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1 Thnk you for prtiipting in Teh It First! This Teh It First Kit ontins Common Core Coh, Mthemtis teher lesson followed y the orresponding student lesson. We re onfident tht using this lesson will help you hieve your ssessment preprtion gols for your entire lss. The Common Core Coh, Mthemtis progrm is sed on the philosophy tht mthemtil skills re uilt on onepts. Mth, mye more thn ny other shool sujet, uilds from onept to onept, one on top of the other, over severl yers. When students understnd onepts nd how they onnet to skills, they re etter equipped to solve prolems tht they enounter in the rel world. This progrm is 100% ligned to the Common Core Stte Stndrds nd provides set of lessons for eh of the five CCSS domins, with eh lesson ligning to one or more stndrd together, lessons over ll the domin s stndrds. Conept Lessons egin with n underlying onept tht onnets diretly to the skill or skills tught in tht lesson. Students will use four-step prolem-solving proess Red, Pln, Solve, Chek to pproh ny mthemtil prolem. Intertive questions follow emples nd sk students to disuss topi, model sitution, try to solve prolem on their own, or hek their work. With this instrutionl nhor, you n implement the Common Core Stte Stndrds with onfidene. We re hppy to provide you this omplimentry smple nd would love to know wht you think. One you hve red through this lesson, do wht you do est present it to your students. Then, don t forget to omplete quik survey y going to Regrds, Triumph Lerning Join the onverstion out Common Core tody y visiting ommonore.om, the ple where tehers, prents, nd eperts ome together to shre est prties nd prtil informtion for suessfully implementing Common Core stndrds in the lssroom. 136 Mdison Avenue New York, NY p: f :
2 25 Eplining the Pythgoren Theorem LESSON Lerning Ojetives Students will understnd proof of the Pythgoren theorem nd its onverse. Students will use the Pythgoren theorem to find the length of missing side of right tringle nd the onverse to hek if tringle is right tringle. Common Core Stte Stndrd 8.G.6 Eplin proof of the Pythgoren Theorem nd its onverse. Voulry hypotenuse in right tringle, the side opposite the right ngle leg in right tringle, one of the two shorter sides; it is opposite one of the two ute ngles Pythgoren theorem sttes tht the sum of the squres of the lengths of the legs in right tringle is equl to the squre of the length of the hypotenuse Before the Lesson Review the definition of right tringle nd the speil voulry ssoited with right tringles. Remind students tht the longest side of tringle is opposite the ngle with the gretest mesure. Point out tht the ngle with the gretest mesure in right tringle is the right ngle. Disuss why. Emphsize tht the side opposite the right ngle in right tringle is lled the hypotenuse. The two sides tht re djent to the right ngle re the legs. Understnd Connet To help develop oneptul understnding, disuss the proof in detil. Emphsize tht euse the figures re ongruent, their res re ongruent. The proof relies on this ft. The ongruent res re lgerilly simplified to produe the Pythgoren theorem. Chek tht students understnd the formul for the re of eh figure. Work through the steps tht re used to simplify eh form of the re. Cll ttention to the epressions for eh re. Point out tht students n lso find the res of the 4 qudrilterls of Figure 1 to determine its re: the smll squre hs n re of 2, the two retngles hve res of, nd the lrge squre hs n re of 2. The re of Figure 1 is the sum of the res of the qudrilterls, Hve students eplin in their own words why the epressions re equted to find the Pythgoren theorem nd how it simplifies to the theorem. Point out tht the onverse of the Pythgoren theorem strts with the speil reltionship etween the sum of the squres of tringle nd sys tht if this reltionship is met, then the tringle is right tringle. To onnet the onept to proedurl understnding, sk, Why n the Pythgoren theorem e used to find the missing side length? (The tringle is right tringle.) Emphsize tht the Pythgoren theorem is only for right tringles. Disuss the importne of distinguishing etween the legs nd the hypotenuse of right tringle to pply the theorem. Ask: Whih side is lwys the Dupliting ny prt of this ook is prohiited y lw. 72
