Pythagoras Theorem. Pythagoras Theorem. Curriculum Ready ACMMG: 222, 245.
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1 Pythgors Theorem Pythgors Theorem Curriulum Redy ACMMG:, 45
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3 Fill in these spes with ny other interesting fts you n find out Pythgors. In the world of Mthemtis, Pythgors is legend. He lived from 580 BC 500 BC. One of his most reognised disoveries ws the reltionship etween the side lengths of ll right-ngled tringles. Give this go! The numers 3, 4, 5 hve the following reltionship: Find nother group of three whole numers tht inludes the numer 4 nd hs the sme reltionship. psst! the other two numers re somewhere etween 45 nd 55! Work through the ook for gret wy to do this Pythgors Theorem Mthletis Pssport 3P Lerning I SERIES TOPIC
4 How does it work? Pythgors Theorem Right-ngled tringles These speil tringles ll hve right-ngle (ngle of size 90 o ) s one of the internl ngles. Short side opposite Hypotenuse (Longest side) Right ngle Other short side Perpendiulr 90 o The two shorter sides re lwys perpendiulr to eh other. For eh of these right-ngled tringles, nme the hypotenuse nd then drw in the right-ngle. (i) (ii) Sides re lower se nd orners re CAPITALS Y X Z The hypotenuse is the longest side hypotenuse side The right-ngle is the ngle opposite the hypotenuse The hypotenuse is the longest side hypotenuse side XZ The hypotenuse is lwys opposite the right-ngle opposite hypotenuse X hypotenuse opposite Y Z I Pythgors Theorem SERIES TOPIC Mthletis Pssport 3P Lerning
5 How does it work? Your Turn Pythgors Theorem Right-ngled tringles For eh of these right-ngled tringles, nme the hypotenuse nd drw the right-ngle in the orret position. y D E x z Hypotenuse is side: Hypotenuse is side: F Q d k j l Hypotenuse is side: P R Hypotenuse is side: TRIANGLES Nme the hypotenuse for eh of these dly drwn tringles: M RIGHT-ANGLED TRIANGLES RIGHT-ANGLED.../.../0... Hypotenuse is side: L Hypotenuse is side: N Pythgors Theorem Mthletis Pssport 3P Lerning I SERIES TOPIC 3
6 How does it work? Pythgors Theorem Squres nd right-ngled tringles When squres re drwn using eh side length of right-ngled tringle, something interesting hppens. For the tringle elow: (i) Use the side lengths in the tringle to rete three squres. 4 units 5 units 3 units 5 units 4 units 3 4 units 5 units 3 units 3 units Are of squre (side length) (ii) Clulte the re of eh squre formed nd write reltionship etween them. Are 4 # Are 3 # Are + Are Are 3 Are 3 5 # units + 9 units 5 units 4 I Pythgors Theorem SERIES TOPIC Mthletis Pssport 3P Lerning
7 How does it work? Your Turn Pythgors Theorem Squres nd right-ngled tringles Show tht the reltionship etween the res of the squres formed using eh side length works for these right-ngled tringles: 5 units 3 units 3 units.../.../0... SQUARES AND RIGHT- ANGLED TRIANGLES SQUARES AND RIGHT- ANGLED TRIANGLES units 3 Are 5 units Are Are 3 units 0 units 6 units Are 8 units 3 Are 0 units Are 3 6 units 8 units 3 Try the jigsw puzzle t the k of this ooklet to see nother wy of showing this property. Pythgors Theorem Mthletis Pssport 3P Lerning I SERIES TOPIC 5
8 How does it work? Pythgors Theorem Pythgors Theorem for right-ngled tringles The squres nd right-ngled tringles setion showed tht reltionship exists etween the side lengths of right-ngled tringles. This reltionship is lled Pythgors Theorem. hypotenuse other short side short side (short side) + (other short side) (longest side) + lwys the hypotenuse If the rule does not work, then it is not right-ngled tringle. Use Pythgors Theorem to determine whih of the following tringles re right-ngled or not. (i) 8 (ii) Sustitute lengths into Pythgors Theorem (short side) + (other short side) (longest side) ! 64 not right-ngled tringle is right-ngled tringle 6 I Pythgors Theorem SERIES TOPIC Mthletis Pssport 3P Lerning
