The Pythagorean Theorem and Its Converse Is That Right?

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "The Pythagorean Theorem and Its Converse Is That Right?"

Transcription

1 The Pythgoren Theorem nd Its Converse Is Tht Right? SUGGESTED LEARNING STRATEGIES: Activting Prior Knowledge, Mrking the Text, Shred Reding, Summrize/Prphrse/Retell ACTIVITY 3.6 How did Pythgors get theorem nmed fter him? Although mny exmples of the Pythgoren Theorem were known nd used y the Bylonins, Chinese, Hindu nd Egyptins well efore Pythgors ws orn (out 570 BCE), he is given credit for eing the first to formlly prove the theorem. Mny others since Pythgors time, including young mn nmed Jmes Grfield who would go on to e President of the United Sttes, hve lso offered forml proofs of the well known theorem. Exmine one proof of the Pythgoren Theorem tht is credited to Pythgors himself. Begin with squre hving edges of length +. In the squre, four right tringles with legs nd hve een drwn. M R N O P ACADEMIC VOCABULARY THE PYTHAGOREAN THEOREM In ny right tringle, the squre of the length of the hypotenuse is equl to the sum of the squres of the lengths of the legs. If nd re the lengths of the legs nd c is the length of the hypotenuse then, c 2 = S T Q 2010 College Bord. All rights reserved. 1. Ech of the four right tringles in the digrm ove re congruent. Wht tringle congruence method justifies this sttement? Explin your nswer. 2. Since the four right tringles re congruent, we know their hypotenuses, RN, TR, PT nd NP, re congruent.. Wht reson cn e used to justify this?. Lel ech hypotenuse in the digrm, c. Unit 3 Similrity, Right Tringles, nd Trigonometry 235

2 ACTIVITY 3.6 The Pythgoren Theorem nd Its Converse Is Tht Right? SUGGESTED LEARNING STRATEGIES: Activting Prior Knowledge, Look for Pttern, Quickwrite 3. MNR is right tringle nd MNR SRT.. Wht is the reltionship etween MRN nd MNR? How do you know?. Use the congruence sttement, MNR SRT. Wht does this indicte out the reltionship etween MRN nd SRT? Explin your resoning. c. Wht kind of ngle is NRT? How do you know? d. Wht re the mesures of RTP, TPN nd PNR? Justify your nswer. 4. Wht specil qudrilterl is formed y the four hypotenuses? Justify your nswer College Bord. All rights reserved. 236 SpringBord Mthemtics with Mening Geometry

3 The Pythgoren Theorem nd Its Converse Is Tht Right? ACTIVITY 3.6 SUGGESTED LEARNING STRATEGIES: Think/Pir/Shre, Crete Representtions 5. It cn e ssumed from the digrm tht the re of the lrge outside squre is equl to the sum of the res of the four tringles nd qudrilterl PNRT. Write n eqution, in terms of,, nd c tht represents this sttement. 6. Use lgeric properties to simplify oth sides of the eqution. 7. Solve the simplified eqution for c 2. You hve now verified lgericlly, much s Pythgors is thought to hve done, The Pythgoren Theorem nd cn use it to solve prolems College Bord. All rights reserved. 8. How high up verticl wll will 24 foot ldder rech if the foot of the ldder is plced 10 feet from the wll? Drw sketch nd show the clcultions tht support your nswer. Unit 3 Similrity, Right Tringles, nd Trigonometry 237

4 ACTIVITY 3.6 The Pythgoren Theorem nd Its Converse Is Tht Right? SUGGESTED LEARNING STRATEGIES: Activting Prior Knowledge, Close Reding, Mrking the Text, Think/Pir/ Shre, Self/Peer Revision 9. Find the re of rectngulr rug if the width of the rug is 13 feet nd the digonl mesures 20 feet. Drw sketch nd show the clcultions tht support your nswer. The Pythgoren Theorem sttes tht, in right tringle, the squre of the length of the hypotenuse is equl to the sum of the squres of the lengths of the legs. Is the converse of this sttement true? 10. Write the Pythgoren Theorem in if-then form. 11. Write the converse of the Pythgoren Theorem in if-then form. 12. Cn the converse of the Pythgoren Theorem e proven? Assume you hve ABC where c 2 = 2 + 2, s shown elow. Complete the following to try to prove ABC is right tringle. Use right DEF, with legs nd nd hypotenuse f. B C c A E F f D 2010 College Bord. All rights reserved. 238 SpringBord Mthemtics with Mening Geometry

5 The Pythgoren Theorem nd Its Converse Is Tht Right? ACTIVITY 3.6 SUGGESTED LEARNING STRATEGIES: Activting Prior Knowledge, Close Reding, Group Presenttion, Quickwrite. It is known tht f 2 = Give reson for this sttement.. It ws ssumed tht in ABC, c 2 = So, the sttement c = f cn e mde. Why is this true? c. ABC DEF y wht reson? d. C is right ngle. Give reson for this sttement. e. ABC is right tringle. Wht reson justifies this sttement? 2010 College Bord. All rights reserved. 13. You exmined the converse of the Pythgoren Theorem. Now, tke look t the inverse.. Write the inverse of the Pythgoren Theorem in if-then form.. Is the inverse true sttement? Why or why not? Unit 3 Similrity, Right Tringles, nd Trigonometry 239