3 hypotenuse in right tringle? (The side opposite the right ngle.) How does the length of the hypotenuse ompre to the length of the legs? (It is lwys the longest side.) Disuss how to use this ft to hek tht the Pythgoren theorem hs een orretly nd urtely used. Suggest tht students lwys write the Pythgoren theorem s the first step when using it to find missing length. Then sustitute nd solve. Review how to use the order of opertions to solve for. DISCUSS MP2 MP4 Disuss how to reognize tht given tringle is right tringle. No, the Pythgoren theorem nnot e used. Eplntions my vry. Possile eplntion: The Pythgoren theorem only reltes the lengths of the sides in right tringle. Tringle MNP is n otuse tringle, so the Pythgoren theorem nnot e used. Emples EXAMPLE A This emple introdues students to rel-life prolem tht is solved y using the Pythgoren theorem. Disuss how the digrm illustrtes the informtion in the prolem. DISCUSS MP1 MP5 Disuss how to evlute the squre root of greter numer. Answers my vry. Possile nswer. One wy is to use guess nd hek. For emple, I know tht , so I would guess numer suh s , whih is still too low. I would guess higher numer, suh s , whih is too high. However, now I know tht 576 is etween 20 nd 25. If I guessed 24 net, I would find out tht , so Prtie As students re working, py speil ttention to prolems 4 9, whih provide n opportunity for students to use the onverse of the Pythgoren theorem to determine if the given sides form right tringle. For nswers, see pges 107 nd 108. Common Errors Students my inorretly pply the Pythgoren theorem y not following the order of opertions in their omputtions. Review the order of opertions nd how they pply in prtiulr to prolems tht use the Pythgoren theorem to find the length of missing side of right tringle. Domin 4 Dupliting ny prt of this ook is prohiited y lw. EXAMPLE B In this emple students use the onverse of the Pythgoren theorem to determine if tringle is right tringle. Review how to hoose the lengths of the sides to test the theorem. TRY MP3 MP6 Point out tht Ros only hnges the length of the longest penil. The other two penils sty the sme length. Disuss why this mens the Pythgoren theorem n e used to find the desired length of the longest penil. 17 entimeters
4 25 LESSON Eplining the Pythgoren Theorem UNDERSTAND The Pythgoren theorem sttes tht, in ny right tringle, the sum of the squres of the lengths of the legs is equl to the squre of the length of the hypotenuse. The two ongruent squres shown elow were uilt using ongruent right tringles nd squres with lengths,, nd. Use Figures 1 nd 2 to prove the Pythgoren theorem. leg hypotenuse leg Figure 1 Figure 2 Let ( 1 ) 2, or ( 1 )( 1 ), represent the re of Figure 1. Simplify. A of Figure 15 ( 1 )( 1 ) Use the distriutive property to simplify. 5 ()( 1 ) 1 ( 1 ) Comine like terms Figure 2 ws uilt using 4 ongruent right tringles with se nd height, nd squre with sides, so: A of Figure 2 5 [4 3 (A of eh tringle)] 1 (A of squre) Sustitute Multiply Sine the figures re ongruent, set the epressions for their res equl. Simplify. (A of Figure 1) 5 (A of Figure 2) Sustitute Sutrt 2 from oth sides The onverse of the Pythgoren theorem sttes tht if tringle hs sides of length,, nd suh tht , then the tringle is right tringle with right ngle opposite. Dupliting ny prt of this ook is prohiited y lw. 136 Domin 4: Geometry
