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

Contents TRIGONOMETRIC METHODS PROBABILITY DISTRIBUTIONS

Contents TRIGONOMETRIC METHODS PROBABILITY DISTRIBUTIONS ontents UNIT 7 TRIGONOMETRI METHODS Lesson 1 Trigonometric Functions................... 462 1 onnecting ngle Mesures nd Liner Mesures.............. 463 2 Mesuring Without Mesuring.........................

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

INVESTIGATION 2. What s the Angle?

INVESTIGATION 2. What s the Angle? INVESTIGATION 2 Wht s the Angle? In the previous investigtion, you lerned tht when the rigidity property of tringles is comined with the ility to djust the length of side, the opportunities for useful

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

Chp. 3_4 Trigonometry.notebook. October 01, Warm Up. Pythagorean Triples. Verifying a Pythagorean Triple... Pythagorean Theorem

Chp. 3_4 Trigonometry.notebook. October 01, Warm Up. Pythagorean Triples. Verifying a Pythagorean Triple... Pythagorean Theorem Chp. 3_4 Trigonometry.noteook Wrm Up Determine the mesure of the vrile in ech of the following digrms: x + 2 x x 5 x + 3 Pythgoren Theorem - is fundmentl reltionship mongst the sides on RIGHT tringle.

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

St Ac Ex Sp TOPICS (Text and Practice Books) 4.1 Triangles and Squares Pythagoras' Theorem - -

St Ac Ex Sp TOPICS (Text and Practice Books) 4.1 Triangles and Squares Pythagoras' Theorem - - MEP: Demonstrtion Projet UNIT 4 Trigonometry N: Shpe, Spe nd Mesures e,f St Ex Sp TOPIS (Text nd Prtie ooks) 4.1 Tringles nd Squres - - - 4. Pythgors' Theorem - - 4.3 Extending Pythgors' Theorem - - 4.4

More information

Geometry Proofs: Chapter 7, Sections 7.1/7.2

Geometry Proofs: Chapter 7, Sections 7.1/7.2 Pythgoren Theorem: Proof y Rerrngement of re Given: Right tringle with leg lengths nd, nd hypotenuse length. Prove: 2 2 2 = + Proof #1: We re given figures I nd II s ongruent right tringles III with leg

More information

Apply the Law of Sines. You solved right triangles. You will solve triangles that have no right angle.

Apply the Law of Sines. You solved right triangles. You will solve triangles that have no right angle. 13.5 pply te Lw of Sines TEKS.1,.4, 2.4.; P.3.E efore Now You solved rigt tringles. You will solve tringles tt ve no rigt ngle. Wy? So you n find te distne etween frwy ojets, s in Ex. 44. Key Voulry lw

More information

In any right-angle triangle the side opposite to the right angle is called the Label the Hypotenuse in each diagram above.

In any right-angle triangle the side opposite to the right angle is called the Label the Hypotenuse in each diagram above. 9 Ademi Mth Dte: Pythgoren Theorem RIGHT ANGLE TRIANGLE - A right tringle is tringle with one 90 0 ngle. For exmple: In ny right-ngle tringle the side opposite to the right ngle is lled the Lbel the Hypotenuse

More information

1 Measurement. What you will learn. World s largest cylindrical aquarium. Australian Curriculum Measurement and Geometry Using units of measurement

1 Measurement. What you will learn. World s largest cylindrical aquarium. Australian Curriculum Measurement and Geometry Using units of measurement Austrlin Curriulum Mesurement nd Geometry Using units of mesurement hpter 1 Mesurement Wht you will lern 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Conversion of units Perimeter Cirumferene Are Are of irle Surfe

More information

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

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

More information

Chapter 5. Triangles and Vectors

Chapter 5. Triangles and Vectors www.ck12.org Chpter 5. Tringles nd Vectors 5.3 The Lw of Sines Lerning Objectives Understnd how both forms of the Lw of Sines re obtined. Apply the Lw of Sines when you know two ngles nd non-included side