6 ACTIVITY 3.6 The Pythgoren Theorem nd Its Converse Is Tht Right? SUGGESTED LEARNING STRATEGIES: Use Mnipultives Since you hve shown the Converse of the Pythgoren Theorem is true, little more explortion follows. 14. Use ech of the following sets of tringle side lengths to uild tringles using the mnipultives (strws) provided y your techer. Step 1: Step 2: Step 3: Step 4: Cut mnipultives into 5 cm, 6 cm, 12 cm, 13 cm, nd 15 cm lengths. Build ech tringle on centimeter grid pper. Identify ech tringle s right, cute or otuse. Complete the tle. Tringle side lengths Type of tringle c , 12, 13 6, 6, 12 5, 6, 12 5, 12, 15 5, 12, 12 6, 12, 13 6, 12, College Bord. All rights reserved. 240 SpringBord Mthemtics with Mening Geometry

7 The Pythgoren Theorem nd Its Converse Is Tht Right? ACTIVITY 3.6 SUGGESTED LEARNING STRATEGIES: Think/Pir/Shre, Look for Pttern 15. Wht does your work in Item 14 suggest out the reltionship etween 2, 2, c 2 nd the type of tringle? 16. Use the Converse of the Pythgoren Theorem to determine whether ech of the following sets of side lengths forms right tringle. If right tringle is not possile, tell whether n cute or otuse tringle cn e formed. Show the method you use to determine your nswers.. 12, 34, , 7 8, 10 7 c. 20, 42, College Bord. All rights reserved. Unit 3 Similrity, Right Tringles, nd Trigonometry 241

8 ACTIVITY 3.6 The Pythgoren Theorem nd Its Converse Is Tht Right? SUGGESTED LEARNING STRATEGIES: Activting Prior Knowledge Another wy to prove the Pythgoren theorem is y using tringle similrity. In right tringle ABC elow, n ltitude is drwn to hypotenuse AB, forming two right tringles tht re similr to tringle ABC. C h A x c x B Corresponding sides of similr tringles re in proportion, so you cn write these proportions involving sides of the tringles. x = c c x = c 17. Use the proportions ove nd lger to prove = c 2. c CHECK YOUR UNDERSTANDING Write your nswers on on noteook pper pper. or on Show grid your work. 3. Tell whether tringle cn e formed hving pper. Show your work. the following side lengths. If tringle cn e formed tell whether it is right, cute or otuse. 1. If television screen is rectngle with 53 inch digonl nd width of 45 inches, wht is the height of the screen? 2. A stndrd sell dimond is squre 90 feet on ech side. Find the distnce of throw mde from the ctcher 3 feet ehind home plte in n ttempt to throw out runner trying to stel second se. Round to the nerest whole numer.. 93 feet. 124 feet c. 130 feet d. 183 feet. 4, 6, 8. 8, 8, MATHEMATICAL REFLECTION The Pythgoren Theorem ws thought of y the erly Greeks s the following: The re of the squre on the hypotenuse of right tringle is equl to the sum of the res of the squres on the legs. Drw digrm to illustrte this sttement. Explin how your digrm illustrtes the Pythgoren Theorem College Bord. All rights reserved. 242 SpringBord Mthemtics with Mening Geometry

Apply the Pythagorean Theorem

Apply the Pythagorean Theorem 8. Apply the Pythgoren Theorem The Pythgoren theorem is nmed fter the Greek philosopher nd mthemtiin Pythgors (580500 B.C.E.). Although nient texts indite tht different iviliztions understood this property

More information

Grade 6. Mathematics. Student Booklet SPRING 2011 RELEASED ASSESSMENT QUESTIONS. Record your answers on the Multiple-Choice Answer Sheet.

Grade 6. Mathematics. Student Booklet SPRING 2011 RELEASED ASSESSMENT QUESTIONS. Record your answers on the Multiple-Choice Answer Sheet. Grde 6 Assessment of Reding, Writing nd Mthemtics, Junior Division Student Booklet Mthemtics SPRING 211 RELEASED ASSESSMENT QUESTIONS Record your nswers on the Multiple-Choice Answer Sheet. Plese note:

More information

Skills Practice Skills Practice for Lesson 4.1

Skills Practice Skills Practice for Lesson 4.1 Skills Prctice Skills Prctice for Lesson.1 Nme Dte Interior nd Exterior Angles of Tringle Tringle Sum, Exterior Angle, nd Exterior Angle Inequlity Theorems Vocbulry Write the term tht best completes ech

More information

Math commonly used in the US Army Pathfinder School

Math commonly used in the US Army Pathfinder School Mth commonly used in the US Army Pthfinder School Pythgoren Theorem is used for solving tringles when two sides re known. In the Pthfinder Course it is used to determine the rdius of circulr drop zones

More information

Why? DF = 1_ EF = _ AC

Why? DF = 1_ EF = _ AC Similr Tringles Then You solved proportions. (Lesson 2-) Now 1Determine whether two tringles re similr. 2Find the unknown mesures of sides of two similr tringles. Why? Simon needs to mesure the height

More information

Lesson 12.1 Right Triangle Trigonometry

Lesson 12.1 Right Triangle Trigonometry Lesson 12.1 Right Tringle Trigonometr 1. For ech of the following right tringles, find the vlues of sin, cos, tn, sin, cos, nd tn. (Write our nswers s frctions in lowest terms.) 2 15 9 10 2 12 2. Drw right

More information

Name Class Date SAMPLE. Complete the missing numbers in the sequences below. 753, ,982. The area of the shape is approximately cm 2

Name Class Date SAMPLE. Complete the missing numbers in the sequences below. 753, ,982. The area of the shape is approximately cm 2 End of term: TEST A You will need penil. Yer 5 Nme Clss Dte 1 2 Complete the missing numers in the sequenes elow. 200 3926 4926 400 500 700 7926 753,982 553,982 Estimte the re of the shpe elow. The re

More information

MATHEMATICAL PRACTICES In the Solve It, you used what you know about triangles to find missing lengths. Key Concept Law of Sines