5 Connet Determine the length of KL in njkl. K 6 J 8 L 1 Wht kind of tringle is njkl? The right-ngle symol shows tht /J is right ngle. n JKL is right tringle. 2 Identify the lengths of the legs nd the hypotenuse. 3 Use the formul for the Pythgoren theorem. The legs hve lengths of 6 units nd 8 units. The hypotenuse, KL, is units long. Dupliting ny prt of this ook is prohiited y lw. Sustitute 6 for nd 8 for. Solve for So, equls 10. The length of KL is 10 units. DISCUSS Could you use the Pythgoren theorem to find the missing length of NP in this tringle? Eplin. N 6 M 8 P Lesson 25: Eplining the Pythgoren Theorem 137
6 EXAMPLE A A guy wire tht is 26 feet long is tthed to the top of pole. The wire is tthed to the ground t point tht is 10 feet from the se of the pole. Determine h, the height of the pole. 26 ft h 10 ft 1 How will you solve this prolem? The right-ngle symol shows tht the wire, the pole, nd the distne from the se of the pole to the ple where the wire is tthed form right tringle. Use the formul for the Pythgoren theorem. 2 Identify the legs nd the hypotenuse. 3 Sustitute those lengths into the formul nd solve for h. The legs hve lengths of 10 feet nd h feet. The hypotenuse is 26 feet long h Sustitute h Sutrt 100 from oth sides. h h h 5 24 The height of the pole is 24 feet. DISCUSS If you hve lultor, finding the integer vlue of 576 is simple. How ould you evlute 576 if you do not hve lultor? Dupliting ny prt of this ook is prohiited y lw. 138 Domin 4: Geometry
7 EXAMPLE B Ros hs three penils, eh different length. The lengths re 18 entimeters, 15 entimeters, nd 8 entimeters. Could she form right tringle using these penils s the sides? 1 Wht does the onverse of the Pythgoren theorem stte? It sttes tht if tringle hs side lengths,, nd suh tht , then the tringle is right tringle. 2 Identify the shorter penil lengths nd the longest penil length. 3 Test the vlues in the formul entimeters, 15 entimeters, nd 18 entimeters re not the side lengths of right tringle. It is not possile for Ros to use those penils to mke right tringle. The shorter lengths re 8 entimeters nd 15 entimeters. Those would represent the shorter sides of the tringle. Let 5 8 nd The longest length is 18 entimeters. Let Dupliting ny prt of this ook is prohiited y lw. TrY Ros finds tht if she shrpens her longest penil nd shortens its length, she n use it to form right tringle with the other two penils. Wht length would she need to mke the longest penil? Lesson 25: Eplining the Pythgoren Theorem 139
8 Prtie Write n eqution tht shows the reltionship etween the given side lengths of these right tringles. Simplify if possile. 1. d 2. m p n 3. y REMEMBER The sum of the squres of the leg lengths equls the squre of the hypotenuse length. Use the onverse of the Pythgoren theorem to determine whether or not tringle with the given side lengths is right tringle. Show your work in., 4 in., 5 in yd, 7 yd, 11 yd 6. 5 m, 10 m, 15 m 7. 9 m, 39 m, 41 m mm, 99 mm, 101 mm 9. 4 m, 7.5 m, 8.5 m Choose the est nswer. 10. A right tringle hs legs mesuring 15 meters nd 20 meters. Wht is the length of the hypotenuse? A. 13 meters B. 17 meters C. 25 meters D. 35 meters 11. A right tringle hs leg mesuring 24 units nd hypotenuse mesuring 25 units. Wht is the length of the other leg? A. 7 units B. 9 units C. 35 units D. 49 units Dupliting ny prt of this ook is prohiited y lw. 140 Domin 4: Geometry
9 Find, the missing side length in eh right tringle. Show your work Prove. Use entimeter ruler nd protrtor, s needed. 18. PROVE Given: nfgh hs sides,, nd units long, nd it is true tht To prove tht the onverse of the Pythgoren theorem is true, show tht nfgh is right tringle. F Dupliting ny prt of this ook is prohiited y lw. H G To do this, mesure lengths nd, in entimeters. Then drw right tringle JKL, with legs nd units long, nd hypotenuse leled d. Mke /J the right ngle. Eplin how you know tht d 2. Then use sequene of rigid motions to prove tht nfgh is right tringle. How does this show tht nfgh is right tringle? Lesson 25: Eplining the Pythgoren Theorem 141
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