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

MTH 112: Elementary Functions

MTH 112: Elementary Functions 1/14 MTH 112: Elementry Functions Section 8.1: Lw of Sines Lern out olique tringles. Derive the Lw os Sines. Solve tringles. Solve the miguous cse. 8.1:Lw of Sines. 2/14 Solving olique tringles Solving

More information

Lesson 2 PRACTICE PROBLEMS Using Trigonometry in Any Triangle

Lesson 2 PRACTICE PROBLEMS Using Trigonometry in Any Triangle Nme: Unit 6 Trigonometri Methods Lesson 2 PRTIE PROLEMS Using Trigonometry in ny Tringle I n utilize the Lw of Sines nd the Lw of osines to solve prolems involving indiret mesurement in non-right tringles.

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 Right Tringle Trigonometry To the ncient Greeks, trigonometry ws the study of right tringles. Trigonometric functions (sine, cosine, tngent, cotngent, secnt, nd cosecnt) cn be defined s right tringle rtios

More information

17.3 Find Unknown Side Lengths

17.3 Find Unknown Side Lengths ? Nme 17.3 Find Unknown Side Lenths ALGEBRA Essentil Question How cn you find the unknown lenth of side in polyon when you know its perimeter? Geometry nd Mesurement 3.7.B MATHEMATICAL PROCESSES 3.1.A,

More information

Special Right Triangles

Special Right Triangles Pge of 5 L E S S O N 9.6 Specil Right Tringles B E F O R E Now W H Y? Review Vocbulr hpotenuse, p. 465 leg, p. 465 You found side lengths of right tringles. You ll use specil right tringles to solve problems.

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

Right Triangles and Trigonometry. Right Triangles and Trigonometry

Right Triangles and Trigonometry. Right Triangles and Trigonometry Right Tringles nd Trigonometr hpter Overview nd Pcing PING (ds) Regulr lock sic/ sic/ verge dvnced verge dvnced Geometric Men (pp. ) 0. 0. Find the geometric men etween two numers. Solve prolems involving

More information

SUMMER ASSIGNMENT FOR FUNCTIONS/TRIGONOMETRY Due September 7 th

SUMMER ASSIGNMENT FOR FUNCTIONS/TRIGONOMETRY Due September 7 th SUMMER ASSIGNMENT FOR FUNCTIONS/TRIGONOMETRY Due Septemer 7 th This summer ssignment is designed to prepre ou for Functions/Trigonometr. Nothing on the summer ssignment is new. Everthing is review of topics

More information

Starter. The Cosine Rule. What the Cosine Rule is and how to apply it to triangles. I can write down the Cosine Rule from memory.

Starter. The Cosine Rule. What the Cosine Rule is and how to apply it to triangles. I can write down the Cosine Rule from memory. Strter 1) Find the re of the green tringle. 12.8m 2) 2 4 ( + ) x 3 5 3 2 54.8 o 9.7m The Cosine Rule Tody we re lerning... Wht the Cosine Rule is nd how to pply it to tringles. I will know if I hve een

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

Recall that the area of a triangle can be found using the sine of one of the angles.

Recall that the area of a triangle can be found using the sine of one of the angles. Nme lss Dte 14.1 Lw of Sines Essentil Question: How n you use trigonometri rtios to find side lengts nd ngle mesures of non-rigt tringles? Resoure Loker Explore Use n re Formul to Derive te Lw of Sines

More information

Chapter 31 Pythagoras theorem and trigonometry (2)

Chapter 31 Pythagoras theorem and trigonometry (2) HPTR 31 86 3 The lengths of the two shortest sides of right-ngled tringle re m nd ( 3) m respetively. The length of the hypotenuse is 15 m. Show tht 2 3 108 Solve the eqution 2 3 108 Write down the lengths