9 How does it work? Your Turn Pythgors Theorem Pythgors Theorem for right-ngled tringles Use Pythgors Theorem to lulte whih of the following tringles re right-ngled or not Right-ngled Not right-ngled Right-ngled Not right-ngled. 3.4 d Right-ngled Not right-ngled Right-ngled Not right-ngled e f FOR RIGHT-ANGLED TRIANGLES PYTHAGORAS THEOREM + FOR RIGHT-ANGLED TRIANGLES.../.../0... PYTHAGORAS THEOREM Right-ngled Not right-ngled Right-ngled Not right-ngled Pythgors Theorem Mthletis Pssport 3P Lerning I SERIES TOPIC
10 How does it work? Your Turn Pythgors Theorem Pythgors Theorem for right-ngled tringles Nme ll the right-ngled tringles pitured elow nd mrk where the right-ngle is with the orret symol. J A Rememer, tringles re nmed y their verties D 6 ΔDEF 0 K I 4.5 H F E B M K J L A 5 C N H 0 G The right-ngled tringles re: 3 Ern n wesome pssport with this one! Nme ll the right-ngled tringles in this imge nd mrk where the right-ngles re with the orret symol. R 36 5 T S * AWESOME *.../.../0... * AWESOME * P 5 Q 5 U The right-ngled tringles re: 6 V Digrm not drwn to sle. 8 I Pythgors Theorem SERIES TOPIC Mthletis Pssport 3P Lerning
11 How does it work? Your Turn Pythgors Theorem PYTHAGORAS THEOREM BY MEASUREMENT PYTHAGORAS THEOREM BY MEASUREMENT Pythgors Theorem y mesurement For eh of these right-ngled tringles: (i) Use ruler to refully mesure the length of eh side to the nerest whole millimetre. (ii) Use the mesurements to omplete the tle t the ottom of the pge..../.../ Pythgors Theorem Mthletis Pssport 3P Lerning I SERIES TOPIC 9
12 Where does it work? Pythgors Theorem Clulting the length of the hypotenuse We n use Pythgors Theorem to lulte the length of the hypotenuse if the two shorter sides of the right-ngled tringle re lredy known. + The order tht we put the short side vlues into the formul does not mtter. Clulte the length of the hypotenuse for this right-ngled tringle. 4 Hypotenuse short side + other short side Let s lel the hypotenuse Stop here if sked for nswer in ext form units or 4 + short side order does not mtter To lulte the length, squre root this vlue Write the positive nswer only euse it s length Rounded off deiml vlues re pproximte nswers only, so the ' ' symol should e used. Clulte the length of the hypotenuse for this right-ngled tringle urte to deiml ples. m 3.4 units 8.6 units Hypotenuse short side + other short side Lel the hypotenuse for esy referening m m m To lulte the length m, squre root this vlue ext form m units Answer in squre root form m m. 84. units Write full lultor reding efore rounding Approximte nswer rounded to deiml ples 0 I Pythgors Theorem SERIES TOPIC Mthletis Pssport 3P Lerning
13 Where does it work? Your Turn Pythgors Theorem Clulting the length of the hypotenuse Complete these Pythgors Theorem lultions to find the length of the hypotenuse in eh tringle. 5 6 g g g + g ext form ext form g units g units Use Pythgors Theorem to lulte the length of the hypotenuse in eh of these tringles d Pythgors Theorem Mthletis Pssport 3P Lerning I SERIES TOPIC
14 Where does it work? Your Turn Pythgors Theorem Clulting the length of the hypotenuse 3 Clulte the length of the hypotenuse in eh of these tringles, leving nswers in ext form h 35 n 4 Clulte the length of the hypotenuse in eh of these tringles, rounding nswers to deiml ples. psst! Rememer to use the for rounded nswers. 0 units 9 units 5.9 units p 3.4 units I Pythgors Theorem SERIES TOPIC Mthletis Pssport 3P Lerning
15 CALCULATING THE LENGTH OF THE HYPOTENUSE Where does it work? Your Turn Pythgors Theorem Clulting the length of the hypotenuse 5 Clulte the totl length of the 3-stge flight pth over the hills shown elow urte to deiml ple. psst! You need to do 3 hypotenuse lultions first. Flight pth Stge Stge 3 36m 39m Stge 98m Lunh pd 40m 5m 360m +.../.../0... Pythgors Theorem Mthletis Pssport 3P Lerning I SERIES TOPIC 3