MATHEMATICAL PRACTICES In the Solve It, you used what you know about triangles to find missing lengths. Key Concept Law of Sines 8-5 -20-5 Lw of Sines ontent Stndrds G.SRT.11 Understnd nd ppl the Lw of Sines... to find unknown mesurements in right nd non-right tringles... lso G.SRT.10 Ojetives To ppl the Lw of Sines 66 ft 35 135

More information

Right Triangle Trigonometry

Right Triangle Trigonometry ONDENSED 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 information

SAMPLE EVALUATION ONLY

SAMPLE EVALUATION ONLY mesurement nd geometry topic 15 Pythgors theorem 15.1 Overview Why lern this? Pythgors ws fmous mthemtiin who lived out 2500 yers go. He is redited with eing the fi rst person to prove tht in ny rightngled

More information

Parallel Lines Cut by a Transversal

Parallel Lines Cut by a Transversal Name Date Class 11-1 Parallel Lines Cut by a Transversal Parallel Lines Parallel Lines Cut by a Transversal A line that crosses parallel lines is a transversal. Parallel lines never meet. Eight angles

More information

The Pythagorean Theorem Diamond in the Rough

The Pythagorean Theorem Diamond in the Rough The Pythagorean Theorem SUGGESTED LEARNING STRATEGIES: Shared Reading, Activating Prior Knowledge, Visualization, Interactive Word Wall Cameron is a catcher trying out for the school baseball team. He

More information

7.2 Assess Your Understanding

7.2 Assess Your Understanding 538 HPTER 7 pplitions of Trigonometri Funtions 7. ssess Your Understnding re You Prepred? nswers re given t the end of these exerises. If you get wrong nswer, red the pges listed in red. 1. The differene

More information

ERRATA for Guide for the Development of Bicycle Facilities, 4th Edition (GBF-4)

ERRATA for Guide for the Development of Bicycle Facilities, 4th Edition (GBF-4) Dvid Bernhrdt, P.E., President Commissioner, Mine Deprtment of Trnsporttion Bud Wright, Executive Director 444 North Cpitol Street NW, Suite 249, Wshington, DC 20001 (202) 624-5800 Fx: (202) 624-5806 www.trnsporttion.org

More information

Student Outcomes. Lesson Notes. Classwork. Discussion (20 minutes)

Student Outcomes. Lesson Notes. Classwork. Discussion (20 minutes) Student Outcomes Students explain a proof of the converse of the Pythagorean Theorem. Students apply the theorem and its converse to solve problems. Lesson Notes Students had their first experience with

More information

Math-3. Lesson 6-5 The Law of Sines The Ambiguous Case

Math-3. Lesson 6-5 The Law of Sines The Ambiguous Case Math-3 Lesson 6-5 The Law of Sines The miguous Case Quiz 6-4: 1. Find the measure of angle θ. Ө = 33.7 2. What is the cosecant ratio for ϴ? Csc Ө = 2 5 5 3. standard position angle passes through the point

More information

Special Right Triangles

Special Right Triangles GEOMETRY Special Right Triangles OBJECTIVE #: G.SRT.C.8 OBJECTIVE Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. *(Modeling Standard) BIG IDEA (Why is

More information

BASICS OF TRIGONOMETRY

BASICS OF TRIGONOMETRY Mathematics Revision Guides Basics of Trigonometry Page 1 of 9 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Foundation Tier BASICS OF TRIGONOMETRY Version: 1. Date: 09-10-015 Mathematics Revision

More information

Name Date PD. Pythagorean Theorem

Name Date PD. Pythagorean Theorem Name Date PD Pythagorean Theorem Vocabulary: Hypotenuse the side across from the right angle, it will be the longest side Legs are the sides adjacent to the right angle His theorem states: a b c In any

More information

Congruence Axioms. Data Required for Solving Oblique Triangles. 1 of 8 8/6/ THE LAW OF SINES

Congruence Axioms. Data Required for Solving Oblique Triangles. 1 of 8 8/6/ THE LAW OF SINES 1 of 8 8/6/2004 8.1 THE LAW OF SINES 8.1 THE LAW OF SINES Congrueny and Olique Triangles Derivation of the Law of Sines Appliations Amiguous Case Area of a Triangle Until now, our work with triangles has

More information

Length, Perimeter and Area

Length, Perimeter and Area Length, Perimeter n Are Teher Book - Series Mthletis Instnt Workooks Copyright Length, Perimeter n Are - Series Contents Setion Answers (p. ) units of length trvelling fr perimeter re 9 7 5 Setion Assessment

More information

CH 34 MORE PYTHAGOREAN THEOREM AND RECTANGLES

CH 34 MORE PYTHAGOREAN THEOREM AND RECTANGLES CH 34 MORE PYTHAGOREAN THEOREM AND RECTANGLES 317 Recalling The Pythagorean Theorem a 2 + b 2 = c 2 a c 90 b The 90 angle is called the right angle of the right triangle. The other two angles of the right

More information

Chapter 10. Right Triangles

Chapter 10. Right Triangles Chapter 10 Right Triangles If we looked at enough right triangles and experimented a little, we might eventually begin to notice some relationships developing. For instance, if I were to construct squares

More information

Listening & Speaking. Grade 1. Supports. instructi GRADE. Develops oral and receptive language. 15- to 20-minute daily activities

Listening & Speaking. Grade 1. Supports. instructi GRADE. Develops oral and receptive language. 15- to 20-minute daily activities Grde 1 to Stte Correlted Stndrds GRADE Develops orl nd receptive lnguge 1 EMC 2416 15- to 20-minute dily ctivities Listening & Home School Connection resources Supports t s r i F g n i Red ding E- bo ok

