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm

More information

Lesson 5.1 Polygon Sum Conjecture

Lesson 5.1 Polygon Sum Conjecture Lesson 5.1 olgon Sum onjeture me eriod te In erises 1 nd 2, find eh lettered ngle mesure. 1.,,, 2.,,, d, e d, e, f d e e d 97 f 26 85 44 3. ne eterior ngle of regulr polgon mesures 10. Wht is the mesure

More information

MAT 1275: Introduction to Mathematical Analysis

MAT 1275: Introduction to Mathematical Analysis 1 MT 1275: Intrdutin t Mtemtil nlysis Dr Rzenlyum Slving Olique Tringles Lw f Sines Olique tringles tringles tt re nt neessry rigt tringles We re ging t slve tem It mens t find its si elements sides nd

More information

Basic Angle Rules 5. A Short Hand Geometric Reasons. B Two Reasons. 1 Write in full the meaning of these short hand geometric reasons.

Basic Angle Rules 5. A Short Hand Geometric Reasons. B Two Reasons. 1 Write in full the meaning of these short hand geometric reasons. si ngle Rules 5 6 Short Hnd Geometri Resons 1 Write in full the mening of these short hnd geometri resons. Short Hnd Reson Full Mening ) se s isos Δ re =. ) orr s // lines re =. ) sum s t pt = 360. d)

More information

Topics Covered: Pythagoras Theorem Definition of sin, cos and tan Solving right-angle triangles Sine and cosine rule

Topics Covered: Pythagoras Theorem Definition of sin, cos and tan Solving right-angle triangles Sine and cosine rule Trigonometry Topis overed: Pythgors Theorem Definition of sin, os nd tn Solving right-ngle tringles Sine nd osine rule Lelling right-ngle tringle Opposite (Side opposite the ngle θ) Hypotenuse (Side opposite

More information

Section 13.1 Right Triangles

Section 13.1 Right Triangles Section 13.1 Right Tringles Ojectives: 1. To find vlues of trigonometric functions for cute ngles. 2. To solve tringles involving right ngles. Review - - 1. SOH sin = Reciprocl csc = 2. H cos = Reciprocl

More information

3.1 Review of Sine, Cosine and Tangent for Right Angles

3.1 Review of Sine, Cosine and Tangent for Right Angles Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles 125 3.1 Review of Sine, osine nd Tngent for Right ngles The word trigonometry is derived from the Greek words trigon,

More information

Project 6: Minigoals Towards Simplifying and Rewriting Expressions

Project 6: Minigoals Towards Simplifying and Rewriting Expressions MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy

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., 3., 31 82 p 98 q 28 53 17 79 23 50 4. r, s, 5., 6. t t s r 100 85 100 30 4 7 31 7. s 8. m 9. m s 76 35

More information

Geometry. Trigonometry of Right Triangles. Slide 2 / 240. Slide 1 / 240. Slide 3 / 240. Slide 4 / 240. Slide 6 / 240.

Geometry. Trigonometry of Right Triangles. Slide 2 / 240. Slide 1 / 240. Slide 3 / 240. Slide 4 / 240. Slide 6 / 240. Slide 1 / 240 Slide 2 / 240 New Jerse enter for Tehing nd Lerning Progressive Mthemtis Inititive This mteril is mde freel ville t www.njtl.org nd is intended for the non-ommeril use of students nd tehers.

More information

Pythagoras Theorem. Pythagoras Theorem. Curriculum Ready ACMMG: 222, 245.

