6.2 The Pythagorean Theorems

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1 PythgorenTheorems nb The Pythgoren Theorems One of the best known theorems in geometry (nd ll of mthemtics for tht mtter) is the Pythgoren Theorem. You hve probbly lredy worked with this theorem in your lgebr studies. The theorem is ttributed to Pythgors, Greek mthemticin nd philospher, who proved this theorem bout twenty-five hundred yers go. In order to understnd the proof we must first investigte geometric mens. Proportions in which the mens re equl occur frequently in geometry. For ny two positive numbers nd b, the geometric men, of nd b is the positive number x such tht Exmple 1: Find the geometric men of 4 nd 8. Solution: 4 ÅÅÅÅ x = ÅÅÅÅ x 8 Write the proportion ÅÅÅÅ x = ÅÅÅÅ x b. x 2 = 32 Use the Cross-Product Property x = è!!!!! 32 Find the positive squre root x = 4 è!!! 2 Write in simplest rdicl form The following theorem shows us how severl geometric mens "pop up" when you drw the ltitude to the hypotenuse of right tringle. Theorem: The ltitude to the hypotenuse of right tringle divides the tringle into two tringles tht re similr to the originl tringle nd to ech other..

2 PythgorenTheorems nb 2 Look t the figure below. We re given right DABC with êêêêê CD s the ltitude to the hypotenuse. C A D B DABC ~ DACD ~ DCBD This theorem cn be esily proved by the AA Similrity Postulte since both smller tringles shre n cute ngle with DABC nd ll three tringles re right tringles. Since both smller tringles re similr to DABC, their corresponding ngles re congruent. Thus the two smller tringles re similr to ech other. There re two importnt corollries of this theorem one of which will be used in the proof of the Pythgoren Theorem. Corollry 1: When the ltitude is drwn to the hypotenuse of right tringle, the length of the ltitude is the geometric men between the segments of the hypotenuse. Corollry 2: When the ltitude is drwn to the hypotenuse of right tringle, ech leg is the geometric men between the hypotenuse nd the segment of the hypotenuse tht is djcent to the leg.

3 PythgorenTheorems nb 3 Let's look t our tringle gin. C A D B The first corollry sttes tht CD is the geometric men of AD nd DB. Thus, AD ÅÅÅÅÅÅÅÅÅ CD = ÅÅÅÅÅÅÅÅ CD DB. The second corollry sttes tht AC is the geometric men of AB nd AD (the segment of the hypotenuse djcent to AC) nd tht CB is the geometric men of AB nd DB. Thus, AD ÅÅÅÅÅÅÅÅÅ AC = ÅÅÅÅÅÅÅÅ AC AB nd DB ÅÅÅÅÅÅÅÅ CB = ÅÅÅÅÅÅÅÅ CB AB Now we re redy to stte nd prove the Pythgoren Theorem. Pythgoren Theorem: In right tringle, the squre of the hypotenuse is equl to the sum of the squres of the legs.

4 PythgorenTheorems nb 4 Given: Right DABC; C is right. Prove: c 2 = 2 + b 2 C b A d e B c Proof: Sttements êêêêê 1. Drw perpendiculr from C to AB Resons 1. Through point outside line, there is exctly one line perp. to the line. 2. c ÅÅÅÅ = ÅÅÅÅ e ; ÅÅÅÅ c b = b d ÅÅÅÅ 2. Corollry 2 from bove. 3. ce = 2 ; cd = b 2 3. A property of proportions 4. ce + cd = 2 + b 2 4. Addition property 5. c(e + d) = 2 + b 2 5. Distributive property 6. c 2 = 2 + b 2 6. Substitution property Let's look t some exmples.

5 PythgorenTheorems nb 5 Exmple 2: Find the vlue of x. 3 x 6 Solution: x 2 = x 2 = x 2 = 45 x = è!!!!! 45 x = 3 è!!! 5 Note: we only used the positive squre root since x represents length. Exmple 3: Find the vlue of x. 10 x x+2 Solution: x 2 + Hx + 2L 2 = 10 2 x 2 + x x + 4 = x x - 96 = 0 x x - 48 = 0 (x + 8)(x - 6) = 0 x = -8 or x = 6. Since x represents length we discrd the negtive solution nd x = 6.

6 PythgorenTheorems nb 6 The converse of the Pythgoren Theorem is lso true nd we present it, without proof, below. The Converse of the Pythgoren Theorem: If the squre of one side of tringle is equl to the sum of the squres of the other two sides, then the tringle is right tringle. Exmple 4: Determine if the tringle with side lengths 5, 12, nd 13 is right tringle. Solution: = = 169 nd 13 2 = 169 hence = 13 2 nd by the converse of the Pythgoren Theorem the tringle is right tringle. Three integers (like 5,12, nd 13) tht stisfy the conditions of the Pythgoren Theorem re clled Pythgoren Triples. If the three integers re reltively prime (mening they hve no common fctors) then the three integers re know nd Primitive Pythgoren Triples. Some common Primitive Pythgoren Triples nd their multiples re listed below. Memorizing some of these triples cn mke your computtions esier in the future. Some Common Pythgoren Triples 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 6, 8, 10 10, 24, 26 9, 12, 15 12, 16, 20 When tringle is not right tringle, the squres of the sides cn be used to determine whether the tringle is obtuse or cute. Look t the theorems below. Theorem: If the squre of the longest side of tringle is greter thn the sums of the squres of the other two sides, then the tringle is n obtuse tringle.

7 PythgorenTheorems nb 7 A little common sense cn tell us why this is true s well. Picture right tringle. c b The conclusion of the Pythgoren Theorem tells us tht c 2 = 2 + b 2. Now picture the right ngle opening like door without chnging lengths nd b. d b Obviously, the new longest side, d, is longer thn the length of c in the first tringle. So, it mkes sense tht d 2 > c 2 nd thus d 2 > 2 + b 2. Theorem: If the squre of the longest side of tringle is less thn the sum of the squres of the other two sides, then the tringle is n cute tringle. Now picture the right ngle closing. e b

8 PythgorenTheorems nb 8 Obviously, in this cse, e < c nd thus e 2 < c 2 nd e 2 < 2 + b 2. So we now hve wy to determine if tringle is right tringle, n obtuse tringle, or n cute tringle, if we know the lengths of ll three sides of the tringle. Exmple 5: Determine whether tringle formed with sides hving the lengths nmed is cute, right, or obtuse. (A) 9, 40, 41 (B) 6, 7, 8 (C) 8, 10, 14 Solutions: (A) ? ? = 1681 The tringle is right. (B) ? ? > 64 The tringle is cute. (C) ? ? < 196 The tringle is obtuse. Now we re redy to look t some specil right tringles in the next section.