More information

bark bark bat bat Multiple Meaning Words: Kindergarten to Grade 2 More Teaching Tools at harsh sound made by a dog

bark bark bat bat Multiple Meaning Words: Kindergarten to Grade 2 More Teaching Tools at  harsh sound made by a dog the brk the brk bt bt hrsh sound mde by dog Mx, stop brking! outside cover of the trunks, brnches, nd roots of woody plnts The brk of this tree is very rough. club of wood or metl used to hit the bll in

More information

OVERVIEW Similarity Leads to Trigonometry G.SRT.6

OVERVIEW Similarity Leads to Trigonometry G.SRT.6 OVERVIEW Similarity Leads to Trigonometry G.SRT.6 G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric

More information

BASKETBALL SPEED AND AGILITY

BASKETBALL SPEED AND AGILITY SKETLL SPEED ND GILITY Off court Speed and gility Work: ox gility Drills: cone set up 5 yards apart, read and follow description Drill 1 : (1-2-3-4) Sprint around cones, make hard cuts Drill 2: 1-2 Sprint,

More information

13.7 Quadratic Equations and Problem Solving

13.7 Quadratic Equations and Problem Solving 13.7 Quadratic Equations and Problem Solving Learning Objectives: A. Solve problems that can be modeled by quadratic equations. Key Vocabulary: Pythagorean Theorem, right triangle, hypotenuse, leg, sum,

More information

2014 Victorian Shooting Championship

2014 Victorian Shooting Championship 2014 Victorin Shooting Chmpionship VPCI, in conjunction with the Stte Coches nd the Stte Umpires invite ll PFA licensed petnque plyers in the Stte of Victori to tke prt in the 2014 Victorin Shooting Chmpionship.

More information

HCI Examination Please answer in Swedish or English

HCI Examination Please answer in Swedish or English HCI Exmintion 02.06.04 8.45-12.45 Plese nswer in Swedish or English PLEASE HAND IN FIRST PAGE OF EXAMINATION SHEET (TES) IF YOU ANSWER MULITPLE CHOICE HERE PART I: NECESSARY FOR PASS (GODKÄNT) 1. Multiple

More information

Put in simplest radical form. (No decimals)

Put in simplest radical form. (No decimals) Put in simplest radical form. (No decimals) 1. 2. 3. 4. 5. 6. 5 7. 4 8. 6 9. 5 10. 9 11. -3 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 3 28. 1 Geometry Chapter 8 - Right Triangles

More information

8-1. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary

8-1. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary 8-1 he Pythagorean heorem and Its Converse Vocabulary Review 1. Write the square and the positive square root of each number. Number Square Positive Square Root 9 81 3 1 4 1 16 1 2 Vocabulary Builder leg

More information

Bicycle wheel and swivel chair

Bicycle wheel and swivel chair Aim: To show conservtion of ngulr momentum. To clrify the vector chrcteristics of ngulr momentum. (In this demonstrtion especilly the direction of ngulr momentum is importnt.) Subjects: Digrm: 1Q40 (Conservtion

More information

PCT MINIMUM DOCUMENTATION

PCT MINIMUM DOCUMENTATION Ref.: PCT inimum Documenttion pge: 4.1.1 PCT INIU TION INVENTY S CCDING TO PCT RULE 34.1 (PERIOD FRO 1920 TO 1996) Explntory Notes 1. On the following pges is given the inventory of ptent documents, covering

More information

Pythagorean Theorem in Sports

Pythagorean Theorem in Sports Name Date Pythagorean Theorem in Sports Activity 1: Pythagorean Theorem in Baseball Directions: Measure the distance between each of the bases using the yard stick provided. Then convert your measurements

More information

Lesson 3: Using the Pythagorean Theorem. The Pythagorean Theorem only applies to triangles. The Pythagorean Theorem + = Example 1

Lesson 3: Using the Pythagorean Theorem. The Pythagorean Theorem only applies to triangles. The Pythagorean Theorem + = Example 1 Lesson 3: Using the Pythagorean Theorem The Pythagorean Theorem only applies to triangles. The Pythagorean Theorem + = Example 1 A sailboat leaves dock and travels 6 mi due east. Then it turns 90 degrees

More information

Design and Calibration of Submerged Open Channel Flow Measurement Structures: Part 3 - Cutthroat Flumes

Design and Calibration of Submerged Open Channel Flow Measurement Structures: Part 3 - Cutthroat Flumes Uth Stte University DigitlCommons@USU Reports Uth Wter Reserch Lbortory Jnury 1967 Design nd Clibrtion of Submerged Open Chnnel Flow Mesurement Structures: Prt 3 - Cutthrot Flumes Gylord V. Skogerboe M.

More information

Parking Lot HW? Joke of the Day: What do you get when you combine a flat, infinite geometric figure with a beef patty?

Parking Lot HW? Joke of the Day: What do you get when you combine a flat, infinite geometric figure with a beef patty? Parking Lot HW? Joke of the Day: What do you get when you combine a flat, infinite geometric figure with a beef patty? a plane burger Agenda 1 23 hw? Finish Special Right Triangles L8 3 Trig Ratios HW:

More information

Application of Geometric Mean

Application of Geometric Mean Section 8-1: Geometric Means SOL: None Objective: Find the geometric mean between two numbers Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse

More information

MORE TRIGONOMETRY

MORE TRIGONOMETRY MORE TRIGONOMETRY 5.1.1 5.1.3 We net introduce two more trigonometric ratios: sine and cosine. Both of them are used with acute angles of right triangles, just as the tangent ratio is. Using the diagram

More information

Small-scale observations of atypical fire spread caused by the interaction of wind, terrain and fire

Small-scale observations of atypical fire spread caused by the interaction of wind, terrain and fire Smll-scle oservtions of typicl fire spred cused y the interction of wind, terrin nd fire Json J. Shrples Applied nd Industril Mthemtics Reserch Group, School of Physicl Environmentl nd Mthemticl Sciences,

More information

Parking Lot HW? Joke of the Day: What do you call a leg that is perpendicular to a foot? Goals:

Parking Lot HW? Joke of the Day: What do you call a leg that is perpendicular to a foot? Goals: Parking Lot Joke of the Day: HW? What do you call a leg that is perpendicular to a foot? a right ankle Goals: Agenda 1 19 hw? Course Recommendations Simplify Radicals skill practice L8 2 Special Right

More information

Date: Period: Directions: Answer the following questions completely on a separate sheet of paper.

