8th Grade. Slide 1 / 145 Slide 2 / 145. Slide 3 (Answer) / 145. Slide 3 / 145. Slide 5 / 145. Slide 4 / 145. Pythagorean Theorem, Distance & Midpoint

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1 Slide 1 / 14 Slide 2 / 14 8th Grade Pthagorean Theorem, Distance & Midpoint Slide 3 / 14 Table of ontents Slide 3 () / 14 Table of ontents Proofs lick on a topic to go to that section Proofs lick on a topic to go to that section Pthagorean Theorem Distance Formula Midpoints Pthagorean Theorem Vocabular Words are bolded in the presentation. The tet bo the word is in is then Distance Formula linked to the page at the end of the presentation with the Midpoints word defined on it. Teacher Notes Glossar & Standards Glossar & Standards Slide 4 / 14 Slide / 14 Introduction to Proofs Proofs Lab: Introduction to Proofs lick to return to the table of contents

2 Teacher Notes Slide () / 14 Introduction to Proofs Have the students brainstorm the steps required to make a sandwich in small groups (2-4 students per group). The questions can be completed either in their groups or as a class discussion Lab: (located Introduction on the net to 6 Proofs slides). The questions in this lab address MP.3: sk: What are some possibilities here? How can ou prove that our answer is correct? Slide 6 / 14 1 Determine which tpe of sandwich ou would like to start. hoose onl one. Peanut utter & Jell Ham and heese Fluffernutter D Tuna melt LT E hicken Salad LT F Peanut utter & anana G ologna and heese H Grilled heese Slide 6 () / 14 Slide 7 / 14 1 Determine which tpe of sandwich ou would like to start. hoose onl one. Peanut utter & Jell Fill in the blank on the net Ham and heese page with the winner of this Fluffernutter poll. Teacher Notes D Tuna melt LT If ou wanted to, ou could E hicken Salad LT make the class prepare our lunch sandwich that F Peanut utter & anana da and adjust the wording G ologna and heese on the net few slides. H Grilled heese 2) Fill in the blank with the winner of the class poll. Everda for lunch, ou make ourself a sandwich. The sandwich that ou are making toda is. 3) What steps do ou need to take in order to make our sandwich? List all of the thoughts of our group in the space provided on the Lab WS. Slide 7 () / 14 2) Fill in the blank with the winner of the class poll. Everda for lunch, ou make ourself a sandwich. The sandwich that ou are making toda is. Teacher Notes Split the students up into groups of 2-4. Give them about 3) What steps do ou need to take in order to make our sandwich? List all of minutes the thoughts to of our group in the space provided on the Lab WS. brainstorm their ideas to make the sandwich. Slide 8 / 14 4) List an additional thoughts provided b the rest of the groups in the class.

3 Slide 8 () / 14 4) List an additional thoughts provided b the rest of the groups in the class. Teacher Notes This is where ou list all of the ideas that our students come up with. The wording of the question corresponds to the Lab WS for the students. Slide 9 / 14 ) List all of the thoughts contributed b our group and the rest of the class in the chronological order decided upon b the class. Slide 9 () / 14 ) List all of the thoughts contributed b our group and the rest of the class in the chronological order decided upon b the class. Slide / 14 6) Was the sandwich made correctl? Wh or wh not? If not, what steps do ou need to add? Teacher Notes Use this slide to chronologicall list all of the student thoughts. Slide () / 14 6) Was the sandwich made correctl? Wh or wh not? If not, what steps do ou need to add? Slide 11 / 14 7) Write down the revised list all of the thoughts contributed b our group and the rest of the class in the chronological order. Teacher Notes Use this slide to list an additional steps required.

4 Slide 11 () / 14 7) Write down the revised list all of the thoughts contributed b our group and the rest of the class in the chronological order. Teacher Notes Use this slide to chronologicall list the revised list of the student thoughts. Slide 12 / 14 Proofs proof involves writing reasoned, logical eplanations that use definitions, algebraic properties, postulates, and previousl proven theorems to arrive at a conclusion. postulate is a propert that is accepted without proof, usuall because one can easil understand wh it is true. One of the classic postulates of Geometr states that "Through an two points, there is eactl one line". Slide 13 / 14 Proofs The lab that ou completed about making a sandwich has the same general steps as a proof. You brainstorm and write down various ideas based on what ou know, and organize them into the proper order that will lead ou to the final conclusion. You can also use proofs when solving algebraic equations. s ou write each step/idea, ou also write down the definition, propert, postulate, etc. that justifies our step/idea. The eample shown below is written as a two-column proof. Given: = 14 Prove: = 4 Statements = Given Reasons 2. 3 = Subtraction 3. = 4 3. Division Note: The first step to solving a proof is alwas "Given" to ou. Slide 1 / 14 Slide 14 / 14 Proofs When writing two-column proofs, ou can also combine the rules of algebra and properties of geometr. Eample: Given: m = (4 + 3), m = ( + 4), ( + 4) m = ( + 23) Prove: Δ is an acute triangle (4 + 3) ( + 23) Slide 16 / 14 Proofs Proofs Given: m = (4 + 3), m = ( + 4), ( + 4) m = ( + 23) Prove: Δ is an acute triangle (4 + 3) ( + 23) Statements Reasons 1. m = (4 + 3), m = ( + 4), m = ( + 23) 1. Given 2. m + m + m = ll of the interior angles of a triangle sum to = Substitution = ombine Like Terms. =. Subtraction 6. = 1 6. Division 7. m = 4(1) + 3 = 63, m = (1) + 4 = Substitution m = (1) + 23 = Δ is an acute triangle 8. Definition of an acute Δ Eample: Write a two-column proof. Given: 2( + 4) + 4( - ) = -42 Prove: = Statements Reasons

5 Slide 16 () / 14 Slide 17 / 14 Proofs Eample: Write a two-column proof. Given: 2( + 4) + 4( - ) = -42 Prove: = Statements Reasons 1. 2( + 4) + 4( - ) = Given Statements = Distributive Reasons Propert = ombine Like Terms Simplif 4. 6 = ddition. =. Division Proofs E Eample: Given: m D = (4 + 7), m E = (9 + ), (9 + ) m F = (2 + 3) Prove: ΔDEF is an obtuse Δ (4 + 7) (2 + 3) D Statements Reasons F Slide 17 () / 14 Slide 18 / 14 Proofs E Eample: tatements Given: m D = (4 + 7), Reasons m E = (9 + ), (9 + ) + 7), m E = (9 + ), m F = ( Given 3) + 3) Prove: ΔDEF is an obtuse Δ m E + m F = The interior angles of (4 + 7) (2 + 3) a triangle sum to 180D = Substitution 1 = 180 Statements 4. ombine Like Terms Reasons 16. Subtraction 6. Division (11) + 7 = 1, 11) + = 4, 11) + 3 = 2 s an obtuse Δ 7. Substitution 8. Definition of an obtuse Δ F Given: -6( - 3) + 3( - ) = -18. Prove: = 7 Proofs The net Response Questions are all based on the problem below. If needed, fill in the two-column proof as ou answer each question. Statements Reasons Slide 19 / 14 2 Given: -6( - 3) + 3( - ) = -18. Prove: = 7 What is the first statement? = -18 Slide 19 () / 14 2 Given: -6( - 3) + 3( - ) = -18. Prove: = 7 What is the first statement? = = -18-6( - 3) + 3( - ) = -18 D = = -18-6( - 3) + 3( - ) = -18 D = -18