More information

77.1 Apply the Pythagorean Theorem

77.1 Apply the Pythagorean Theorem Right Triangles and Trigonometry 77.1 Apply the Pythagorean Theorem 7.2 Use the Converse of the Pythagorean Theorem 7.3 Use Similar Right Triangles 7.4 Special Right Triangles 7.5 Apply the Tangent Ratio

More information

11.4 Apply the Pythagorean

11.4 Apply the Pythagorean 11.4 Apply the Pythagorean Theorem and its Converse Goal p and its converse. Your Notes VOCABULARY Hypotenuse Legs of a right triangle Pythagorean theorem THE PYTHAGOREAN THEOREM Words If a triangle is

More information

6 TRIGONOMETRY TASK 6.1 TASK 6.2. hypotenuse. opposite. adjacent. opposite. hypotenuse 34. adjacent. opposite. a f

6 TRIGONOMETRY TASK 6.1 TASK 6.2. hypotenuse. opposite. adjacent. opposite. hypotenuse 34. adjacent. opposite. a f 1 6 TIGONOMETY TK 6.1 In eh tringle elow, note the ngle given nd stte whether the identified side is in the orret position or not. 1. 4. opposite 41 2. djent 3. 58 63 djent 32 hypotenuse 5. 68 djent 6.

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

Chapter 8: Right Triangles (page 284)

Chapter 8: Right Triangles (page 284) hapter 8: Right Triangles (page 284) 8-1: Similarity in Right Triangles (page 285) If a, b, and x are positive numbers and a : x = x : b, then x is the between a and b. Notice that x is both in the proportion.

More information

5.5 The Law of Sines

5.5 The Law of Sines 434 HPTER 5 nlyti Trigonometry 5.5 Te Lw of Sines Wt you ll lern out Deriving te Lw of Sines Solving Tringles (S, S) Te miguous se (SS) pplitions... nd wy Te Lw of Sines is powerful extension of te tringle

More information

Michelangelo,

Michelangelo, Michelngelo Red Section 3 to lern bout this Renissnce figure. Renissnce Figure 7 Michelngelo, 1475 1564 Techers Curriculum Institute Leding Figures of the Renissnce 1 Titin Red Section 4 to lern bout this

More information

SUMMER ASSIGNMENT FOR FUNCTIONS/TRIGONOMETRY Bring to school the 1 st day of class!

SUMMER ASSIGNMENT FOR FUNCTIONS/TRIGONOMETRY Bring to school the 1 st day of class! SUMMER ASSIGNMENT FOR FUNCTIONS/TRIGONOMETRY Bring to school the st d of clss! This summer ssignment is designed to prepre ou for Functions/Trigonometr. Nothing on the summer ssignment is new. Everthing

More information

Lesson 5.1 Polygon Sum Conjecture

Lesson 5.1 Polygon Sum Conjecture Lesson 5.1 Polgon Sum onjeture In Eerise 1, find eh lettered ngle mesure. 1.,,, d, e e d 97 26 2. ne eterior ngle of regulr polgon mesures 10. Wht is the mesure of eh interior ngle? How mn sides does the

More information

8.1 Right Triangle Trigonometry; Applications

8.1 Right Triangle Trigonometry; Applications SECTION 8.1 Right Tringle Trigonometry; pplitions 505 8.1 Right Tringle Trigonometry; pplitions PREPRING FOR THIS SECTION efore getting strted, review the following: Pythgoren Theorem (ppendix, Setion.,

More information

5-8 Applying Special Right Triangles

5-8 Applying Special Right Triangles 5-8 Applying Special Right Triangles Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up For Exercises 1 and 2, find the value of x. Give your answer in simplest radical form. 1. 2. Simplify each

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

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

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

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 The Pythagorean Theorem and Its Converse Vocabulary Review 1. Write the square and the positive square root of each number. Number 9 Square Positive Square Root 1 4 1 16 Vocabulary Builder leg (noun)