Date: Period: Directions: Answer the following questions completely on a separate sheet of paper. Name: Right Triangle Review Sheet Date: Period: Geometry Honors Directions: Answer the following questions completely on a separate sheet of paper. Part One: Simplify the following radicals. 1) 2) 3) 4)

More information

Week 11, Lesson 1 1. Warm Up 2. Notes Sine, Cosine, Tangent 3. ICA Triangles

Week 11, Lesson 1 1. Warm Up 2. Notes Sine, Cosine, Tangent 3. ICA Triangles Week 11, Lesson 1 1. Warm Up 2. Notes Sine, Cosine, Tangent 3. ICA Triangles HOW CAN WE FIND THE SIDE LENGTHS OF RIGHT TRIANGLES? Essential Question Essential Question Essential Question Essential Question

More information

Graphic Standards Guide

Graphic Standards Guide Grphic Stndrds Guide YOGA LOVE RUN PEACE 2 ABOUT This Grphic Stndrds Guide covers the bsic guidelines for the Lululemon Athletic s new grphic identity. The Guide provides summry of the primry fetures nd

More information

Lateral Earth Pressure on Lagging in Soldier Pile Wall Systems

Lateral Earth Pressure on Lagging in Soldier Pile Wall Systems Lterl Erth Pressure on Lgging in Soldier Pile Wll Systems Howrd A. Perko, Ph.D, P.E., CTL Thompson, Fort Collins, CO, USA hperko@ctlthompson.com John J Boulden, SGM, Inc., Grnd Junction, CO johnb@sgm-inc.com

More information

Coroutines in Propeller Assembly Language

Coroutines in Propeller Assembly Language www.prllxsemiondutor.om sles@prllxsemiondutor.om support@prllxsemiondutor.om phone: 916 632 4664 fx:916 624 8003 pplition Note oroutines in Propeller ssemly Lnguge strt: The multiore P8X32 does not require

More information

Unit 2: Right Triangle Trigonometry RIGHT TRIANGLE RELATIONSHIPS

Unit 2: Right Triangle Trigonometry RIGHT TRIANGLE RELATIONSHIPS Unit 2: Right Triangle Trigonometry This unit investigates the properties of right triangles. The trigonometric ratios sine, cosine, and tangent along with the Pythagorean Theorem are used to solve right

More information

Working Paper: Reversal Patterns

Working Paper: Reversal Patterns Remember to welcome ll ides in trding. AND remember to reserve your opinion until you hve independently vlidted the ide! Working Pper: Reversl Ptterns Working Pper In this pper I wnt to review nd (hopefully)

More information

Areas of Parallelograms and Triangles 7-1

Areas of Parallelograms and Triangles 7-1 Areas of Parallelograms and Triangles 7-1 Parallelogram A parallelogram is a quadrilateral where the opposite sides are congruent and parallel. A rectangle is a type of parallelogram, but we often see

More information

8.3 Trigonometric Ratios-Tangent. Geometry Mr. Peebles Spring 2013

8.3 Trigonometric Ratios-Tangent. Geometry Mr. Peebles Spring 2013 8.3 Trigonometric Ratios-Tangent Geometry Mr. Peebles Spring 2013 Bell Ringer 3 5 Bell Ringer a. 3 5 3 5 = 3 5 5 5 Multiply the numerator and denominator by 5 so the denominator becomes a whole number.

More information

RS γ γ. * Kevin D. Dayaratna is a doctoral candidate in Mathematical Statistics at the

RS γ γ. * Kevin D. Dayaratna is a doctoral candidate in Mathematical Statistics at the The Pythgoren Won-Loss Formul nd Hockey: A Sttisticl Justifiction for Using the Clssic Bsebll Formul s n Evlutive Tool in Hockey By Kevin D. Dyrtn nd Steven J. Miller * October 16, 2013 Abstrct Originlly

More information

Math Section 4.1 Special Triangles

Math Section 4.1 Special Triangles Math 1330 - Section 4.1 Special Triangles In this section, we ll work with some special triangles before moving on to defining the six trigonometric functions. Two special triangles are 30 60 90 triangles

More information

Acoustic measurements of bubbles in the wake of ship model in tank

Acoustic measurements of bubbles in the wake of ship model in tank Acoustic mesurements of bubbles in the wke of ship model in tnk A. Sutin,b, A. Benilov, H.-S. Roh nd Y.I. Nh c Stevens Institute of Technology, Cstle Point on Hudson, Hoboken, NJ 07030, USA b Artnn Lbortories,

More information

CS 188: Artificial Intelligence Spring Announcements

CS 188: Artificial Intelligence Spring Announcements CS 188: Artificil Intelligence Spring 2011 Lecture 19: Dynmic Byes Nets, Nïve Byes 4/6/2011 Pieter Aeel UC Berkeley Slides dpted from Dn Klein. Announcements W4 out, due next week Mondy P4 out, due next

More information

Trigonometry. terminal ray

Trigonometry. terminal ray terminal ray y Trigonometry Trigonometry is the study of triangles the relationship etween their sides and angles. Oddly enough our study of triangles egins with a irle. r 1 θ osθ P(x,y) s rθ sinθ x initial

