9.1 Day 1 Warm Up. Solve the equation = x x 2 = March 1, 2017 Geometry 9.1 The Pythagorean Theorem 1
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1 9.1 Dy 1 Wrm Up Solve the eqution = x x 2 = x 2 = x 2 = 12 2 Mrh 1, 2017 Geometry 9.1 The Pythgoren Theorem 1
2 9.1 Dy 2 Wrm Up Use the Pythgoren Theorem to find the missing side length Mrh 1, 2017 Geometry 9.1 The Pythgoren Theorem 2
3 Geometry 9.1 The Pythgoren Theorem
4 9.1 Essentil Question How n you determine if tringle is right tringle, ute tringle, or otuse tringle? Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 4
5 Gols Prove the Pythgoren Theorem. Solve tringles using the theorem. Solve prolems using the theorem. Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 5
6 This is nient history. The Egyptin Pyrmid uilders used it to mke squre orners. Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 6
7 Leg Terminology The two legs form the right ngle. The hypotenuse is ross from the right ngle. Leg Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 7
8 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 1, 2017 Geometry 9.2 The Pythgoren Theorem 8
9 Eulid s Digrm Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 9
10 Chinese Proof Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 10
11 Chinese Proof Prt 1: Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 11 Are of the smll squre: A = 2 Are of one tringle: A = (½) Are of 4 tringles: A4 = 2
12 Chinese Proof Prt 1: Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 12 Are of the smll squre: A = 2 Are of 4 tringles: A4 = 2 Are Sum A + A
13 Chinese Proof Prt 2: + Are Sum Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 13
14 Chinese Proof Prt 2: Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 14 Are Sum Are of the lrge squre: A ( ) ( )( )
15 Chinese Proof Prt 3: Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 15 Are Sum or These res re equl.
16 Chinese Proof Prt 3: Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 16
17 Proof y President Grfield (1876) 20 th President of the United Sttes Are of Trpezoid = Sum of re of three tringles Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 17
18 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 B Mrh 1, 2017 C Geometry 9.3 Converse of the Pythgoren Theorem 18 A
19 Exmple 1 Solve Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 19
20 Exmple 2 Solve = = = 25 = 25 = 5 Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 20
21 Exmple 3 x 20 x Solve x x 400 x 200 x x 200 x 10 2 x Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 21
22 Your Turn = = = 75 = 75 = 5 3 Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 22
23 Pythgoren Triples nd re Pythgoren Triples. Eh side is n integer. A Pythgoren Triple multiplied y whole numer results in nother Pythgoren Triple 23 Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem
24 Pythgoren Triples Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 24
25 Exmple 4 Is Pythgoren Triple? = 20 2? = 400? 200 = 400? Flse! Not Pythgoren Triple. Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 25
26 Exmple 5 Is Pythgoren Triple? = 29 2? = 841? 841 = 841 True It is Pythgoren Triple. Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 26
27 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 1, 2017 Geometry 9.2 The Pythgoren Theorem 27
28 Exmple Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 28
29 Exmple in in. Find the digonl mesure of the LCD sreen to the nerest inh. Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 29
30 Exmple in in. Find the digonl mesure of the LCD sreen to the nerest inh. Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 30
31 Exmple 7 Find the digonl mesure of the LCD sreen to the nerest inh in in Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 31
32 Exmple 7 Find the digonl mesure of the LCD sreen to the nerest inh in in Aout 42 inhes 2 Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 32
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 1, 2017 Geometry 9.2 The Pythgoren Theorem 33
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 1, 2017 Geometry 9.2 The Pythgoren Theorem 34
35 Gols Determine if tringle is right tringle. Use the Pythgoren inequlities to determine if tringle is ute or otuse. Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 35
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 B C Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 36 A
37 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, then ABC is right tringle. B C Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 37 A
38 Exmple 8 Is POD right tringle? 2 2? P O Longest Side D ? Yes! Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 38
39 Exmple 9 Is SAD right tringle? 9 S 20 Whih segment is the longest? SD 2 2? ? A 12 No! D Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 39
40 Reminder x 2 = x 3x 2 = 3 2 x 2 = 9x = 5 3 x 2 = 3 2 x 2 = 9x 17 2 = = = 9 3 = 27 3x 2 = 3x = = 16 5 = 80 Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 40
41 Exmple 10 Is HUG right tringle? H Whih segment is the longest? HG? ? U 10 Yes! G Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 41?
42 Your Turn. Is RST right? 10 S ? ? R 26 T Yes it is. Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 42
43 Tringle Inequlity Theorem In tringle, the sum of ny two sides is greter thn the third side > > > 4 Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 43
44 Tringle Inequlity Theorem 5 4 This is not tringle sine < Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 44
45 Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 45
46 Begin with right tringle = 2 Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 46
47 Rotte side in. 2 = < nd hve not hnged hs not hnged. got smller. 2 got smller. nd The right ngle gets smller: it is ute. Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 47
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 < C B Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 48
49 Tke nother right tringle = 2 Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 49
50 Rotte side out. 2 = > nd hve not hnged hs not hnged. got lrger. 2 got lrger. nd The right ngle gets lrger: it is otuse. Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 50
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 > C B Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 51
52 Exmple 11 The sides of tringle mesure 5, 7, nd 11. Clssify it s ute, right, or otuse. Solution: The longest side is ? ? > 74 Otuse Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 52
53 Exmple 12 The sides of tringle re 17, 20, nd 25. Clssify the tringle. Solution: 25 2? ? < 689 Aute Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 53
54 Exmple 13 Clssify this tringle ? ? Right Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 54
55 Exmple 14 Clssify this tringle It isn t tringle! 6 +8 < 16. Mrh 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 55
56 Summry In right tringle, the hypotenuse is the longest side. If tringle is right tringle, then = 2. If the three sides re ll integers, they form Pythgoren Triple. Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 56
57 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 1, 2017 Geometry 9.3 Converse of the Pythgoren Theorem 57
58 Homework Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 58
59 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 1, 2017 Geometry 9.2 The Pythgoren Theorem 59
60 Generting Pythgoren Triples Exmple: Choose = 4 nd = = = = 2(4)(3) = = = 7. 7, 24, 25 is Pythgoren Triple. Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 60
61 Generting Pythgoren Triples 7, 24, 25 is Pythgoren Triple. Chek: = 25 2? = 625? 625 = 625 Tht s triple! Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 61
62 Pythgoren Triples nd re reltively prime. > Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 62
63 Try it. Using = 8 nd = 3, find the Pythgoren Triple. Answer: = 64 9 = 55 2(8)(3) = = = 73 2? = 5329? 5329 = 5329 heks. Are Mrh 1, 2017 Geometry 9.2 The Pythgoren Theorem 63
= x x 2 = 25 2
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
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