6 Slide 20 / 14 3 Given: -6( - 3) + 3( - ) = -18. Prove: = 7 Slide 20 () / 14 3 Given: -6( - 3) + 3( - ) = -18. Prove: = 7 What is the first reason? Distributive Propert ddition Multiplication D Given What is the first reason? D Distributive Propert Statements ddition 1. -6( - 3) + 3( - ) = -18 Multiplication D Given Reasons 1. Given Slide 21 / 14 4 Given: -6( - 3) + 3( - ) = -18. Prove: = 7 What is the second statement? = -18 Slide 21 () / 14 4 Given: -6( - 3) + 3( - ) = -18. Prove: = 7 What is the second statement? = = -18-6( - 3) + 3( - ) = -18 D = = -18-6( - 3) + 3( - ) = -18 D = -18 Slide 22 / 14 Given: -6( - 3) + 3( - ) = -18. Prove: = 7 What is the second reason? Distributive Propert Slide 22 () / 14 Given: -6( - 3) + 3( - ) = -18. Prove: = 7 What is the second reason? Distributive Propert ddition Multiplication D Given ddition Statements 1. -6( Multiplication - 3) + 3( - ) = = -18 D Given Reasons 1. Given 2. Distributive Propert

7 Slide 23 / 14 6 Given: -6( - 3) + 3( - ) = -18. Prove: = 7 What is the third statement? -3-3 = -18 Slide 23 () / 14 6 Given: -6( - 3) + 3( - ) = -18. Prove: = 7 What is the third statement? -3-3 = = = = = -18 D D = -18 D = -18 Slide 24 / 14 7 Given: -6( - 3) + 3( - ) = -18. Prove: = 7 What is the third reason? Simplif or ombine Like Terms Slide 24 () / 14 7 Given: -6( - 3) + 3( - ) = -18. Prove: = 7 What is the third reason? Simplif or ombine Like Terms ddition Division D Subtraction ddition Statements 1. -6( Division - 3) + 3( - ) = = -18 D Subtraction = -18 Reasons 1. Given 2. Distributive Propert 3. Simplif or ombine Like Terms Slide 2 / 14 8 Given: -6( - 3) + 3( - ) = -18. Prove: = 7 What is the fourth statement? -3 = -1 Slide 2 () / 14 8 Given: -6( - 3) + 3( - ) = -18. Prove: = 7 What is the fourth statement? -3 = -1-3 = = -1-3 = = -1 D 9 = -21 D 9 = -21

8 Slide 26 / 14 9 Given: -6( - 3) + 3( - ) = -18. Prove: = 7 What is the fourth reason? Slide 26 () / 14 9 Given: -6( - 3) + 3( - ) = -18. Prove: = 7 D What is the fourth reason? Simplif or ombine Like Terms ddition Division D Subtraction Simplif or ombine Like Terms Statements Reasons ddition 1. -6( - 3) + 3( - ) = Given = Distributive Division Propert = Simplif or D Subtraction ombine Like Terms = Subtraction Slide 27 / 14 Slide 27 () / 14 Given: -6( - 3) + 3( - ) = -18. Prove: = 7 What is the fifth statement? = 7 Given: -6( - 3) + 3( - ) = -18. Prove: = 7 What is the fifth statement? = 7 = = - D = = = - D = Slide 28 / Given: -6( - 3) + 3( - ) = -18. Prove: = 7 What is the fifth reason? Slide 28 () / Given: -6( - 3) + 3( - ) = -18. Prove: = 7 What is the fifth reason? Simplif or ombine Like Terms ddition Division D Subtraction Simplif or ombine Like Terms Statements Reasons ddition 1. -6( - 3) + 3( - ) = Given = Distributive Division Propert = Simplif or D Subtraction ombine Like Terms = Subtraction. = 7. Division

9 Slide 29 / 14 Proofs The net 8 Response Questions are all based on the problem below. If needed, fill in the two-column proof as ou answer each question. Given: and D are supplementar Prove: EFG. Statements 1. and D are supplementar 2. m + m D = = = = = 7. m = 6() + 7 = 67, m EFG = 7() - 3 = m EFG Slide 30 () / 14 (6 + 7) (11 + 3) F E D (7-3) G Reasons Given: and D are supplementar Prove: EFG What is the first reason? Definition of supplementar angles 7. m = 6() + 7 = 67, m EFG = 7() - 3 = 67 F8. Given m EFG G Subtraction F Statements Reasons ddition 1. and D are supp. 1. Given 2. m + m D = Substitution = = 180 D. Division = =. E Definition of congruent 6. angles H ombine Like Terms Slide 31 () / Given: and D are supplementar Prove: EFG What is the second reason? Definition of supplementar angles ddition Statements Reasons 1. and D are supp. 1. Given Substitution 2. m + m D = Def. of supplementar s = D Division = = 170 E Definition 6. =. of congruent angles F Given 8. m EFG G Subtraction 7. m = 6() + 7 = 67, m EFG = 7() - 3 = 67 H ombine Like Terms Slide 30 / Given: and D are supplementar Prove: EFG What is the first reason? Definition of supplementar angles ddition Substitution D Division E Definition of congruent angles F Given G Subtraction H ombine Like Terms Slide 31 / Given: and D are supplementar Prove: EFG What is the second reason? Definition of supplementar angles ddition Substitution D Division E Definition of congruent angles F Given G Subtraction H ombine Like Terms Slide 32 / Given: and D are supplementar Prove: EFG What is the third reason? Definition of supplementar angles ddition Substitution D Division E Definition of congruent angles F Given G Subtraction H ombine Like Terms