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

Section 8: Right Triangles

Section 8: Right Triangles The following Mathematics Florida Standards will be covered in this section: MAFS.912.G-CO.2.8 Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition

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

Unit 2 Day 4 Notes Law of Sines

Unit 2 Day 4 Notes Law of Sines AFM Unit 2 Day 4 Notes Law of Sines Name Date Introduction: When you see the triangle below on the left and someone asks you to find the value of x, you immediately know how to proceed. You call upon your

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

CCM8 Unit 7: Pythagorean Theorem Vocabulary

CCM8 Unit 7: Pythagorean Theorem Vocabulary CCM8 Unit 7: Pythagorean Theorem Vocabulary Base Exponent Hypotenuse Legs Perfect Square Pythagorean Theorem When a number is raised to a power, the number that is used as a factor The number that indicates

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

Chapter. Similar Triangles. Copyright Cengage Learning. All rights reserved.

Chapter. Similar Triangles. Copyright Cengage Learning. All rights reserved. Chapter 5 Similar Triangles Copyright Cengage Learning. All rights reserved. 5.4 The Pythagorean Theorem Copyright Cengage Learning. All rights reserved. The Pythagorean Theorem The following theorem will

More information

Chapter 7. Right Triangles and Trigonometry

Chapter 7. Right Triangles and Trigonometry Chapter 7 Right Triangles and Trigonometry 4 16 25 100 144 LEAVE IN RADICAL FORM Perfect Square Factor * Other Factor 8 20 32 = = = 4 *2 = = = 75 = = 40 = = 7.1 Apply the Pythagorean Theorem Objective:

More information

PCT MINIMUM DOCUMENTATION

PCT MINIMUM DOCUMENTATION Ref.: PCT Minimum Documenttion pge: 4.1.1 PCT MINIMUM TION INVENTORY S CCORDING TO PCT RULE 34.1 (PERIOD FROM 1920 TO 2000) Explntory Notes 1. On the following pges is given the inventory of ptent documents,

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

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

* SEE ANCHOR SCHEDULE SHEET 7

* SEE ANCHOR SCHEDULE SHEET 7 "-20 MLE PNELMTE W/ WINGNUT -1/2" O.C. "-20 X 1/2" MCHINE OLT & NUT -1/2" O.C 04/0/15 JH UPDTE TO 5TH EDITION (2014) FC 8/14/1 Y DTE SPCING ERROR OR MSONRY POWERS CLK-IN W/ "-20 SIDEWLK OLT -1/2" O.C.

More information

Geometry 1A Multiple Choice Final Exam Practice

Geometry 1A Multiple Choice Final Exam Practice Name Date: Per: Geometry 1 Multiple hoice Final Eam Practice 1. Let point E be between points F and G. Solve for r. FE = 6r 20 EG = 5r 24 FG = 55 [] r = 14 [] r = 5 [] r = 4 [D] r = 9 2. m JHI = ( 2 7)

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

Learning Goal: I can explain when to use the Sine, Cosine and Tangent ratios and use the functions to determine the missing side or angle.

Learning Goal: I can explain when to use the Sine, Cosine and Tangent ratios and use the functions to determine the missing side or angle. MFM2P Trigonometry Checklist 1 Goals for this unit: I can solve problems involving right triangles using the primary trig ratios and the Pythagorean Theorem. U1L4 The Pythagorean Theorem Learning Goal:

More information

Skills Practice Skills Practice for Lesson 3.1

Skills Practice Skills Practice for Lesson 3.1 Skills Practice Skills Practice for Lesson.1 Name Date Get Radical or (Be) 2! Radicals and the Pythagorean Theorem Vocabulary Write the term that best completes each statement. 1. An expression that includes

More information

TeeJay Publishers Homework for Level C book Ch 12 - Length & Area

TeeJay Publishers Homework for Level C book Ch 12 - Length & Area Chpter 12 Exerise Perentges 1 Length & Are 1. Would you use ruler, tpe mesure or r odometer to mesure : your tehers height the length of 5 note the length of your edroom d the distne from Glsgow to Crlisle?