Pythagoras Theorem. Pythagoras Theorem. Curriculum Ready ACMMG: 222, 245. Pythgors Theorem Pythgors Theorem Curriulum Redy ACMMG:, 45 www.mthletis.om Fill in these spes with ny other interesting fts you n find out Pythgors. In the world of Mthemtis, Pythgors is legend. He lived

More information

m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r

m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r CO-ORDINTE GEOMETR II I Qudrnt Qudrnt (-.+) (++) X X - - - 0 - III IV Qudrnt - Qudrnt (--) - (+-) Region CRTESIN CO-ORDINTE SSTEM : Retngulr Co-ordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr

More information

2.1 ANGLES AND THEIR MEASURE. y I

2.1 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 information

MEP Practice Book ES3. 1. Calculate the size of the angles marked with a letter in each diagram. None to scale

MEP Practice Book ES3. 1. Calculate the size of the angles marked with a letter in each diagram. None to scale ME rctice ook ES3 3 ngle Geometr 3.3 ngle Geometr 1. lculte the size of the ngles mrked with letter in ech digrm. None to scle () 70 () 20 54 65 25 c 36 (d) (e) (f) 56 62 d e 60 40 70 70 f 30 g (g) (h)

More information

Comparing the Pre-image and Image of a Dilation

Comparing the Pre-image and Image of a Dilation hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) ngle-ngle Similrity Theorem (.2) Side-Side-Side Similrity

More information

Proving the Pythagorean Theorem

Proving the Pythagorean Theorem Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul 2 + 2 = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or

More information

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

More information

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the

More information

Perimeter, area and volume

Perimeter, area and volume 6 Perimeter, re nd volume Syllus topi M. Perimeter, re nd volume This topi will develop your skills to ompetently solve prolems involving perimeter, re, volume nd pity. Outomes Clulte the re of irles nd

More information

Find the value of x. Give answers as simplified radicals.

Find the value of x. Give answers as simplified radicals. 9.2 Dy 1 Wrm Up Find the vlue of. Give nswers s simplified rdicls. 1. 2. 3 3 3. 4. 10 Mrch 2, 2017 Geometry 9.2 Specil Right Tringles 1 Geometry 9.2 Specil Right Tringles 9.2 Essentil Question Wht is the

More information

Pythagoras theorem and surds

Pythagoras theorem and surds HPTER Mesurement nd Geometry Pythgors theorem nd surds In IE-EM Mthemtis Yer 8, you lernt out the remrkle reltionship etween the lengths of the sides of right-ngled tringle. This result is known s Pythgors

More information

UNCORRECTED. Australian curriculum MEASUREMENT AND GEOMETRY

UNCORRECTED. Australian curriculum MEASUREMENT AND GEOMETRY 3 3 3C 3D 3 3F 3G 3H 3I 3J Chpter Wht you will lern Pythgors theorem Finding the shorter sides pplying Pythgors theorem Pythgors in three dimensions (tending) Trigonometri rtios Finding side lengths Solving

More information

The Ellipse. is larger than the other.

The Ellipse. is larger than the other. The Ellipse Appolonius of Perg (5 B.C.) disovered tht interseting right irulr one ll the w through with plne slnted ut is not perpendiulr to the is, the intersetion provides resulting urve (oni setion)

More information

*GMT41* *24GMT4101* Mathematics. Unit T4 (With calculator) Higher Tier [GMT41] WEDNESDAY 6 JUNE 9.15 am am. 2 hours.

*GMT41* *24GMT4101* Mathematics. Unit T4 (With calculator) Higher Tier [GMT41] WEDNESDAY 6 JUNE 9.15 am am. 2 hours. entre Numer ndidte Numer Mtemtics Generl ertificte Secondry Eduction 0 Unit T4 (Wit clcultor) Higer Tier [GMT4] WEDNESDAY 6 JUNE 9.5 m.5 m *GMT4* *GMT4* TIME ours. INSTRUTIONS TO ANDIDATES Write your entre

More information

Plotting Ordered Pairs Using Integers

Plotting Ordered Pairs Using Integers SAMPLE Plotting Ordered Pirs Using Integers Ple two elsti nds on geoord to form oordinte xes shown on the right to help you solve these prolems.. Wht letter of the lphet does eh set of pirs nme?. (, )