16 Where does it work? Pythgors Theorem Clulting the length of short side To lulte short side length in right-ngled tringle, the formul needs little djusting. - Sutrt the given short side squred wy from the hypotenuse squred. Clulte the length of the missing side for this right-ngled tringle. 5 units units Short side hyponenuse - other short side Let s lel the short side units 9 units or - Alwys (longest side) (smller side) To lulte the length of, squre root this vlue Write the positive nswer only euse it s length Answers left in squre root form re not pproximtions, so the n still e used. Clulte the length of side k for this right-ngled tringle, leving nswer in ext form..3 k.9 Hypotenuse short side + other short side or k - k k k k units Lel the hypotenuse for referening To lulte the length k, squre root this vlue Answer in ext form 4 I Pythgors Theorem SERIES TOPIC Mthletis Pssport 3P Lerning
17 CALCULATING THE LENGTH OF THE SHORT SIDE Where does it work? Your Turn Pythgors Theorem Clulting the length of short side Fill the gps in these lultions to find the length of the missing short side in eh tringle units units Use Pythgors Theorem to lulte the length of the missing short side in eh of these tringles. j 8 units units -.../.../0... Pythgors Theorem Mthletis Pssport 3P Lerning I SERIES TOPIC 5
18 Where does it work? Your Turn Pythgors Theorem Clulting the length of short side 3 Clulte the length of the missing short side in eh of these tringles, leving nswers in squre root form w 4 Clulte the length of the missing short side in eh of these tringles, rounding nswers to deiml ple. psst! Rememer to use the. for rounded off nswers. 3.4 y 3.8 x I Pythgors Theorem SERIES TOPIC Mthletis Pssport 3P Lerning
19 Where does it work? Your Turn Pythgors Theorem Comintion of hypotenuse nd short side lultions Mth the tringles with the orret side length on the right to revel the missing nswer. The speil nme given right-ngled tringle whih is extly one hlf of n equilterl tringle: COMBO TIME COMBO TIME COMBO TIME.../.../ tringle I M 4. E 3.4 d 4.9 Q e E 6 6. h H 30 g 60 6 Pythgors Theorem Mthletis Pssport 3P Lerning I SERIES TOPIC
20 Where does it work? Pythgors Theorem Applitions of Pythgors Theorem Distnes tht re diffiult to mesure n e solved using Pythgors Theorem. Clulte how fr 5 m support will reh up wll if stnding 9 m wy from its se. Support 5 m h 9 m Wlls nd uildings re perpendiulr to the ground Let s lel the height up the wll h This is short side of right-ngled tringle h 5-9 h 5-8 h 44 h m Squre root this vlue to find h Write the positive nswer only euse it s length Pythgors Theorem is often used to lulte unknown lengths in perimeter nd re lultions. Clulte the perimeter of the grden shped like right-ngled tringle shown elow. Rememer: Perimeter is the totl distne round the outside m 35 m 35 + First need the distne long the hypotenuse Squre root this vlue 369 m 3m Perimeter of the grden 35m+ m+ 3 m Add ll the side distnes together 84 m 8 I Pythgors Theorem SERIES TOPIC Mthletis Pssport 3P Lerning
21 Where does it work? Your Turn Pythgors Theorem Applitions of Pythgors Theorem One end of 3 m stright wire is tthed to flg pole m ove the ground. How fr wy from the se of the flg pole (x) will the other end e tthed to the ground s support? PYTHAGORAS THEOREM APPLICATIONS OF PYTHAGORAS THEOREM.../.../0... APPLICATIONS OF 3 m m x Gini hs mde pudding in lrge 4 m y 34 m try. If she first uts the pudding digonlly from one orner to the other, how long ws the ut Gini mde to the nerest whole m? 4 m 34 m 3 To void going through muddy swmp, Mil wlks. km west nd then 3.9 km South. (i) How fr is Mil wy from where she strted t the end of this wlk? Round nswer to deiml ples. psst! West nd South diretions re perpendiulr (90 o ) to eh other.. km Strt 3.9 km Finish (ii) How muh further did Mil hve to wlk to void the swmp? Pythgors Theorem Mthletis Pssport 3P Lerning I SERIES TOPIC 9
22 Where does it work? Your Turn Pythgors Theorem Applitions of Pythgors Theorem 4 (i) Clulte the se length of the pinted tringle elow. (ii) Use the se length to lulte the re of the tringle. psst! The re equls (se # height) ' (i).6 m m (ii) Bse length 5 The mouse wnts to run the shortest pth from point A to point C ross the floor shown. Clulte the shortest pth etween these two points if orner B loks the diret pth. C B 54.4 m 3.3 m A 8 m Digrm not drwn to sle. 6 (i) Clulte the length of the side mrked y. (ii) Clulte the perimeter of the trpezium. m (i) y 0 m (ii) 3 m 0 I Pythgors Theorem SERIES TOPIC Mthletis Pssport 3P Lerning