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

Two Special Right Triangles

Two Special Right Triangles Page 1 of 7 L E S S O N 9.3 In an isosceles triangle, the sum of the square roots of the two equal sides is equal to the square root of the third side. Two Special Right Triangles In this lesson you will

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

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

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

10mm SHOWER PANEL SIZES 1000 & 1200 SIZES 500, 600, 700, 800 & 900 SIZES 1000 & 1200 OPTION 1 - PAGES 2-4 OPTION 2 - PAGES 2-5 OPTION 3 - PAGES 6-7

10mm SHOWER PANEL SIZES 1000 & 1200 SIZES 500, 600, 700, 800 & 900 SIZES 1000 & 1200 OPTION 1 - PAGES 2-4 OPTION 2 - PAGES 2-5 OPTION 3 - PAGES 6-7 SHOWER PANEL 785 / Issue / 5 SIZES 5, 6, 7, 8 & 9 SIZES & SIZES & OPTION - PAGES - OPTION - PAGES - 5 OPTION 3 - PAGES 6-7 Plese red these instrutions refully nd keep for future referene. Inorret fitting

More information

Practice 9-1. The Real Numbers. Write all names that apply to each number

Practice 9-1. The Real Numbers. Write all names that apply to each number Chapter 9 Practice 9-1 The Real Numbers Write all names that apply to each number. 1. 3.2 2. 2 5 3. 12 4. 4 2 5. 20 6. 16 7. 7 8 8. 0.15 9. 18 2 10. 45 11. 25 12. 6.75 State if the number is rational,

More information

Unit 4. Triangle Relationships. Oct 3 8:20 AM. Oct 3 8:21 AM. Oct 3 8:26 AM. Oct 3 8:28 AM. Oct 3 8:27 AM. Oct 3 8:27 AM

Unit 4. Triangle Relationships. Oct 3 8:20 AM. Oct 3 8:21 AM. Oct 3 8:26 AM. Oct 3 8:28 AM. Oct 3 8:27 AM. Oct 3 8:27 AM Unit 4 Triangle Relationships 4.1 -- Classifying Triangles triangle -a figure formed by three segments joining three noncollinear points Classification of triangles: by sides by angles Oct 3 8:20 AM Oct

More information

Step #2 Liner Type. Enter the pool shape you selected from STEP 2 of the MEASURING INSTRUCTIONS:

Step #2 Liner Type. Enter the pool shape you selected from STEP 2 of the MEASURING INSTRUCTIONS: LINER MESURING FORM 6385 150th ve. N. Clearwater, FL 33760 1-877-747-7946 Fax: 1-888-858-7946 Please refer to the MESURING INSTRUCTIONS for details on accurately completing this form! o EDED o LOW HUNG

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

Math 154 Chapter 7.7: Applications of Quadratic Equations Objectives:

Math 154 Chapter 7.7: Applications of Quadratic Equations Objectives: Math 154 Chapter 7.7: Applications of Quadratic Equations Objectives: Products of numbers Areas of rectangles Falling objects Cost/Profit formulas Products of Numbers Finding legs of right triangles Finding

More information

Chapter 4 Group of Volunteers

Chapter 4 Group of Volunteers CHAPTER 4 SAFETY CLEARANCE, FREEBOARD AND DRAUGHT MARKS 4-1 GENERAL 4-1.1 This chpter specifies the minimum freebord for inlnd wterwy vessels. It lso contins requirements concerning the indiction of the

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

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

Chapter88. Measurement. Contents: A Length B Perimeter C Area D The area of a rectangle E Volume F Capacity G Mass

Chapter88. Measurement. Contents: A Length B Perimeter C Area D The area of a rectangle E Volume F Capacity G Mass Chpter88 Mesurement Contents: A Length B Perimeter C Are D The re of rectngle E Volume F Cpcity G Mss 164 MEASUREMENT (Chpter 8) Opening prolem Byron is out to ply gme of soccer on pitch like the one longside.

