.5 Proving the Pythgoren Theorem Proving the Pythgoren Theorem nd the Converse of the Pythgoren Theorem Lerning Gols In this lesson, you will: Prove the Pythgoren Theorem using similr tringles. Prove the Converse of the Pythgoren Theorem using lgeri resoning. The Pythgoren Theorem is one of the most fmous theorems in mthemtis. And the proofs of the theorem re just s fmous. It my e the theorem with the most different proofs. The ook Pythgoren Proposition lone ontins 370 proofs. The srerow in the film The Wizrd of Oz even tries to reite the Pythgoren Theorem upon reeiving his rin. He proudly sttes, 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! Oh, rpture! I ve got rin! Sdly the srerow s version of the theorem is wrong so muh for tht rin the wizrd gve him! Crnegie Lerning 311
Prolem 1 Proving the Pythgoren Theorem with Similr Tringles Use the Right Tringle/Altitude Similrity Theorem to prove the Pythgoren Theorem. C A B Given: Tringle ABC with right ngle C Prove: AC 2 1 CB 2 5 AB 2 1. Construt ltitude CD to hypotenuse AB. 2. Applying the Right Tringle/Altitude Similrity Theorem, wht n you onlude? 3. Write proportionl sttement desriing the reltionship etween the longest leg nd hypotenuse of tringle ABC nd tringle CBD.. Rewrite the proportionl sttement you wrote in Question 3 s produt. 5. Write proportionl sttement desriing the reltionship etween the shortest leg nd hypotenuse of tringle ABC nd tringle ACD. Crnegie Lerning 312 Chpter Similrity Through Trnsformtions
6. Rewrite the proportionl sttement you wrote in Question 5 s produt. 7. Add the sttement in Question to the sttement in Question 6. 8. Ftor the sttement in Question 7. 9. Wht is equivlent to DB 1 AD? 10. Sustitute the nswer to Question 9 into the nswer to Question 8 to prove the Pythgoren Theorem. Crnegie Lerning.5 Proving the Pythgoren Theorem nd the Converse of the Pythgoren Theorem 313
Prolem 2 Proving the Pythgoren Theorem with Algeri Resoning Use the digrm shown nd the following questions to prove the Pythgoren Theorem. 1. Wht is the re of the lrger squre? 2. Wht is the totl re of the four right tringles? 3. Wht is the re of the smller squre?. Wht is the reltionship etween the re of the four right tringles, the re of the smller squre, nd the re of the lrger squre? Crnegie Lerning 31 Chpter Similrity Through Trnsformtions
Prolem 3 Proving the Converse of the Pythgoren Theorem Use the digrm shown nd the following questions to prove the Converse of the Pythgoren Theorem. A lrge squre is omposed with four identil right tringles in its orners. 2 3 1 1 3 2 1 3 2 Rell, the Converse of the Pythgoren Theorem sttes: If 2 + 2 = 2, then nd re the lengths of the legs of right tringle nd is the length of the hypotenuse. 2 3 1 1. Wht n you onlude out m 1 1 m 2 1 m 3? 2. Use the Tringle Sum Theorem to determine m 1 1 m 2. 3. Knowing m 1 1 m 2, wht n you onlude out m 3?. Wht does m 3 tell you out the qudrilterl inside of the lrge squre? Crnegie Lerning 5. Wht is the re of one of the right tringles?.5 Proving the Pythgoren Theorem nd the Converse of the Pythgoren Theorem 315
6. Wht is the re of the qudrilterl inside the lrge squre? 7. Write n expression tht represents the omined res of the four right tringles nd the qudrilterl inside the lrge squre. Use your nswers from Questions 5 nd 6. 8. Write n expression to represent the re of the lrge squre, given tht one side is expressed s ( 1 ). Simplify your nswer. 9. Write n eqution using the two different expressions representing the re of the lrge squre from Questions 7 nd 8. Then, solve the eqution to prove the Converse of the Pythgoren Theorem. Crnegie Lerning Be prepred to shre your solutions nd methods. 316 Chpter Similrity Through Trnsformtions