More information

Performance Comparison of Dynamic Voltage Scaling Algorithms for Hard Real-Time Systems

Performance Comparison of Dynamic Voltage Scaling Algorithms for Hard Real-Time Systems Performnce Comprison of Dynmic Voltge Scling Algorithms for Hrd Rel-Time Systems Woonseok Kim Λ Dongkun Shin y Hn-Sem Yun y Jihong Kim y Sng Lyul Min Λ School of Computer Science nd Engineering Seoul Ntionl

More information

8.1 The Law of Sines Congruency and Oblique Triangles Using the Law of Sines The Ambiguous Case Area of a Triangle

8.1 The Law of Sines Congruency and Oblique Triangles Using the Law of Sines The Ambiguous Case Area of a Triangle Chapter 8 Applications of Trigonometry 8-1 8.1 The Law of Sines Congruency and Oblique Triangles Using the Law of Sines The Ambiguous Case Area of a Triangle A triangle that is not a right triangle is

More information

Daytime Habitat Selection by Resident Golden Eagles in Southern Idaho, USA

Daytime Habitat Selection by Resident Golden Eagles in Southern Idaho, USA Dytime Hbitt Selection by Resident Golden Egles in Southern Idho, USA Environmentl & Sttisticl Consultnts Chd LeBeu 1-04-14 Ryn Nielson, Eric Hllingstd, nd Dve Young cwlebeu@west-inc.com Introduction Risk

More information

Unit 2. Looking for Pythagoras. Investigation 5: Using the Pythagorean Theorem: Analyzing Triangles and Circles

Unit 2. Looking for Pythagoras. Investigation 5: Using the Pythagorean Theorem: Analyzing Triangles and Circles I can understand and apply the Pythagorean Theorem. Investigation 5 Unit 2 Looking for Pythagoras Investigation 5: Using the Pythagorean Theorem: Analyzing Triangles and Circles Lesson 1: Stopping Sneaky

More information

Right is Special 1: Triangles on a Grid

Right is Special 1: Triangles on a Grid Each student in your group should have a different equilateral triangle. Complete the following steps: Using the centimeter grid paper, determine the length of the side of the triangle. Write the measure

More information

Practice A. Congruent Figures. Are there any congruent figures in each picture? If there are, describe them

Practice A. Congruent Figures. Are there any congruent figures in each picture? If there are, describe them Name Date Class Practice A Are there any congruent figures in each picture? If there are, describe them. Determine the unknown measure in each set of congruent polygons. 7. 8. 9. 10. Name Date Class Practice

More information

ABSTRACT EXPERIMENTAL METHOD AND MODEL

ABSTRACT EXPERIMENTAL METHOD AND MODEL Proceedings of The Twelfth () Interntionl Offshore nd Polr Engineering Conference Kitkyushu, Jpn, My 31, Copyright by The Interntionl Society of Offshore nd Polr Engineers ISBN 1-53-5-3 (Set); ISSN 19-19

More information

Data Compression. Reduce the size of data. Reduces time to retrieve and transmit data. Compression ratio = original data size/compressed data size

Data Compression. Reduce the size of data. Reduces time to retrieve and transmit data. Compression ratio = original data size/compressed data size Dt Compression Reduce the size of dt. Reduces storge spce nd hence storge cost. Compression rtio = originl dt size/compressed dt size Reduces time to retrieve nd trnsmit dt. Lossless And Lossy Compression

More information

Comparison of Uniconazole Products on Greenhouse Crops

Comparison of Uniconazole Products on Greenhouse Crops A Report Sumitted to Fine Americs, Inc. Comprison of Uniconzole Products on Greenhouse Crops Spring 2006 Mtthew Blnchrd, Roerto Lopez, Mike Olrich, nd Erik Runkle Michign Stte University OBJECTIVE The

More information

A Measurement Framework for National Key Performance Measures

A Measurement Framework for National Key Performance Measures MINISTERIAL COUNCIL ON EDUCATION, EMPLOYMENT, TRAINING AND YOUTH AFFAIRS Performnce Mesurement nd Reporting Tskforce Working for qulity eduction outcomes A Mesurement Frmework for Ntionl Key Performnce

More information

Warm Up Find what numbers the following values are in between.

Warm Up Find what numbers the following values are in between. Warm Up Find what numbers the following values are in between. 1. 30 2. 14 3. 55 4. 48 Color squares on each side of the triangles with map pencils. Remember A square has 4 equal sides! Looking back at

More information

INTERNATIONAL WOMEN S CRICKET COUNCIL PLAYING CONDITIONS. Except as varied hereunder the Laws of Cricket (2000 Code 2nd Edition ) shall apply.

INTERNATIONAL WOMEN S CRICKET COUNCIL PLAYING CONDITIONS. Except as varied hereunder the Laws of Cricket (2000 Code 2nd Edition ) shall apply. INTERNATIONAL WOMEN S CRICKET COUNCIL PLAYING CONDITIONS Except s vried hereunder the Lws of Cricket (2000 Code 2nd Edition - 2003) shll pply. 1 Lw 1 - The Plyers 1.1 Lw 1.2 Nomintion of Plyers Lw 1.2

More information

Unit 3 Trigonometry. 3.1 Use Trigonometry to Find Lengths

Unit 3 Trigonometry. 3.1 Use Trigonometry to Find Lengths Topic : Goal : Unit 3 Trigonometry trigonometry I can use the primary trig ratios to find the lengths of sides in a right triangle 3.1 Use Trigonometry to Find Lengths In any right triangle, we name the

More information

Object preference by walking fruit flies, Drosophila melanogaster, is mediated by vision and graviperception

