Geometry Proofs: Chapter 7, Sections 7.1/7.2

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1 Pythgoren Theorem: Proof y Rerrngement of re Given: Right tringle with leg lengths nd, nd hypotenuse length. Prove: = + Proof #1: We re given figures I nd II s ongruent right tringles III with leg lengths nd, nd hypotenuse I II length. So the re of figures I nd II is ½. We know tht figure III is n isoseles right tringle sine ngles 1,2,3 mke stright ngle while ngles 1 nd 3 re omplementry (orollry to Tringle Sum Th.) nd thus ngle 2 is right ngle. The re of figure III is ½ 2. We know tht figures I,II,III re ordered y trpezoid (Lines to Trnsversl Th.) whose re is ½ ( + )( + ). y the re ddition Postulte, the re of the trpezoid equls the sums of the res of figures I,II,III: Geometry Proofs: hpter 7, Setions 1/2 onverse of the 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. Given: with longest side, 2 = Prove: Right 1. with longest side, 2 = onstrut Right PQR with 2. Perpendiulr & Segment Right R nd leg lengths nd onstrution Pythgoren Theorem Sustitution (#1,3) Property of Squre Roots Proof #2: We re given figures I,II,III,IV s 1 2 ongruent right tringles with leg lengths nd, II nd hypotenuse length. So the re of figures I I,II,III,IV is ½. We know tht figure V is V squre ( Lines Form 4 Right ngles) with side lengths ( ) with n re of ( ) 2. III We know tht figures I,II,III,IV,V re ordered y IV squre sine ngles 1 nd 2 re omplementry (orollry to Tringle Sum Th.) nd thus re right ngles with side lengths of whose re is 2. y the re ddition Postulte, the re of the lrge squre equls the sums of the res of figures I,II,III,IV,V: 6. PQ; QR; RP 6. PQR m = m R m R = ef. of Right ngle 1 1 ef. of Right Tringle

2 Theorem 4: If the squre of the length of the longest side of tringle is less thn the sum of the squres of the lengths of the other two sides, then the tringle is n ute tringle. Given: with longest side, 2 < Prove: is ute 1. with longest side, 2 < onstrut right PQR w/ Right R nd leg lengths nd QR; RP 2. Perpendiulr & Segment onstrution Pythgoren Theorem < r Prop. of Squre Roots m < m R ef. of Right ngle Sustitution (#7,8) 10. is ute is n ute tringle 11. Theorem 5: If the squre of the length of the longest side of tringle is greter thn the sum of the squres of the lengths of the other two sides, then the tringle is n otuse tringle. Given: with longest side, 2 > Prove: is otuse 1. with longest side, 2 > onstrut right PQR w/ Right R nd leg lengths nd QR; RP 2. Perpendiulr & Segment onstrution Pythgoren Theorem > r Prop. of Squre Roots m > m R ef. of Right ngle Sustitution (#7,8) 10. is otuse is n otuse tringle 11.

3 Geometry Proofs: hpter 7, Setion 3 Th. #5: If the ltitude is drwn to the hypotenuse of right tringle, then the two tringles formed re similr to the originl tringle nd to eh other. GMT: The ltitude drwn from the right ngle to the hypotenuse of right tringle is the geometri men of the two hypotenuse prts. Given: Right (with right ) with ltitude Prove: ~ ~ Given: Right Prove: = with ltitude 1. Right with ltitude Right (with right ) 1. with ltitude ef. of ltitude, re right s If the ltitude is drwn to the hypotenuse of right tringle, then the two tringles formed re similr to the originl tringle nd to eh other. = 4., Reflexive Property of ongruene 6. ~, ~ 6. Trnsitive Property of Similr Tringles

4 GMLT: The leg of right tringle is the geometri men of the djent hypotenuse length when n ltitude is drwn from the right ngle to the hypotenuse nd the whole hypotenuse. Given: Right Prove: = with ltitude Pythgoren Theorem (Using GMLT) 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 of the right tringle. Given: Right, is right ngle Prove: 2 = e f Given 1. Right with ltitude If the ltitude is drwn to the hypotenuse of right tringle, then the two tringles formed re similr to the originl tringle nd to eh other. = 2. rw 2. f = ; e = 4. f= 2 ; e= f + e = = e + f 6. (f + e) = Sustitution (#6,7) 9. 2 =

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