More information

Lesson 8.1 Graphing Parametric Equations

Lesson 8.1 Graphing Parametric Equations Lesson 8.1 Grphing Prmetric Equtions 1. rete tle for ech pir of prmetric equtions with the given vlues of t.. x t 5. x t 3 c. x t 1 y t 1 y t 3 y t t t {, 1, 0, 1, } t {4,, 0,, 4} t {4, 0,, 4, 8}. Find

More information

Calculating Tank Wetted Area Saving time, increasing accuracy

Calculating Tank Wetted Area Saving time, increasing accuracy Clulting Tnk Wetted Are ving time, inresing ur B n Jones, P.., P.E. C lulting wetted re in rtillfilled orizontl or vertil lindril or ellitil tnk n e omlited, deending on fluid eigt nd te se of te eds (ends)

More information

STRAND I: Geometry and Trigonometry. UNIT 32 Angles, Circles and Tangents: Student Text Contents. Section Compass Bearings

STRAND I: Geometry and Trigonometry. UNIT 32 Angles, Circles and Tangents: Student Text Contents. Section Compass Bearings ME Jmi: STR I UIT 32 ngles, irles n Tngents: Stuent Tet ontents STR I: Geometry n Trigonometry Unit 32 ngles, irles n Tngents Stuent Tet ontents Setion 32.1 ompss erings 32.2 ngles n irles 1 32.3 ngles

More information

Precalculus Notes: Unit 6 Law of Sines & Cosines, Vectors, & Complex Numbers. A can be rewritten as B

Precalculus Notes: Unit 6 Law of Sines & Cosines, Vectors, & Complex Numbers. A can be rewritten as B Date: 6.1 Law of Sines Syllaus Ojetie: 3.5 Te student will sole appliation prolems inoling triangles (Law of Sines). Deriing te Law of Sines: Consider te two triangles. a C In te aute triangle, sin and

More information

8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1

8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1 8. The Hperol Some ships nvigte using rdio nvigtion sstem clled LORAN, which is n cronm for LOng RAnge Nvigtion. A ship receives rdio signls from pirs of trnsmitting sttions tht send signls t the sme time.

More information

Pythagoras Theorem. The area of the square on the hypotenuse is equal to the sum of the squares on the other two sides

Pythagoras Theorem. The area of the square on the hypotenuse is equal to the sum of the squares on the other two sides Pythgors theorem nd trigonometry Pythgors Theorem The hypotenuse of right-ngled tringle is the longest side The hypotenuse is lwys opposite the right-ngle 2 = 2 + 2 or 2 = 2-2 or 2 = 2-2 The re of the

More information

Standard Trigonometric Functions

Standard Trigonometric Functions CRASH KINEMATICS For ngle A: opposite sine A = = hypotenuse djent osine A = = hypotenuse opposite tngent A = = djent For ngle B: opposite sine B = = hypotenuse djent osine B = = hypotenuse opposite tngent

More information

Identifying and Classifying 2-D Shapes

Identifying and Classifying 2-D Shapes Ientifying n Clssifying -D Shpes Wht is your sign? The shpe n olour of trffi signs let motorists know importnt informtion suh s: when to stop onstrution res. Some si shpes use in trffi signs re illustrte

More information

SAMPLE. Trigonometry. Naming the sides of a right-angled triangle

SAMPLE. Trigonometry. Naming the sides of a right-angled triangle H P T E R 7 Trigonometry How re sin, os nd tn defined using right-ngled tringle? How n the trigonometri rtios e used to find the side lengths or ngles in right-ngled tringles? Wht is ment y n ngle of elevtion

More information

15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions )

15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions ) - TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the

More information

Perimeter and Area. Mathletics Instant Workbooks. Copyright

Perimeter and Area. Mathletics Instant Workbooks. Copyright Perimeter nd Are Student Book - Series J- L B Mthletis Instnt Workooks Copyright Student Book - Series J Contents Topis Topi - Plne shpes Topi 2 - Perimeter of regulr shpes Topi 3 - Perimeter of irregulr