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

United States Patent (19)

United States Patent (19) United Sttes Ptent (19) Mobley (11) 45) Ptent Number: Dte of Ptent: Nov. 19, 1991 (54) BICYCLE RACK FORMOUNTING ON A VAN 76 Inventor: Mrk H. Mobley, P.O. Box 636, Norris, Tenn. 37828 21 Appl. No.: 436,492

More information

CH 21 THE PYTHAGOREAN THEOREM

CH 21 THE PYTHAGOREAN THEOREM 121 CH 21 THE PYTHAGOREAN THEOREM The Right Triangle A n angle of 90 is called a right angle, and when two things meet at a right angle, we say they are perpendicular. For example, the angle between a

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

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

Paradoxical Euler or Integration by Differentiation:

Paradoxical Euler or Integration by Differentiation: Prdoicl Euler or Integrtion by Differentition: A Synopsis of E36 Andrew Fbin nd Hieu Nguyen Rown University The Euler Society 008 Annul Conference July 1, 008 E36 Eposition de quelques prdoes dns le clcul

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

Math Practice Use Clear Definitions

Math Practice Use Clear Definitions Prllel Lines nd Trnsversls How cn you descrie ngles formed y rllel lines nd rnsversls? Trnsverse When n ojec is rnsverse, i is lying or exending cross somehing ACTIVITY: A Proery of Prllel Lines Work wih

More information

Lesson 8: Application Technology

Lesson 8: Application Technology The type of ppliction equipment used must suit the type of ppliction. In this module, you ll lern the prts of the most common types of ppliction equipment used by ssistnt pplictors, s well s how to properly

More information

8.7 Extension: Laws of Sines and Cosines

8.7 Extension: Laws of Sines and Cosines www.ck12.org Chapter 8. Right Triangle Trigonometry 8.7 Extension: Laws of Sines and Cosines Learning Objectives Identify and use the Law of Sines and Cosines. In this chapter, we have only applied the

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

First Aid in School Policy

First Aid in School Policy First Aid in School Policy Approved y the Governing Body of West Monkton Church of Englnd Primry School: Reviewed y Governors nd Agreed: Septemer 2017 At West Monkton we im to e n inspirtionl community,

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

Simplifying Radical Expressions and the Distance Formula

Simplifying Radical Expressions and the Distance Formula 1 RD. Simplifying Radical Expressions and the Distance Formula In the previous section, we simplified some radical expressions by replacing radical signs with rational exponents, applying the rules of

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

Name: Class: Date: Geometry Chapter 4 Test Review

Name: Class: Date: Geometry Chapter 4 Test Review Name: Class: Date: ID: C Geometry Chapter 4 Test Review. 1. Determine the measure of angle UPM in the following figure. Explain your reasoning and show all your work. 3. Determine the side length of each

More information

Nanobiophysical exploration of transthyretin amyloid fibrils Final report

Nanobiophysical exploration of transthyretin amyloid fibrils Final report Nnoiophysicl explortion of trnsthyretin myloid firils Finl report Summry In this work we used AFM to follow the myloidogenetic pthwy of trnsthyretin (TTR) y imging the events leding to the formtion of

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

Similar Right Triangles

Similar Right Triangles MATH 1204 UNIT 5: GEOMETRY AND TRIGONOMETRY Assumed Prior Knowledge Similar Right Triangles Recall that a Right Triangle is a triangle containing one 90 and two acute angles. Right triangles will be similar

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

Workfit -SR, Dual Monitor Short Surface

Workfit -SR, Dual Monitor Short Surface User's Guide Workfit -SR, Dul Monitor Short Surfce Do not throw wy! Crdord locks needed for instlltion. Visit http://www.ergotron.com/workfi t-sr-instll for instlltion instructionl video. NOTE: 10 feet

More information