10 Slide 32 () / Given: and D are supplementar Prove: EFG What is the third reason? Definition of supplementar angles ddition Statements Reasons 1. and D are supp. 1. Given 2. m + m D = 180 Substitution 2. Def. of supplementar s = Substitution = 180 D Division = =. E Definition of congruent 6. angles F Given 8. m EFG G Subtraction 7. m = 6() + 7 = 67, m EFG = 7() - 3 = 67 H ombine Like Terms Slide 33 () / 14 1 Given: and D are supplementar Prove: EFG What is the fourth reason? Definition of supplementar angles ddition Statements Reasons 1. and D are supp. 1. Given Substitution 2. m + m D = Def. of supplementar s = Substitution D Division = ombine Like Terms. 17 = 170 E Definition 6. = of congruent. angles F Given 8. m EFG G Subtraction H 7. m = 6() + 7 = 67, m EFG = 7() - 3 = 67 H ombine Like Terms Slide 34 () / Given: and D are supplementar Prove: EFG What is the fifth reason? Definition of supplementar angles Statements Reasons ddition 1. and D are supp. 1. Given 2. m + m D = Def. of supplementar s Substitution = Substitution = ombine Like Terms D Division. 17 = =. Subtraction E Definition 7. m = of 6() congruent + 7 = 67, 6. angles m EFG = 7() - 3 = F Given 8. m EFG 8. G Subtraction G H ombine Like Terms Slide 33 / 14 1 Given: and D are supplementar Prove: EFG What is the fourth reason? Definition of supplementar angles ddition Substitution D Division E Definition of congruent angles F Given G Subtraction H ombine Like Terms Slide 34 / Given: and D are supplementar Prove: EFG What is the fifth reason? Definition of supplementar angles ddition Substitution D Division E Definition of congruent angles F Given G Subtraction H ombine Like Terms Slide 3 / Given: and D are supplementar Prove: EFG What is the sith reason? Definition of supplementar angles ddition Substitution D Division E Definition of congruent angles F Given G Subtraction H ombine Like Terms

11 Slide 3 () / Given: and D are supplementar Prove: EFG What is the sith reason? Definition of supplementar angles ddition Statements Substitution D Division = = 170 E Definition 6. =. Subtraction of congruent angles F Given 8. m EFG G Subtraction D 1. and D are supp. 2. m + m D = = m = 6() + 7 = 67, m EFG = 7() - 3 = 67 H ombine Like Terms Reasons 1. Given 2. Def. of supplementar s 3. Substitution 4. ombine Like Terms 6. Division Slide 36 () / Given: and D are supplementar Prove: EFG What is the seventh reason? Definition of supplementar angles Statements ddition Substitution = = 180 D Division. 17 = = E Definition of congruent 6. angles Division F Given 8. m EFG G Subtraction 1. and D are supp. 2. m + m D = m = 6() + 7 = 67, m EFG = 7() - 3 = 67 H ombine Like Terms Reasons 1. Given 2. Def. of supplementar s 3. Substitution 4. ombine Like Terms. Subtraction 7. Substitution 8. Slide 37 () / Given: and D are supplementar Prove: EFG What is the seventh reason? Definition of supplementar angles ddition Statements Reasons 1. and D are supp. 1. Given Substitution 2. m + m D = Def. of supplementar s = Substitution = 180 D Division 4. ombine Like Terms. 17 = =. Subtraction E Definition 7. of congruent angles m = 6() + 7 = 67, 6. Division m EFG = 7() - 3 = Substitution F Given 8. m EFG 8. Def. of angles G Subtraction E Slide 36 / Given: and D are supplementar Prove: EFG What is the seventh reason? Definition of supplementar angles ddition Substitution D Division E Definition of congruent angles F Given G Subtraction H ombine Like Terms Slide 37 / Given: and D are supplementar Prove: EFG What is the seventh reason? Definition of supplementar angles ddition Substitution D Division E Definition of congruent angles F Given G Subtraction H ombine Like Terms Slide 38 / 14 Pthagorean Theorem lick to return to the table of contents H ombine Like Terms

12 Slide 39 / 14 Pthagorean Theorem Pthagorean theorem is used for right triangles. It was first known in ancient ablon and Egpt beginning about However, it was not widel known until Pthagoras stated it. a c Slide 40 / 14 Labels for a right triangle Hpotenuse click to reveal - Opposite the right angle - Longest click of the to 3 reveal sides Pthagoras lived during the 6th centur.. on the island of Samos in the egean Sea. He also lived in Egpt, ablon, and southern Ital. He was a philosopher and a teacher. Legs click to reveal b - 2 sides that form the right angle click to reveal Slide 41 / 14 Pthagorean Theorem Proofs Slide 42 / 14 Proof of the Pthagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs (a and b) is equal to the square of the length of the hpotenuse (c). a 2 + b 2 = c 2 lick on the links below to see several animations of the proof Lab: Proof of the Pthagorean Theorem Water demo Move slider to show c 2 Moving of squares Slide 42 () / 14 Proof of the Pthagorean Theorem Have the students complete parts #1-3 in their small groups (2-4 students per group). Lab: The Proof "Follow-up of the Pthagorean questions" can Theorem be completed either in their groups or as a class discussion (located on the net 4 slides). Teacher Notes Follow-up Questions: Slide 43 / How is the area of the printed square on page 1 related to the area of the printed square on page 2? Eplain how ou reached that conclusion. 12. How are the 4 right triangles that ou cut out at the beginning of this lab related to each other? Eplain how ou reached that conclusion.

13 Follow-up Questions: Slide 43 () / How is the area of the printed square on page 1 related to the area of the printed square on page 2? Eplain how ou reached that conclusion. Follow-up Questions (cont'd): Slide 44 / How are the areas that ou found in question #6 & question # related to each other? Eplain how ou reached that conclusion. Math Practice Question #11: MP6 & MP7 Question #12: MP 6 & MP7 12. How are the 4 right triangles that ou cut out at the beginning of this lab related to each other? Eplain how ou reached that conclusion. 14. What algebraic epression represents the side length of the printed squares. Eplain how ou reached that conclusion. Follow-up Questions (cont'd): Slide 44 () / How are the areas that ou found in question #6 & question # related to each other? Eplain how ou reached that conclusion. Follow-up Questions (cont'd): Slide 4 / Multipl these side lengths together and simplif the epression. Math Practice Question #13: MP6 & MP7 Question #14: MP2 & MP3 14. What algebraic epression represents the side length of the printed squares. Eplain how ou reached that conclusion. 16. Using the arrangement of the shapes from question #, write an algebraic epression to represent the area of the entire figure. Follow-up Questions (cont'd): Slide 4 () / Multipl these side lengths together and simplif the epression. Follow-up Questions (cont'd): Slide 46 / Set the epressions from question #1 & question #16 equal to one another and simplif the equation. Math Practice Question #1: MP2 Question #16: MP2 16. Using the arrangement of the shapes from question #, write an algebraic epression to represent the area of the entire figure.

