9.1 Wrm Up Solve the eqution. 1. 4 2 + 3 2 = x 2 2. 13 2 + x 2 = 25 2 3. 3 2 2 + x 2 = 5 2 2 4. 5 2 + x 2 = 12 2 Mrh 7, 2016 Geometry 9.1 The Pythgoren Theorem 1
Geometry 9.1 The Pythgoren Theorem
9.1 Essentil Question How n I use the Pythgoren Theorem to find lengths of the sides of Right Tringles? Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 3
Gols Prove the Pythgoren Theorem. Solve tringles using the theorem. Solve prolems using the theorem. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 4
This is nient history. The Egyptin Pyrmid uilders used it to mke squre orners. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 5
Leg Terminology The two legs form the right ngle. The hypotenuse is ross from the right ngle. Leg Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 6
Hypotenuse = strethed ginst 3 5 4 Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 7
Proof Proofs of the Pythgoren Theorem re numerous well over 300 known. Disovered in mny nient ultures. Eulid s is mong the most diffiult to understnd. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 8
Eulid s Digrm Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 9
Chinese Proof Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 10
Chinese Proof Prt 1: Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 11 Are of the smll squre: A = 2 Are of one tringle: A = (½) Are of 4 tringles: A = 2
Chinese Proof Prt 1: Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 12 Are of the smll squre: A = 2 Are of 4 tringles: A = 2 Are Sum 2 + 2
? Chinese Proof Prt 2: + Are Sum 2 + 2 Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 13? +
Chinese Proof Prt 2: Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 14 Are Sum 2 + 2 Are of the lrge squre: A ( ) ( )( ) 2 2 2 2 2 2
Chinese Proof Prt 3: Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 15 Are Sum 2 + 2 or 2 + 2 + 2 These res re equl.
Chinese Proof Prt 3: 2 2 2 2 2 2 2 2 Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 16
Proof y President Grfield (1876) 20 th President of the United Sttes Are of Trpezoid = Sum of re of three tringles 1 1 1 1 2 2 2 2 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 17
Pythgoren Theorem 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. If ABC is right tringle, then 2 + 2 = 2 B Mrh 7, 2016 C Geometry 9.3 Converse of the Pythgoren Theorem A 18
Exmple 1 Solve. 2 5 2 6 2 5 25 36 61 6 61 7.81 Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 19
Exmple 2 Solve. 2 2 2 2 10 2 4 100 10 2 96 96 16 6 2 4 6 9.80 Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 20
x Exmple 3 20 x Solve. 2 2 2 x 2 2 20 2x 400 x 200 x x 200 x 10 2 x 14.14 Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 21
Your Turn 3 5 4 3 4 2 2 2 12 5 12 2 2 2 2 9 16 2 25 144 2 25 5 2 169 13 Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 22
Pythgoren Triples 3 5 5 13 4 12 3 4 5 nd 5 12 13 re Pythgoren Triples. Eh side is n integer. A Pythgoren Triple multiplied y whole numer results in nother Pythgoren Triple Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 23
Pythgoren Triples Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 24
Exmple 4 Is 10-10-20 Pythgoren Triple? 10 2 + 10 2 = 20 2? 100 + 100 400 200 400 Flse! Not Pythgoren Triple. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 25
Exmple 5 Is 20-21-29 Pythgoren Triple? 20 2 + 21 2 = 29 2? 400 + 441 = 841? 841 = 841 True It is Pythgoren Triple. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 26
Exmple 6 The distne etween ses on sell dimond is 90 feet. A ther throws the ll from home se to 2 nd se. Wht is the distne? Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 27
Exmple 6 90 90 2 2 2 2 2 90 90 8100 8100 16200 16200 127.3 Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 28
Exmple 7 36.8 in. 20.7 in. Find the digonl mesure of the LCD sreen to the nerest inh. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 29
Exmple 7 36.8 in. 20.7 in. Find the digonl mesure of the LCD sreen to the nerest inh. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 30
Exmple 7 Find the digonl mesure of the LCD sreen to the nerest inh. 2 2 2 36.8 in. 20.7 in. 36.8 20.7 1354.24 428.49 1782.73 42.22 2 2 Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 31
Exmple 7 Find the digonl mesure of the LCD sreen to the nerest inh. 36.8 in. 20.7 in. 36.8 20.7 2 2 2 1354.24 428.49 1782.73 42.22 2 2 Aout 42.2 inhes Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 32
Dy 1 Summry In right tringle, the hypotenuse is the longest side. The sum of the squres of the legs is equl to the squre of the hypotenuse. If the three sides re ll integers, they form Pythgoren Triple. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 33
True or Flse? The sum of the squre roots of ny two sides of n isoseles tringle is equl to the squre root of the remining side. Oh joy! Rpture! I got rin! + =? Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 34