More information

MATHEMATICS PAPER & SOLUTION

MATHEMATICS PAPER & SOLUTION MATHEMATICS PAPER & SOLUTION Code: SS--Mtemtis Time : Hours M.M. 8 GENERAL INSTRUCTIONS TO THE EXAMINEES:. Cndidte must write first is / er Roll No. on te question pper ompulsorily.. All te questions re

More information

GEOMETRY OF THE CIRCLE TANGENTS & SECANTS

GEOMETRY OF THE CIRCLE TANGENTS & SECANTS Geometry Of The ircle Tngents & Secnts GEOMETRY OF THE IRLE TNGENTS & SENTS www.mthletics.com.u Tngents TNGENTS nd N Secnts SENTS Tngents nd secnts re lines tht strt outside circle. Tngent touches the

More information

Lesson 2.1 Inductive Reasoning

Lesson 2.1 Inductive Reasoning Lesson 2.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 12, 16,, 2. 400, 200, 100, 50, 25,, 3. 1 8, 2 7, 1 2, 4, 5, 4. 5, 3, 2,

More information

Lesson 4.1 Triangle Sum Conjecture

Lesson 4.1 Triangle Sum Conjecture Lesson.1 ringle um onjecture Nme eriod te n ercises 1 9, determine the ngle mesures. 1. p, q., 3., 31 8 p 98 q 8 53 17 79 3 50. r, s, 5.,. t t 85 s 100 r 30 100 7 31 7. s 8. m 9. m s 7 35 m c c 10. Find

More information

Lesson 2.1 Inductive Reasoning

Lesson 2.1 Inductive Reasoning Lesson 2.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 12, 16,, 2. 400, 200, 100, 50, 25,, 3. 1 8, 2 7, 1 2, 4, 5, 4. 5, 3, 2,

More information

Two Triads of Congruent Circles from Reflections

Two Triads of Congruent Circles from Reflections Forum Geometriorum Volume 8 (2008) 7 12. FRUM GEM SSN 1534-1178 Two Trids of ongruent irles from Refletions Qung Tun ui strt. Given tringle, we onstrut two trids of ongruent irles through the verties,

More information

Similarity and Congruence

Similarity and Congruence Similrity nd ongruence urriculum Redy MMG: 201, 220, 221, 243, 244 www.mthletics.com SIMILRITY N ONGRUN If two shpes re congruent, it mens thy re equl in every wy ll their corresponding sides nd ngles

More information

HYPERBOLA. AIEEE Syllabus. Total No. of questions in Ellipse are: Solved examples Level # Level # Level # 3..

HYPERBOLA. AIEEE Syllabus. Total No. of questions in Ellipse are: Solved examples Level # Level # Level # 3.. HYPERBOLA AIEEE Sllus. Stndrd eqution nd definitions. Conjugte Hperol. Prmetric eqution of te Hperol. Position of point P(, ) wit respect to Hperol 5. Line nd Hperol 6. Eqution of te Tngent Totl No. of

More information

Algebra: Function Tables - One Step

Algebra: Function Tables - One Step Alger: Funtion Tles - One Step Funtion Tles Nme: Dte: Rememer tt tere is n input nd output on e funtion tle. If you know te funtion eqution, you need to plug in for tt vrile nd figure out wt te oter vrile

More information

Geometry AP Book 8, Part 2: Unit 3

Geometry AP Book 8, Part 2: Unit 3 Geometry ook 8, rt 2: Unit 3 IMRTNT NTE: For mny questions in this unit, there re multiple correct nswers, e.g. line segment cn e written s, RST is the sme s TSR, etc. Where pproprite, techers should e

More information

8Similarity ONLINE PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8.