14 Follow-up Questions (cont'd): Slide 46 () / Set the epressions from question #1 & question #16 equal to one another and simplif the equation. Slide 47 / 14 Pthagorean Theorem How to use the formula to find missing sides. Missing Leg Missing Hpotenuse Math Practice Question #17: MP2 Write Equation Substitute in numbers Square numbers Write Equation Substitute in numbers Square numbers Subtract Find the Square Root dd Find the Square Root Label Label Slide 48 / 14 Pthagorean Theorem Missing Leg Slide 48 () / 14 Pthagorean Theorem Missing Leg a 2 + b 2 = c 2 Write Equation a 2 + b 2 = c 2 Write Equation MP.6: ttend to precision 1 ft 2 + b 2 = b 2 = Substitute in numbers Square numbers Subtract Math Practice sk: How do ou know 2 + b 2 that = 1our 2 answer Substitute in numbers is accurate? 1 To ftget the answer: 2 Talk + b 2 about = 22finding Square the numbers perfect squares before and after the radicand -2 in the problem. -2 Subtract ft b 2 = 200 Find the Square Root Label ft What labels b 2 could = 200ou use? Find the Square Root Label Slide 49 / 14 Pthagorean Theorem Slide 0 / 14 Pthagorean Theorem Missing Leg Missing Hpotenuse 9 in 18 in a 2 + b 2 = c b 2 = b 2 = b 2 = 243 Write Equation Substitute in numbers Square numbers Subtract Find the Square Root Label 7 in 4 in a 2 + b 2 = c 2 Write Equation = c 2 Substitute in numbers = c Square numbers 6 = c 2 dd Find the Square Root & Label

15 20 What is the length of the third side? Slide 1 / What is the length of the third side? Slide 1 () / = = 2 6 = What is the length of the third side? Slide 2 / What is the length of the third side? Slide 2 () / = = = What is the length of the third side? Slide 3 / What is the length of the third side? Slide 3 () / = = 49 2 =

16 23 What is the length of the third side? Slide 4 / What is the length of the third side? Slide 4 () / = = 2 2 = 2 = 4 4 Slide / 14 Pthagorean Triples Slide 6 / 14 Pthagorean Triples 3 4 There are combinations of whole numbers that work in the Pthagorean Theorem. These sets of numbers are known as Pthagorean Triples. 3-4 is the most famous of the triples. If ou recognize the sides of the triangle as being a triple (or multiple of one), ou won't need a calculator! an ou find an other Pthagorean Triples? Use the list of squares to see if an other triples work. 1 2 = = = = = = = = = = = = 76 2 = = = = = = = = = = = = = = = = = = 900 Triples Slide 6 () / 14 Slide 7 / 14 Pthagorean Triples Pthagorean Triples an ou find an other Pthagorean Triples? Use the list of squares 7 to - 24 see - 2 if an other 8-1 triples - 17 work. Multiples of these combinations work too! & Math Practice MP.8: Look for and epress regularit in 1 2 = 1 repeated 11 2 = reasoning = = = = = 9 sk: Is 13 it true 2 = 169 ever time? 23 2 = 29 4What 2 = 16 concepts 14that 2 = 196 have we 24 2 learned = 76 2 before = 2 were 1useful 2 = 22 in solving 2 2 = this = problem? 2 = = ould = 49 this problem 17 2 = 289 help ou 27 2 = solve = 64 another 18 2 = problem? = = = = = = = 900 Triples 24 What is the length of the third side? 8 6

17 24 What is the length of the third side? Slide 7 () / 14 2 What is the length of the third side? Slide 8 / = 2 OR = 2 0 = 2 6 = What is the length of the third side? Slide 8 () / = = = 144 = 12 OR What is the length of the third side? 0 48 Slide 9 / What is the length of the third side? Slide 9 () / Slide 60 / The legs of a right triangle are 7.0 and 3.0, what is the length of the hpotenuse? = = = 196 = 14 OR

18 Slide 60 () / The legs of a right triangle are 7.0 and 3.0, what is the length of the hpotenuse? Slide 61 / The legs of a right triangle are 2.0 and 12, what is the length of the hpotenuse? = = 2 8 = 2 Slide 61 () / The legs of a right triangle are 2.0 and 12, what is the length of the hpotenuse? = = = 2 Slide 62 / The hpotenuse of a right triangle has a length of 4.0 and one of its legs has a length of 2.. What is the length of the other leg? Slide 62 () / The hpotenuse of a right triangle has a length of 4.0 and one of its legs has a length of 2.. What is the length of the other leg? = = 16 2 = 9.7 Slide 63 / The hpotenuse of a right triangle has a length of 9.0 and one of its legs has a length of 4.. What is the length of the other leg?

19 Slide 63 () / The hpotenuse of a right triangle has a length of 9.0 and one of its legs has a length of 4.. What is the length of the other leg? Slide 64 / 14 This is a great problem and draws on a lot of what we've learned. Tr it in our groups. Then we'll work on it step b step together b asking questions that break the problem into pieces = = 81 2 = 60.7 In Δ, D is perpendicular to. The dimensions are shown in centimeters. 8 D What is the length of? From PR EOY sample test calculator #1 Slide 6 / What have we learned that will help solve this problem? Pthagorean Theorem Pthagorean Triples Distance Formula D and onl In Δ, D is perpendicular to. The dimensions are shown in centimeters. Slide 6 () / What have we learned that will help solve this problem? Pthagorean Theorem Pthagorean Triples Distance Formula D and onl In Δ, D is perpendicular to. The dimensions are shown in centimeters. D and onl 8 8 D D What is the length of? What is the length of? Slide 66 / 14 First, notice that we have two right triangles (perpendicular lines make right angles). The triangles are outlined red & blue in the diagram below. In Δ, D is perpendicular to. The dimensions are shown in centimeters. 8 Slide 67 / What is the length of the 3rd side in the red triangle? 3 cm 6 cm 9 cm D 13.4 cm In Δ, D is perpendicular to. The dimensions are shown in centimeters. D 8 What is the length of? D What is the length of?