Flse. It should hve een The sum of the squres of the two legs of right tringle is equl to the squre of the remining side. Oh joy! Rpture! I hve rin! Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 35
Gols Determine if tringle is right tringle. Use the Pythgoren inequlities to determine if tringle is ute or otuse. Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 36
Pythgoren Theorem 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. If ABC is right tringle, then 2 + 2 = 2 B C Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 37 A
Converse of Pythgoren Theorem If the squre of the length of the longest side of tringle is equl to the sum of the squres of the lengths of the other two sides, then the tringle is right tringle. If 2 + 2 = 2, then ABC is right tringle. B C Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 38 A
Exmple 8 Is POD right tringle? 2 2? 2 16 30 34 30 O 16 256 900 1156? P 34 Longest Side D 1156 1156 Yes! Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 39
Exmple 9 Is SAD right tringle? 9 S 20 Whih segment is the longest? SD 2 2? 2 9 12 20 81 144 400? A 12 No! D 225 400 Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 40
Reminder x 2 = x 3x 2 = 3 2 x 2 = 9x 2 5 2 = 5 3 x 2 = 3 2 x 2 = 9x 17 2 = 17 3 3 2 = 3 2 3 2 = 9 3 = 27 3x 2 = 3x 4 5 2 = 4 2 5 2 = 16 5 = 80 Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 41
Exmple 10 Is HUG right tringle? H Whih segment is the longest? HG 2 2 5 10 5 5? 2 5 5 5? 2 2 25 100 5 5 U 10 Yes! G? 125 25 5 125 125 Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 42
Your Turn. 10 Is RST right? S 24 2 2? 2 10 24 26 100 576 676 676 676? R 26 T Yes it is. Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 43
Tringle Inequlity Theorem In tringle, the sum of ny two sides is greter thn the third side. 4 5 7 4 + 7 > 5 4 + 5 > 7 5 + 7 > 4 Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 44
Tringle Inequlity Theorem 5 4 This is not tringle sine 5 + 4 < 10. 10 Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 45
Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 46
Begin with right tringle 2 + 2 = 2 Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 47
Rotte side in. 2 = < 2 + 2 nd hve not hnged. 2 + 2 hs not hnged. got smller. 2 got smller. nd The right ngle gets smller: it is ute. Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 48
Theorem 9.6 If the squre of the length of the longest side of tringle is less thn the sum of the squres of the other two sides, then the tringle is ute. A 2 < 2 + 2 C B Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 49
Tke nother right tringle 2 + 2 = 2 Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 50
Rotte side out. 2 = > 2 + 2 nd hve not hnged. 2 + 2 hs not hnged. got lrger. 2 got lrger. nd The right ngle gets lrger: it is otuse. Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 51
Theorem 9.6 If the squre of the length of the longest side of tringle is greter thn the sum of the squres of the other two sides, then the tringle is otuse. A 2 > 2 + 2 C B Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 52
Exmple 11 The sides of tringle mesure 5, 7, nd 11. Clssify it s ute, right, or otuse. Solution: The longest side is 11. 11 2? 5 2 + 7 2 121? 25 + 49 121 > 74 Otuse Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 53
Exmple 12 The sides of tringle re 17, 20, nd 25. Clssify the tringle. Solution: 25 2? 17 2 + 20 2 625? 689 625 < 689 Aute Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 54
Exmple 13 Clssify this tringle. 7 5 2 2 2 12 7 5? 12 7 5? 12 12 12 Right Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 55
Exmple 14 Clssify this tringle. 16 8 6 It isn t tringle! 6 +8 < 16. Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 56
Summry If 2 = 2 + 2, RIGHT. If 2 < 2 + 2, ACUTE. If 2 > 2 + 2, OBTUSE. The lst two n e very onfusing; don t get them mixed up. Mrh 7, 2016 Geometry 9.3 Converse of the Pythgoren Theorem 57
Homework Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 58
Generting Pythgoren Triples Find two positive integers & whih re reltively prime nd >. Tht is, they hve no ftors in ommon other thn 1. Then the triples re: 2 + 2, 2 nd 2 2. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 59
Generting Pythgoren Triples Exmple: Choose = 4 nd = 3. 2 + 2 = 4 2 + 3 2 = 25. 2 = 2(4)(3) = 24. 2 2 = 4 2 3 2 = 7. 7, 24, 25 is Pythgoren Triple. Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 60
Generting Pythgoren Triples 7, 24, 25 is Pythgoren Triple. Chek: 7 2 + 24 2 = 25 2? 49 + 576 = 625? 625 = 625 Tht s triple! Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 61
Pythgoren Triples nd re reltively prime. > 2 + 2 2 2 2 Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 62
Try it. Using = 8 nd = 3, find the Pythgoren Triple. Answer: 8 2 3 2 = 64 9 = 55 2(8)(3) = 48 8 2 + 3 2 = 73 55 2 + 48 2 = 73 2? 3025 + 2304 = 5329? 5329 = 5329 heks. Are Mrh 7, 2016 Geometry 9.2 The Pythgoren Theorem 63