8Similarity ONLINE PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8. 8.1 Kick off with S 8. Similr ojects 8. Liner scle fctors 8Similrity 8.4 re nd volume scle fctors 8. Review Plese refer to the Resources t in the Prelims section of your eookplus for comprehensive step-y-step

More information

Lesson-5 PROPERTIES AND SOLUTIONS OF TRIANGLES

Lesson-5 PROPERTIES AND SOLUTIONS OF TRIANGLES Leon-5 PROPERTIES ND SOLUTIONS OF TRINGLES Reltion etween the ide nd trigonometri rtio of the ngle of tringle In ny tringle, the ide, oppoite to the ngle, i denoted y ; the ide nd, oppoite to the ngle

More information

Measurement. Chapter. What you will learn

Measurement. Chapter. What you will learn Chpter 11 Mesurement Wht you will lern 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 Mesurement systems Using nd onverting metri lengths Perimeter Ares nd retngles Are of tringle Are of prllelogrm

More information

A P P E N D I X POWERS OF TEN AND SCIENTIFIC NOTATION A P P E N D I X SIGNIFICANT FIGURES

A P P E N D I X POWERS OF TEN AND SCIENTIFIC NOTATION A P P E N D I X SIGNIFICANT FIGURES A POWERS OF TEN AND SCIENTIFIC NOTATION In science, very lrge nd very smll deciml numbers re conveniently expressed in terms of powers of ten, some of wic re listed below: 0 3 0 0 0 000 0 3 0 0 0 0.00

More information

Log1 Contest Round 3 Theta Individual. 4 points each 1 What is the sum of the first 5 Fibonacci numbers if the first two are 1, 1?

Log1 Contest Round 3 Theta Individual. 4 points each 1 What is the sum of the first 5 Fibonacci numbers if the first two are 1, 1? 008 009 Log1 Contest Round Thet Individul Nme: points ech 1 Wht is the sum of the first Fiboncci numbers if the first two re 1, 1? If two crds re drwn from stndrd crd deck, wht is the probbility of drwing

More information

4. Statements Reasons

4. Statements Reasons Chpter 9 Answers Prentie-Hll In. Alterntive Ativity 9-. Chek students work.. Opposite sides re prllel. 3. Opposite sides re ongruent. 4. Opposite ngles re ongruent. 5. Digonls iset eh other. 6. Students

More information

Right Triangles. Ready to Go On? Module Quiz

Right Triangles. Ready to Go On? Module Quiz UNIT 8 Module 27 Right Tringles ontents M9-2.G.SRT.8 M9-2.G.SRT.6 M9-2.G.O.2 27- The Pthgoren Theorem.............................. 820 27-2 ppling Speil Right Tringles......................... 828 Tsk

More information

2) Three noncollinear points in Plane M. [A] A, D, E [B] A, B, E [C] A, B, D [D] A, E, H [E] A, H, M [F] H, A, B

2) Three noncollinear points in Plane M. [A] A, D, E [B] A, B, E [C] A, B, D [D] A, E, H [E] A, H, M [F] H, A, B Review Use the points nd lines in the digrm to identify the following. 1) Three colliner points in Plne M. [],, H [],, [],, [],, [],, M [] H,, M 2) Three noncolliner points in Plne M. [],, [],, [],, [],,

More information

Trigonometry (3A) Quadrant Angle Trigonometry Negative Angle Trigonometry Reference Angle Trigonometry Sinusoidal Waves. Young Won Lim 12/30/14

Trigonometry (3A) Quadrant Angle Trigonometry Negative Angle Trigonometry Reference Angle Trigonometry Sinusoidal Waves. Young Won Lim 12/30/14 Trigonometr (3) Qudrnt ngle Trigonometr Negtive ngle Trigonometr Referene ngle Trigonometr Sinusoidl Wves opright () 2009-2014 Young W. Lim. Permission is grnted to op, distriute nd/or modif this doument

More information

Applications of Trigonometry: Triangles and Vectors

Applications of Trigonometry: Triangles and Vectors 7 Applitions of Trigonometry: Tringles nd Vetors Norfolk, Virgini Atlnti Oen Bermud Bermud In reent dedes, mny people hve ome to elieve tht n imginry re lled the Bermud Tringle, loted off the southestern

More information