20 Slide 67 () / What is the length of the 3rd side in the red triangle? 3 cm 6 cm a = 2 a = 0 9 cm a 2 = 36 D 13.4 cm a = 6 In Δ, D is perpendicular to. The dimensions are shown in centimeters. or 2(3-4) = 6-8, so a = How is D related to D? D > D D < D Slide 68 / 14 D = D D not enough information to relate these segments In Δ, D is perpendicular to. The dimensions are shown in centimeters. 8 D 6 D What is the length of? 33 How is D related to D? D > D D < D Slide 68 () / 14 D = D Two right triangles are D not enough information equal, so to their relate these segments corresponding sides are In Δ, D is perpendicular equal. lso, to. if The ou dimensions use are shown in centimeters. Pthagorean Theorem again to find D, it will also equal D What is the length of? 34 What is the length of? D = D Slide 69 () / In Δ, D is perpendicular to. The 12 dimensions are shown in centimeters. What is the length of? 34 What is the length of? Slide 69 / 14 In Δ, D is perpendicular to. The dimensions are shown in centimeters. 8 D What is the length of? Slide 70 / 14 onverse of the Pthagorean Theorem If a and b are measures of the shorter sides of a triangle, c is the measure of the longest side, and c 2 = a 2 + b 2, then the triangle is a right triangle. If c 2 a 2 + b 2, then the triangle is not a right triangle. This is the onverse of the Pthagorean Theorem. 8 D a = 3 ft c = ft What is the length of? b = 4 ft

21 Slide 71 / 14 onverse of the Pthagorean Theorem Slide 72 / 14 onverse of the Pthagorean Theorem In other words, ou can check to see if a triangle is a right triangle b seeing if the Pthagorean Theorem is true. Test the Pthagorean Theorem. If the final equation is true, then the triangle is right. If the final equation is false, then the triangle is not right. 8 in, 17 in, 1 in a 2 + b 2 = c = = = 289 Yes! Is it a Right Triangle? Write Equation Plug in numbers Square numbers Simplif both sides re the equal? Slide 72 () / 14 Slide 73 / 14 onverse of the Pthagorean Theorem MP.1: Make sense of problems and 8 in, 17 in, 1 persevere in in Is solving it a Right them Triangle? MP.8: Look for and epress regularit in a 2 + b 2 = c 2 repeated reasoning. Write Equation sk: 2 = 17 What 2 is this problem asking? (MP.1) How could ou start Plug this problem? in numbers (MP.1) What = 289 concepts that we have learned before were useful Square in solving numbers this 289 = 289 problem? (MP.8) What generalizations Simplif can ou both make? sides (MP.8) Yes! : If the numbers re the are equal? a Pthagorean [This Triple, object is a (or pull tab] multiple of one) then it's a right triangle. Math Practice 3 Is the triangle a right triangle? Yes No 6 ft 8 ft ft Slide 73 () / 14 3 Is the triangle a right triangle? Slide 74 / Is the triangle a right triangle? Yes No = = 0 6 ft 0 = 0 YES OR ft Pthagorean Triple 8 ft3-4 Yes No 24 ft 36 ft 30 ft

22 Slide 74 () / Is the triangle a right triangle? Slide 7 / Is the triangle a right triangle? Yes No 24 ft 36 ft = = = ft NO Yes No 8 in. 12 in. in. Slide 7 () / Is the triangle a right triangle? Slide 76 / Is the triangle a right triangle? Yes No 8 in. 128in = = = 196 NO in. Yes No ft 12 ft 13 ft Slide 76 () / 14 Slide 77 / Is the triangle a right triangle? Yes No 13 ft ft Yes - Pthagorean Triple! ft 39 an ou construct a right triangle with three lengths of wood that measure 7. in, 18 in and 19. in? Yes No

23 Slide 77 () / an ou construct a right triangle with three lengths of wood that measure 7. in, 18 in and 19. in? Slide 78 / 14 pplications of Pthagorean Theorem Yes No = = = YES Steps to Pthagorean Theorem pplication Problems. 1. Draw a right triangle to represent the situation. 2. Solve for unknown side length. 3. Round to the nearest tenth. Slide 78 () / 14 pplications of Pthagorean Theorem Slide 79 / 14 pplications of Pthagorean Theorem Math Practice The Steps eamples to Pthagorean in this lesson Theorem (net 6 pplication Problems. slides) address MP.4: 1. Draw Model a right with triangle mathematics to represent the situation. MP.: 2. Use Solve appropriate for unknown tools side strategicall. length. 3. Round to the nearest tenth. sk: What do ou alread know about solving this problem? (MP.4) What connections do ou see between this problem and Pthagorean Theorem? (MP.4) How could ou use manipulatives or a drawing to show our thinking? (MP.) Work with our partners to complete: To get from his high school to his home, Jamal travels.0 miles east and then 4.0 miles north. When Sheila goes to her home from the same high school, she travels 8.0 miles east and 2.0 miles south. What is the measure of the shortest distance, to the nearest tenth of a mile, between Jamal's home and Sheila's home? 0 From the New York State Education Department. Office of ssessment Polic, Development and dministration. Internet. vailable from accessed 17, June, Slide 79 () / 14 pplications of Pthagorean Theorem Work with our partners to complete: To get from his high school to his home, Jamal travels.0 miles east and then 4.0 miles north. When Sheila goes to her home from the same high school, she travels 8.0 miles east and 2.0 miles south. What is the measure of the shortest distance, to the nearest tenth of a mile, between Jamal's home and Sheila's home? = = 2 4 0= = Slide 80 / 14 pplications of Pthagorean Theorem Work with our partners to complete: straw is placed into a rectangular bo that is 3 inches b 4 inches b 8 inches, as shown in the accompaning diagram. If the straw fits eactl into the bo diagonall from the bottom left front corner to the top right back corner, how long is the straw, to the nearest tenth of an inch? From the New York State Education Department. Office of ssessment Polic, Development and dministration. Internet. vailable from accessed 17, June, From the New York State Education Department. Office of ssessment Polic, Development and dministration. Internet. vailable from accessed 17, June, 2011.

24 Slide 80 () / 14 pplications of Pthagorean Theorem Work with our partners to complete: straw is placed into a rectangular bo that is 3 inches b 4 inches b 8 inches, as shown in the accompaning diagram. If the straw fits eactl into the bo diagonall from the bottom left 3 front corner to the top right 2 = back 2 corner, how c 2 + d long 2 = e is 2 Pthagorean Triple the straw, to the nearest tenth of an c = inch? = e 2 89 = e = e e c a b d Slide 81 / 14 pplications of Pthagorean Theorem The Pthagorean Theorem can be applied to 3 Dimensional Figures In this figure: a = slant height (height of triangular face) b = 1/2 base length (from midpoint of side of base to center of the base of the pramid) h = height of pramid From the New York State Education Department. Office of ssessment Polic, Development and dministration. Internet. vailable from accessed 17, June, Slide 82 / 14 pplications of Pthagorean Theorem right triangle is formed between the three lengths. If ou know two of the measurements, ou can calculate the third. EXMPLE: Find the slant height of a pramid whose height is cm and whose base has a length of 8cm. Slide 82 () / 14 pplications of Pthagorean Theorem right triangle is formed between the three lengths. If ou know two of the measurements, ou can calculate the third. EXMPLE: Find the slant height of a pramid whose height is cm and whose base has a length of 8cm. Slide 83 / 14 pplications of Pthagorean Theorem Find the slant height of the pramid whose base length is cm and height is 12 cm. Label the diagram with the measurements. Slide 83 () / 14 pplications of Pthagorean Theorem Find the slant height of the pramid whose base length is cm and height is 12 cm. Label the diagram with the measurements.