Object preference by walking fruit flies, Drosophila melanogaster, is mediated by vision and graviperception 2494 The Journl of Experimentl iology 23, 2494-256 2. Published by The ompny of iologists Ltd doi:.242/jeb.4749 Object preference by wlking fruit flies, rosophil melnogster, is medited by vision nd grviperception

More information

Converting Between Measurement Systems. ESSENTIAL QUESTION How can you use ratios and proportions to convert measurements? 7.4.E

Converting Between Measurement Systems. ESSENTIAL QUESTION How can you use ratios and proportions to convert measurements? 7.4.E LESSON 3.1 Converting Between Measurement Systems Proportionality 7.4.E Convert between measurement systems, including the use of proportions and the use of unit rates. Also 7.4.D? ESSENTIAL QUESTION How

More information

ACTIVITY: Finding a Formula Experimentally

ACTIVITY: Finding a Formula Experimentally 8.1 Volumes of Cylinders How can you find the volume of a cylinder? 1 ACTIVITY: Finding a Formula Experimentally Work with a partner. a. Find the area of the face of a coin. b. Find the volume of a stack

More information

Multiple Meaning Words: Kindergarten to Grade 2 More Teaching Tools at

Multiple Meaning Words: Kindergarten to Grade 2 More Teaching Tools at n prk n prk prk prk re of public lnd, s in city, tht is set side for rest nd enjoyment Let s got for wlk in the prk. re devoted specified purpose My prents compny is locted in business prk. field or stdium

More information

5.5 Use Inequalities in a Triangle

5.5 Use Inequalities in a Triangle 5.5 Use Inequalities in a Triangle Goal p Find possible side lengths of a triangle. Your Notes Example 1 Relate side length and angle measure Mark the largest angle, longest side, smallest angle, and shortest

More information

A Universal Zombie RPG Add-On

A Universal Zombie RPG Add-On A Universl Zombie RPG Add-On Brk Blckburn Assuming you re plying gme tht uses dice for tsk resolution, the Zombie Die cn be semlessly integrted into your gme. Zombies! Thnks to George Romero, Zombies re

More information

Year 10 Mathematics, 2007

Year 10 Mathematics, 2007 Student s Name: Teacher s Name: 10 Year 10 athematics, 2007 lgebra Use straightforward algebraic methods and sketch and interpret features of linear graphs Time: 20 minutes. Check that you have entered

More information

Areas of Trapezoids, Rhombuses, and Kites. To find the area of a trapezoid, rhombus, or kite

Areas of Trapezoids, Rhombuses, and Kites. To find the area of a trapezoid, rhombus, or kite 10-2 Areas of Trapezoids, Rombuses, and Kites Common Core State Standards G-MG.A.1 Use geometric sapes, teir measures, and teir properties to describe objects. MP 1, MP 3, MP 4, MP 6 Objective To find

More information

Applications of trigonometry

Applications of trigonometry Applications of trigonometry This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,

More information

Lesson 21: Special Relationships within Right Triangles Dividing into Two Similar Sub-Triangles

Lesson 21: Special Relationships within Right Triangles Dividing into Two Similar Sub-Triangles : Special Relationships within Right Triangles Dividing into Two Similar Sub-Triangles Learning Targets I can state that the altitude of a right triangle from the vertex of the right angle to the hypotenuse

More information

Perimeter. Perimeter is the distance around a shape. You can use grid. Step 1 On grid paper, draw a rectangle that has a length

Perimeter. Perimeter is the distance around a shape. You can use grid. Step 1 On grid paper, draw a rectangle that has a length Lesson 13.1 Perimeter Perimeter is the distance around a shape. You can use grid paper to count the number of units around the outside of a rectangle to find its perimeter. How many feet of ribbon are

More information

A /ense of Place. w-mt/^ortm. i/i the. J p m Q. by MICHAEL HJRTMA/N. A u/ilderness paddler ponders lake trout.

A /ense of Place. w-mt/^ortm. i/i the. J p m Q. by MICHAEL HJRTMA/N. A u/ilderness paddler ponders lake trout. A /ense of Plce w-mt/^ortm. i/i the. J p m Q m by MICHAEL HJRTMA/N A u/ilderness pddler ponders lke trout. In the north, in the spring, there lives little strem. Come summer, it brely flows; nd gret, gry

More information

MAXIMUM CARDIAC PERFORMANCE OF RAINBOW TROUT (ONCORHYNCHUS MYKISS) AT TEMPERATURES APPROACHING THEIR UPPER LETHAL LIMIT

MAXIMUM CARDIAC PERFORMANCE OF RAINBOW TROUT (ONCORHYNCHUS MYKISS) AT TEMPERATURES APPROACHING THEIR UPPER LETHAL LIMIT The Journl of Experimentl Biology 199, 663 672 (1996) Printed in Gret Britin The Compny of Biologists Limited 1996 JEB0137 663 MAXIMUM CARDIAC PERFORMANCE OF RAINBOW TROUT (ONCORHYNCHUS MYKISS) AT TEMPERATURES

More information

A Step, Stride and Heading Determination for the Pedestrian Navigation System

A Step, Stride and Heading Determination for the Pedestrian Navigation System Journl of Globl Positioning Systems (2004) Vol. 3,. 1-2: 273-279 A Step, Stride nd Heding Determintion for the Pedestrin Nvigtion System Jeong Won Kim GNSS Technology Reserch Center, Chungnm Ntionl University,

More information

Functions - Trigonometry

Functions - Trigonometry 10. Functions - Trigonometry There are si special functions that describe the relationship between the sides of a right triangle and the angles of the triangle. We will discuss three of the functions here.