23 Where does it work? Your Turn Pythgors Theorem Applitions of Pythgors Theorem Give these two trikier pplitions go to ern n wesome pssport stmp! Use Pythgors Theorem twie to find the distne etween points X nd Y. psst! Find the differene etween WY nd WX Y * AWESOME *.../.../0... * AWESOME * X W 6 Z 8 Clulte the length of the le support BD on the rne piture elow if CD 9.5 m, AB 6 m nd BC 8.5 m. B 8.5 m 9.5 m C 6 m D A Pythgors Theorem Mthletis Pssport 3P Lerning I SERIES TOPIC
24 Wht else n you do? Pythgors Theorem Pythgoren trids Pythgoren trid is the speil nme given to set of three positive integers tht work in Pythgors Theorem. Pythgoren trid integers represent the side lengths of right-ngled tringle. The integers 3, 4 nd 5 re the est known Pythgoren trid. " 3, 4, 5, Rememer: integers re just whole numers. Bres re used to disply set of integers " 3, 4, 5, is Pythgoren trid euse Integers work in Pythgors Theorem 5 3 They form right-ngled tringle 4 Beuse eh integer is side length for right-ngled tringle, negtive vlues re not llowed. Pythgors Theorem is used to show if set of three integers form Pythgoren trid. Show whether these sets of integers form Pythgoren trid or not. (i) " 8,, 4, (ii) " 9,, 5, Test to see if: (lrgest vlue) (smllest vlue) + (middle vlue) Lrgest vlue 4 Lrgest vlue 5 Middle vlue Middle vlue Smllest vlue 8 Smllest vlue 9 Test: does 4 8 +? ? 56! 548 Is " 8,, 4, Pythgoren trid? You n use the LHS RHS test pproh here too Test: does 5 9 +? ? 5 5 Is " 9,, 5, Pythgoren trid? Yes No Yes No I Pythgors Theorem SERIES TOPIC Mthletis Pssport 3P Lerning
25 PPYTHAGOREAN TRIADS Wht else n you do? Your Turn Pythgors Theorem Pythgoren trids Write the side lengths of these right-ngled tringles s Pythgoren trid set /.../0... YTHAGOREAN TRIADS 5 4 ",6,0, {,, } psst! Note tht they re written in order of size. 35 d {,, } 40 {,, } Show whether these sets of positive integers form Pythgoren trid or not. ", 4, 5, " 4, 48, 50, ", 34, 36, Yes No Yes No Yes No d " 5, 36, 39, e " 6, 60, 63, f ", 30, 3, Yes No Yes No Yes No Pythgors Theorem Mthletis Pssport 3P Lerning I SERIES TOPIC 3
26 Wht else n you do? Pythgors Theorem Eulid s formul for Pythgoren trids Eulid of Alexndri ( Greek mthemtiin) developed this method to find most Pythgoren trids: Step : Choose two positive integers p nd q. When you pik these integers, mke p lrger thn q i.e. p > q Step : Sustitute vlues for p nd q into these to mke Pythgoren trid: " p - q, pq, p + q, smll integer other smll integer lrgest integer Use the vlues p 3 nd q to mke Pythgoren trid. For the vlues p 3 nd q " p - q, pq, p + q, " 3 -,# 3 #,3 +, Sustitute in p nd q vlues " 9-4,, 9+ 4, " 5,, 3, Clulte finl vlues Let s hek tht it works 5 + 3? ? Integers form Pythgoren trid Here is nother exmple with speifi request. Use Eulid s formul to mke Pythgoren trid tht ontins the numer 8. For the vlues p 3 nd q " p - q, pq, p + q, Let pq 8 This will e the esiest to use this time p 4 nd q " 4 -,# 4 #,4 +, p > q nd ensures vlue of 8 Sustitute in p nd q vlues " 6 -, 8, 6 +, " 5, 8,, " 8, 5,, Clulte finl vlues Put vlues into sending order for trid 4 I Pythgors Theorem SERIES TOPIC Mthletis Pssport 3P Lerning
27 Wht else n you do? Your Turn Pythgors Theorem Eulid s formul for Pythgoren trids Complete this tle for the given vlues of p nd q to mke Pythgoren trids. p q p - q pq p + q Trid Mke Pythgoren trids mthing eh of these speifi requests. (i) Find Pythgoren trid in whih p nd p - q is equl to 33. EUCLID S FORMULA FOR PYTHAGOREAN TRAIDS * ", p - q, pq, p + q.../.../0... (ii) Find Pythgoren trid in whih q 5 nd p + q is equl to 6. Pythgors Theorem Mthletis Pssport 3P Lerning I SERIES TOPIC 5