25 Slide 84 / 14 pplications of Pthagorean Theorem Slide 84 () / 14 pplications of Pthagorean Theorem Find the base length of the pramid whose height is 21 m and slant height is 29 m. Label the diagram with the measurements. Find the base length of the pramid whose height is 21 m and slant height is 29 m. Label the diagram with the measurements. Slide 8 / The sizes of television and computer monitors are given in inches. However, these dimensions are actuall the diagonal measure of the rectangular screens. Suppose a 14-inch computer monitor has an actual screen length of 11-inches. What is the height of the screen? Slide 8 () / The sizes of television and computer monitors are given in inches. However, these dimensions are actuall the diagonal measure of the rectangular screens. Suppose a 14-inch computer monitor has an actual screen length of 11-inches. What is the height of the screen? = = = 7 Slide 86 / Find the height of the pramid whose base length is 16 in and slant height is 17 in. Label the diagram with the measurements. Slide 86 () / Find the height of the pramid whose base length is 16 in and slant height is 17 in. Label the diagram with the measurements.

26 Slide 87 / tree was hit b lightning during a storm. The part of the tree still standing is 3 meters tall. The top of the tree is now resting 8 meters from the base of the tree, and is still partiall attached to its trunk. ssume the ground is level. How tall was the tree originall? Slide 87 () / tree was hit b lightning during a storm. The part of the tree still standing is 3 meters tall. The top of the tree is now resting 8 meters from the base of the tree, and is still partiall attached to its trunk. ssume the ground is level. How tall was the tree originall? = = 2 73 = 2 The base of the tree is 3 m, the part that fell is 8. m tall, so the tree was a total of 11. m tall. Slide 88 / Suppose ou have a ladder of length 13 feet. To make it sturd enough to climb ou mct place the ladder eactl feet from the wall of a building. You need to post a banner on the building feet above ground. Is the ladder long enough for ou to reach the location ou need to post the banner? Yes No Slide 88 () / Suppose ou have a ladder of length 13 feet. To make it sturd enough to climb ou mct place the ladder eactl feet from the wall of a building. You need to post a banner on the building feet above ground. Is the ladder long enough for ou to reach the location ou need to post the banner? Yes No (Derived from ( (Derived from ( Slide 89 / You've just picked up a ground ball at 3rd base, and ou see the other team's plaer running towards 1st base. How far do ou have to throw the ball to get it from third base to first base, and throw the runner out? ( baseball diamond is a square) Slide 89 () / You've just picked up a ground ball at 3rd base, and ou see the other team's plaer running towards 1st base. How far do ou have to throw the ball to get it from third base to first base, and throw the runner out? ( baseball diamond is a square) 90 ft. 2nd 90 ft. 90 ft. 2nd = = 90 ft. 2 16,200 = 2 3rd 1st 3rd 1st 90 ft. 90 ft. 90 ft. 90 ft. home home

27 Slide 90 / 14 4 You're locked out of our house and the onl open window is on the second floor, 2 feet above ground. There are bushes along the edge of our house, so ou'll have to place a ladder feet from the house. What length of ladder do ou need to reach the window? Slide 90 () / 14 4 You're locked out of our house and the onl open window is on the second floor, 2 feet above ground. There are bushes along the edge of our house, so ou'll have to place a ladder feet from the house. What length of ladder do ou need to reach the window? = = 2 72 = feet = Slide 91 / Scott wants to swim across a river that is 400 meters wide. He begins swimming perpendicular to the shore, but ends up 0 meters down the river because of the current. How far did he actuall swim from his starting point? Slide 91 () / Scott wants to swim across a river that is 400 meters wide. He begins swimming perpendicular to the shore, but ends up 0 meters down 400 the m river because of the current. How far did he actuall swim 0 from m his starting point? = 2 160,000 +,000 = 2 170,000 = 2 Slide 92 / 14 Distance Formula lick to return to the table of contents Slide 93 / 14 Distance etween Two Points If ou have two points on a graph, such as (,2) and (,6), ou can find the distance between them b simpl counting units on the graph, since the lie in a vertical line. 0 The distance between these two points is 4. The top point is 4 above the lower point.

28 Slide 94 / What is the distance between these two points? Slide 94 () / What is the distance between these two points? The distance is. The blue point is five to the right of the red point. Distance is alwas positive. 0 0 Slide 9 / What is the distance between these two points? Slide 9 () / What is the distance between these two points? Slide 96 / What is the distance between these two points? Slide 97 / 14 Distance etween Two Points Most sets of points do not lie in a vertical or horizontal line. For eample: 0 0 ounting the units between these two points is impossible. So mathematicians have developed a formula using the Pthagorean theorem to find the distance between two points.

29 0 Slide 98 / 14 Distance etween Two Points Draw the right triangle around these two points. Then use the Pthagorean theorem to find the distance in red. c a b Slide 98 () / 14 Distance etween Two Points Draw the right triangle around these two points. Then use the Pthagorean theorem to find the distance in red. c 2 = a 2 + b 2 c 2 = c 2 = c 2 = 2 c b c = The distance between a the two points (2,2) 0 and (,6) is units. Slide 99 / 14 Distance etween Two Points Slide 99 () / 14 Distance etween Two Points Eample: 0 Eample: 0 c 2 = a 2 + b 2 c 2 = c 2 = c 2 = 4 The distance between the two points (-3,8) and (-9,) is approimatel 6.7 units. Slide 0 / 14 Distance etween Two Points Slide 0 () / 14 Distance etween Two Points Tr This: Tr This: c 2 = a 2 + b 2 c 2 = c 2 = c 2 = 22 c = The distance between the two points (, ) and (7, -4) is 1 units.

30 Slide 1 / 14 Distance Formula Slide 2 / 14 Distance Formula Deriving a formula for calculating distance... reate a right triangle around the two points. Label the points as shown. Then substitute into the Pthagorean Theorem. ( 2, 2) d length = 2-1 ( 1, 1) ( 2, 1) 0 length = 2-1 c 2 = a 2 + b 2 d 2 = ( 2-1) 2 + ( 2-1) 2 d = ( 2-1) 2 + ( 2-1) 2 This is the distance formula now substitute in values. Slide 2 () / 14 Distance Formula reate a right triangle around the two points. Label the points as shown. Then substitute into the Pthagorean Theorem. d = ( - 2) 2 + (6-2) 2 d = (3) 2 + (4) 2 ( 2, 2) d = d length = 2-1 d = 2 ( 1, 1) ( 2, 1) 0 d = [This object is length a pull tab] = 2-1 c 2 = a 2 + b 2 d 2 = ( 2-1) 2 + ( 2-1) 2 d = ( 2-1) 2 + ( 2-1) 2 This is the distance formula now substitute in values. Slide 3 / 14 Distance Formula You can find the distance d between an two points ( 1, 1) and ( 2, 2) using the formula below. d = ( 2-1 ) 2 + ( 2-1 ) 2 how far between the -coordinate how far between the -coordinate Slide 4 / 14 When onl given the two points, use the formula. Find the distance between: Point 1 (-4, -7) Point 2 (, -2) Distance Formula Slide 4 () / 14 When onl given the two points, use the formula. Find the distance between: Point 1 (-4, -7) Point 2 (, -2) Distance Formula