More information

Discovering Special Triangles Learning Task

Discovering Special Triangles Learning Task The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still

More information

Pinpoint GPS 90-8M

Pinpoint GPS 90-8M FCC nd IC Complince Sttement PINPOINT KEY FOB, FCC ID MVU10148 ACMA: N2523 IC: 6094A 09291, 6094A 09305 This device complies with prt 15 of the FCC Rules. Opertion is suject to the following two conditions:

More information

Inertial Sensor-Based Methods in Walking Speed Estimation: A Systematic Review

Inertial Sensor-Based Methods in Walking Speed Estimation: A Systematic Review Sensors 2012, 12, 6102-6116; doi:10.3390/s120506102 OPEN ACCESS sensors ISSN 1424-8220 www.mdpi.com/journl/sensors Review Inertil Sensor-Bsed Methods in Wlking Speed Estimtion: A Systemtic Review Shuozhi

More information

Are You Ready? Pythagorean Theorem

Are You Ready? Pythagorean Theorem SKILL Pythagorean Theorem Teahing Skill Objetive Find the length of the hypotenuse of a right triangle. Have students read the Pythagorean Theorem. Restate the theorem in words, as follows: the sum of

More information

UNIT 2 RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios Instruction

UNIT 2 RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios Instruction Prerequisite Skills This lesson requires the use of the following skills: defining and calculating sine, cosine, and tangent setting up and solving problems using the Pythagorean Theorem identifying the

More information

Invading rainbow trout usurp a terrestrial prey subsidy from native charr and reduce their growth and abundance

Invading rainbow trout usurp a terrestrial prey subsidy from native charr and reduce their growth and abundance Oecologi (27) 153:461 47 DOI 1.17/s442-7-743-x COMMUNITY ECOLOGY Invding rinow trout usurp terrestril prey susidy from ntive chrr nd reduce their growth nd undnce Colden V. Bxter Kurt D. Fusch Msshi Murkmi

More information

Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task 3.1.2

Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task 3.1.2 Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic. Pythagorean Theorem; Task 3.. TASK 3..: 30-60 RIGHT TRIANGLES Solutions. Shown here is a 30-60 right triangle that has one leg of length and

More information

RULES OF INDOOR HOCKEY

RULES OF INDOOR HOCKEY RULES OF INDOOR HOCKEY 2007-2008 Rules of Indoor Hokey inluding explntions Effetive from 1 Jnury 2007 for the yers 2007 nd 2008 Copyright FIH 2006 The Interntionl Hokey Federtion Rue du Vlentin 61 CH

More information

9.3 Altitude-on-Hypotenuse Theorems

9.3 Altitude-on-Hypotenuse Theorems 9.3 Altitude-on-Hypotenuse Theorems Objectives: 1. To find the geometric mean of two numbers. 2. To find missing lengths of similar right triangles that result when an altitude is drawn to the hypotenuse

More information

Unit #8 Review Right Triangle Trigonometry. 1. Which of the following could represent the sides of a right triangle?

Unit #8 Review Right Triangle Trigonometry. 1. Which of the following could represent the sides of a right triangle? Name: Date: Unit #8 Review Right Triangle Trigonometry 1. Which of the following could represent the sides of a right triangle? (1) { 6, 8,14 } (2) {, 20, } (3) { 15, 20, } (4) {,15, 20 } 2. Which of the

More information

Owner s Manual For TAC Series Planetary Gear Air Winches ORIGINAL TEXT

Owner s Manual For TAC Series Planetary Gear Air Winches ORIGINAL TEXT Red this Owner s Mnul thoroughly efore operting the equipment. Keep it with the equipment t ll times. Replcements re ville from Thern, Inc., PO Box 347, Winon, MN 55987, 507-454-2996. www.thern.com IMPORTANT:

More information

Transform Your Classroom and Construct Learning with Geometry Golf

Transform Your Classroom and Construct Learning with Geometry Golf Transform Your Classroom and Construct Learning with Geometry Golf Nancy Norem Powell nancynpowell@gmail.com 2012 Conference Common Core State Standards for Mathematics Practice Standards: This lesson

More information

Scarborough Spring 2013 Math Exam I 1. "On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic work.

Scarborough Spring 2013 Math Exam I 1. On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic work. Scarborough Spring 2013 Math 365-501 Exam I 1 Math 365 Exam 1 Spring 2013 Scarborough NEATLY PRINT NAME: STUDENT ID: DATE: "On my honor, as an Aggie, I have neither given nor received unauthorized aid

More information

ANNEX 1 CLASSIFICATION AND LABELLING SUMMARY TABLES

ANNEX 1 CLASSIFICATION AND LABELLING SUMMARY TABLES ANNEX CLASSIFICATION AND LABELLING SUMMARY TABLES ANNEX CLASSIFICATION AND LABELLING SUMMARY TABLES NOTE: The codifiction of hzrd s is further explined in Annex 3 (Section ). The hzrd re intended to be

More information

Assessment of River Herring and Striped Bass in the Connecticut River: Abundance, Population Structure, and Predator/Prey Interactions

Assessment of River Herring and Striped Bass in the Connecticut River: Abundance, Population Structure, and Predator/Prey Interactions University of Connecticut DigitlCommons@UConn EEB Articles Deprtment of Ecology nd Evolutionry Biology 2009 Assessment of River Herring nd Striped Bss in the Connecticut River: Aundnce, Popultion Structure,

More information

AN Wireless Tower Assembly Instructions

AN Wireless Tower Assembly Instructions N Wireless Tower ssembly Instructions RED THOROUGHLY EFORE SSEMLY ND ERETION Thank you for your purchase of an N Wireless Self Supporting Tower. N Wireless Towers are designed and fabricated to be high-strength,

More information