28 Wht else n you do? Your Turn Pythgors Theorem Eulid s formul for Pythgoren trids 3 Find group of three integers tht inludes the numer 4 nd forms Pythgoren trid. Rememer me? 4 Use the spe elow to show why the vlue of p must e greter thn the vlue of q when using Eulid s formul to find Pythgoren trid. Use your own vlues of p nd q to help show your nswer. * AWESOME * hint: Pythgoren trids n e mde using positive integers only. This is definitely worth n wesome stmp!!.../.../0... * AWESOME * 6 I Pythgors Theorem SERIES TOPIC Mthletis Pssport 3P Lerning
29 Wht else n you do? Pythgors Theorem Wheel of Theodorus When squres re drwn using eh side length of right-ngled tringle, something interesting hppens. Strting with this isoseles tringle, eh new right-ngled tringle is uilt using the hypotenuse of the previous one The length of the longest sides form nie squre root numer pttern. How does this work? Using Pythgors Theorem: + + for the first tringle: + for the seond tringle: + ( ) The pttern ontinues in this fshion lwys using s the shortest side vlue. Pythgors Theorem Mthletis Pssport 3P Lerning I SERIES TOPIC
30 WWHEEL OF THEODORUS Wht else n you do? Your Turn Pythgors Theorem Wheel of Theodorus Using the strt mde for you, ontinue the pttern lwys using s the shortest side vlue to rete your own net spirl wheel..../.../0... HEEL OF THEODORUS 8 8 I Pythgors Theorem SERIES TOPIC Mthletis Pssport 3P Lerning
31 Wht else n you do? Your Turn Pythgors Theorem Refletion Time Refleting on the work overed within this ooklet: Wht useful skills hve you gined y lerning Pythgors Theorem? Write out one wy you think you ould pply Pythgors Theorem to rel life sitution. 3 If you disovered or lernt out ny shortuts to help with Pythgors lultions or some other ool fts, jot them down here: Pythgors Theorem Mthletis Pssport 3P Lerning I SERIES TOPIC 9
32 Chet Sheet Pythgors Theorem Here is summry of the things you need to rememer for Pythgors Theorem Right-ngled tringles These speil tringles ll hve right-ngle (ngle of size 90 o ) s one of the internl ngles. The 90 o ngle is lwys opposite the longest side. The two shorter sides re lwys perpendiulr to eh other. Pythgors Theorem for right-ngled tringles The squres nd right-ngled tringles setion showed tht reltionship exists etween the side lengths of right-ngled tringles. This reltionship is lled Pythgors Theorem. hypotenuse other short side short side (short side) + (other short side) (longest side) + lwys the hypotenuse If the rule does not work, then it is not right-ngled tringle. Clulting the length of the hypotenuse Use Pythgors Theorem to lulte the length of the hypotenuse if the two shorter sides of the right-ngled tringle re lredy known. The order tht we put the short side vlues into the formul does not mtter. Clulting the length of short side To lulte short side length in right-ngled tringle, the formul needs little djusting. + - Sutrt the given short side squred wy from the hypotenuse squred. Pythgoren Trids A Pythgoren trid is set of three positive numers tht work in Pythgors Theorem. Mking Pythgoren Trids Step : Choose two positive numers p nd q. When you pik these numers, mke p lrger thn q i.e. p > q Step : Sustitute vlues for p nd q into this to mke Pythgoren trid: " p - q, pq, p + q, Step 3: Write the vlues in sending order. 30 I Pythgors Theorem SERIES TOPIC Mthletis Pssport 3P Lerning
33 Chet Jigsw Sheet Puzzle Pythgors Theorem Squres nd right-ngled tringles: Jigsw Puzzle Step : Cut the two shded squres out from the pge Step : Cut the lrger of these two long the dotted lines. Step 3: Arrnge ll the piees to fit perfetly inside this squre. Step 4: Stik the piees to the pge to show the re of the two smller squres dd together to give the re of this squre on the hypotenuse. Pythgors Theorem Mthletis Pssport 3P Lerning I SERIES TOPIC 3
34 Pythgors Theorem Notes 3 I Pythgors Theorem SERIES TOPIC Mthletis Pssport 3P Lerning
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36 Pythgors Theorem
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