31 Slide / 14 Slide () / 14 0 Find the distance between (2, 3) and (6, 8). Round answer to the nearest tenth. hint Let: 1 = 2 1 = 3 2 = 6 2 = 8 Slide 6 / 14 Slide 6 () / 14 1 Find the distance between (-7, -2) and (11, 3). Round answer to the nearest tenth. hint Let: 1 = -7 1 = -2 2 = 11 2 = 3 Slide 7 / 14 Slide 7 () / 14 2 Find the distance between (4, 6) and (1, ). Round answer to the nearest tenth.

32 Slide 8 / 14 3 Find the distance between (7, ) and (9, -1). Round answer to the nearest tenth. Slide 8 () / 14 3 Find the distance between (7, ) and (9, -1). Round answer to the nearest tenth. Slide 9 / 14 pplications of the Distance Formula How would ou find the perimeter of this rectangle? Slide 9 () / 14 pplications of the Distance Formula How would ou find the perimeter of this rectangle? Either just count the units or find the distance between the points from the ordered pairs. Either just count the units or find the distance between the points from the ordered pairs. length = 8 width = = 28 (0,-1) Slide 1 / 14 pplications of the Distance Formula an we just count how man units long each line segment is in this quadrilateral to find the perimeter? D (3,3) (9,4) (8,0) Slide 1 () / 14 pplications of the Distance Formula MP.1: Make an sense we just of count problems how and man persevere units long each line segment in is solving this quadrilateral them. to find the perimeter? MP.2: Reasoning abstractl and quantitativel. MP.7: Look for and make use of structure. D (3,3) (9,4) sk: What is the problem asking? (MP.1) How could ou start this problem? (MP.1) How can ou represent the problem with smbols and numbers? (MP.2) (8,0) How is finding the perimeter (0,-1) of a polgon in the coordinate plane related to the distance formula? (MP.7) What do ou know about the distance formula that ou can [This appl object to is a this pull tab] situation? (MP.7)

33 Slide 111 / 14 Slide 111 () / 14 pplications of the Distance Formula You can use the Distance Formula to solve geometr problems. D (3,3) (0,-1) (9,4) (8,0) Find the perimeter of D. Use the distance formula to find all four of the side lengths. Then add then together. = = = = D = D = D = D = Slide 112 / 14 Slide 112 () / 14 4 Find the perimeter of ΔEFG. Round the answer to the nearest tenth. F (3,4) G (1,1) E (7,-1) Slide 113 / 14 Find the perimeter of the square. Round answer to the nearest tenth. Slide 113 () / 14 Find the perimeter of the square. Round answer to the nearest tenth. K (-1,3) H (1,) I (3,3) K (-1,3) H (1,) I (3,3) Each side length is #8 So the perimeter is 4 times #8 # 11.3 J (1,1) J (1,1)

34 Slide 114 / 14 Slide 114 () / 14 6 Find the perimeter of the parallelogram. Round answer to the nearest tenth. L (1,2) M (6,2) O (0,-1) N (,-1) Slide 11 / 14 Slide 116 / 14 Midpoint Midpoints 0 0 (2, ) (2, 2) Find the midpoint of the line segment. What is a midpoint? How did ou find the midpoint? What are the coordinates of the midpoint? lick to return to the table of contents Slide 117 / 14 Slide 117 () / 14 Midpoint Midpoint (3, 4) (9, 4) 0 Find the midpoint of the line segment. What are the coordinates of the midpoint? How is it related to the coordinates of the endpoints? Find the midpoint of the line segment. Midpoint = (6, 4) What are the coordinates of the It is in the middle of midpoint? the segment. & Math Practice verage of -coordinates. How is it related to the coordinates of (3, 4) verage (9, of 4) -coordinates. the endpoints? 0 The questions on this slide address MP.7: Look for and make use of structure.

35 Slide 118 / 14 The Midpoint Formula To calculate the midpoint of a line segment with endpoints ( 1, 1) and ( 2, 2) use the formula: ( , 2 2 ) Slide 119 / 14 The midpoint of a segment is the point M on halfwa between the endpoints and. (2,) The Midpoint Formula (8,1) The and coordinates of the midpoint are the averages of the and coordinates of the endpoints, respectivel. See net page for answer Slide 120 / 14 The Midpoint Formula Slide 120 () / 14 The Midpoint Formula The midpoint of a segment is the point M on halfwa between the endpoints and. The midpoint of a segment is the point M on halfwa between the endpoints and. (2,) M (8,1) Use the midpoint formula: , 2 2 ) ( 1 (2,) M (8,1) Use Substitute the midpoint in values: formula: ( , ( 2 2, 2 2 ) Simplif the numerators:, 6 ( 2 2 ) Write fractions in simplest form: (,3) is the midpoint of Slide 121 / 14 The Midpoint Formula Slide 121 () / 14 The Midpoint Formula Find the midpoint of (1, 0) and (, 3). Use the midpoint formula: , 2 2 ) ( 1 Substitute in values: Find the midpoint of (1, 0) and (, 3). 1 +, ( 2 2 ) Use the midpoint formula: Simplif the numerators: ,, 3 ( ) Write fractions in simplest form: (-2,1.) is the midpoint

36 Slide 122 / 14 Slide 122 () / 14 7 What is the midpoint of the line segment that has the endpoints (2, ) and (6,-4)? (3, 4) (4, 7) (4, 3) D (1., 3) 0 7 What is the midpoint of the line segment that has the endpoints (2, ) and (6,-4)? (3, 4) (4, 7) (4, 3) D (1., 3) 0 Slide 123 / 14 Slide 123 () / 14 8 What is the midpoint of the line segment that has the endpoints (4, ) and (-2, 6)? (3, 6.) (1,.) (-1,.) D (1, 0.) 0 8 What is the midpoint of the line segment that has the endpoints (4, ) and (-2, 6)? (3, 6.) (1,.) (-1,.) D (1, 0.) 0 Slide 124 / 14 9 What is the midpoint of the line segment that has the endpoints (-4, -7) and (-12, 2)? Slide 124 () / 14 9 What is the midpoint of the line segment that has the endpoints (-4, -7) and (-12, 2)? (-8, -2.) (-8, -2.) (-4, -4.) (-1, -6.) D (-8, -4) (-4, -4.) (-1, -6.) D (-8, -4)

37 Slide 12 / What is the midpoint of the line segment that has the endpoints (, 9) and (, 3)? Slide 12 () / What is the midpoint of the line segment that has the endpoints (, 9) and (, 3)? (6., 2) (6., 2) (6, 7.) (7., 6) D (1, 12) (6, 7.) (7., 6) D (1, 12) Slide 126 / Find the center of the circle with a diameter having endpoints at (-4, 3) and (0, 2). Which formula should be used to solve this problem? D Pthagorean Formula Distance Formula Midpoint Formula Formula for rea of a ircle Slide 126 () / Find the center of the circle with a diameter having endpoints at (-4, 3) and (0, 2). Which formula should be used to solve this problem? D Pthagorean Formula Distance Formula Since the center is at the Midpoint Formula midpoint of an diameter, find the midpoint of the two given Formula for rea of a ircle endpoints. Slide 127 / Find the center of the circle with a diameter having endpoints at (-4, 3) and (0, 2). (2.,-2) (2,2.) (-2,2.) D (-1,1.) Slide 127 () / Find the center of the circle with a diameter having endpoints at (-4, 3) and (0, 2). (2.,-2) (2,2.) (-2,2.) D (-1,1.)

38 Slide 128 / Find the center of the circle with a diameter having endpoints at (-12, ) and (2, 6). (-7,8) (,8) (,8) D (7,8) Slide 128 () / Find the center of the circle with a diameter having endpoints at (-12, ) and (2, 6). (-7,8) (,8) (,8) D (7,8) Slide 129 / 14 Using Midpoint to Find the Missing Endpoint If point M is the midpoint between the points P and Q. Find the coordinates of the missing point. Q =? M (8, 1) P (8, -6) Use the midpoint formula and solve for the unknown , 2 2 ) Substitute Multipl both sides b 2 dd or subtract ( 1 (8, 8) Slide 130 / 14 Using Midpoint to Find the Missing Endpoint If point M is the midpoint between the points P and Q. Find the coordinates of the missing point. nother method that can be used to find the missing endpoint is to look at the relationship between both the - and -coordinates and use the relationship again to calculate the missing endpoint. Slide 129 () / 14 Using Midpoint to Find the Missing Endpoint If point M is the midpoint between the points P and Q. Find the coordinates of the missing point. Q =? efore moving onto the net slide, ask the class: M (8, 1) "an ou find a shortcut to solve P (8, -6) this problem? How Use the midpoint formula would and solve our for shortcut the unknown. make ( the problem easier?" , The 2 answer 2 to ) this question is shown on the net slide. Substitute Teacher Notes Multipl both sides b 2 dd or subtract (8, 8) Slide 131 / If Point M is the midpoint between the points P and Q. What are the coordinates of the missing point? (-13, -22) (-8., -9.) (-4., -7.) D (-12., -6.) P = (-4,3) M = (-8.,-9.) Q =? Q =? M (8, 1) P (8, -6) Following the pattern, we see that the coordinates for point Q are (8, 8), which is eactl the same answer that we found using the midpoint formula.

39 Slide 131 () / If Point M is the midpoint between the points P and Q. What are the coordinates of the missing point? Slide 132 / 14 6 If Point M is the midpoint between the points P and Q. What are the coordinates of the missing point? (-13, -22) (-8., -9.) (-4., -7.) P = (-4,3) M = (-8.,-9.) Q =? (1, -1) (-13, 19) (-8, 11) Q = (-6, 9) M = (-7, ) P =? D (-12., -6.) D (-19, 8) Slide 132 () / 14 Slide 133 / 14 6 If Point M is the midpoint between the points P and Q. What are the coordinates of the missing point? (1, -1) (-13, 19) (-8, 11) D (-19, 8) Q = (-6, 9) M = (-7, ) P =? Glossar & Standards lick to return to the table of contents Slide 133 () / 14 Teacher Notes Vocabular Words are bolded in the presentation. The tet bo the word is in is then Glossar & Standards linked to the page at the end of the presentation with the word defined on it. Slide 134 / 14 onverse of Pthagorean Theorem If a and b are measures of the shorter sides of a triangle, c is the measure of the longest side, and c squared equals a squared plus b squared, then the triangle is a right triangle. a a 2 +b 2 = c 2 c 4 3 Eample: = = 2 2 = 2 b right triangle lick to return to the table of contents ack to Instruction

40 Slide 13 / 14 Distance Length Measurement of how far two points are through space. Slide 136 / 14 Hpotenuse The longest side of a right triangle that is opposite the right angle. Distance Formula: = a 2 +b 2 = c 2 ack to Instruction ack to Instruction Slide 137 / 14 Leg 2 sides that form the right angle of a right triangle. Slide 138 / 14 Midpoint The middle of something. The point halfwa along a line. a 2 +b 2 = c 2 ( 1 Midpoint Formula: , 2 2 ) ( , 2 2 ) ( , 2 2 ) ( 4, ) ( 6 2, ) ack to Instruction ack to Instruction Through an two points, there is eactl one line. Slide 139 / 14 Postulate propert that is accepted without proof Segment ddition Postulate D D + DE = E E ngle ddition Postulate J F G H m FGJ + m JGH = m FGH Slide 140 / 14 Proof Reasoned, logical eplanations that use definitions, algebraic properties, postulates, and previousl proven theorems to arrive at a conclusion Given: 3-24 = 0 Prove: = 8 Statements Reasons = 0 1. Given 2. 3 = ddition 3. = 8 3. Division ack to Instruction ack to Instruction

41 Slide 141 / 14 Pthagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs (a and b) is equal to the square of the length of hpotenuse (c). Slide 142 / 14 Pthagorean Triples ombinations of whole numbers that work in the Pthagorean Theorem. Formula: 4 Eample: = = 2 2 = = = 2 2 = = = = = = = ack to Instruction ack to Instruction Slide 143 / 14 Right Triangle triangle that has a right angle (90 ). Slide 144 / 14 Two-olumn Proof tool to organize our reasoning into two columns. Statements are written in the left column. Reasons are written in the right column. 30º 60º 4º 4º stair case sail Given: 3-24 = 0 Prove: = 8 Statements Reasons = 0 1. Given 2. 3 = ddition 3. = 8 3. Division ack to Instruction ack to Instruction Slide 14 / 14 Standards for Mathematical Practices MP1 Make sense of problems and persevere in solving them. MP2 Reason abstractl and quantitativel. MP3 onstruct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP Use appropriate tools strategicall. MP6 ttend to precision. MP7 Look for and make use of structure. MP8 Look for and epress regularit in repeated reasoning. lick on each standard to bring ou to an eample of how to meet this standard within the unit.