1Measurement and geometry. Pythagoras theorem

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1 1Mesurement n geometry Pythgors theorem rhiteture, engineering, surveying n mny other fiels. ws known to numer of nient ivilistions, inluing the Egytins who use the rule to onstrut their yrmis s fr k s 3000 BCE, ut Pythgors ws the first erson to write it in mthemtil wy. exresses the reltionshi etween the sies of right-ngle tringle. It is still use toy in

2 NEW CENTURY MATHS for the Austrlin Curriulum8 n Chter outline Profiieny strns 1-01 Squre roots n surs U 1-02 Disovering U F R C 1-03 Fining the hyotenuse U F 1-04 Fining shorter sie U F 1-05 Mixe rolems F 1-06 Testing for right-ngle R tringles U F 1-07 Pythgoren tris U F 1-08 rolems F PS This is Yer 9 toi in the Austrlin Curriulum ut Stge 4 (Yers 7 8) toi in the NSW syllus. n Wornk onverse A rule or sttement turne k-to-front; the reverse sttement hyotenuse The longest sie of right-ngle tringle; the sie oosite the right ngle Pythgors An nient Greek mthemtiin who isovere n imortnt formul out the sies of right-ngle tringle sur A squre root (or other tye of root) whose ext eiml vlue nnot e foun theorem A forml rule or formul tri A grou of three relte ojets

3 Chter n In this hter you will: investigte the onet of irrtionl numers estlish the reltionshi etween the lengths of the sies of right-ngle tringle in rtil wys, inluing using igitl tehnologies. investigte n its lition to solving simle rolems involving rightngle tringles solve rtil rolems involving, roximting the nswer s eiml or giving n ext nswer s sur use the onverse of to estlish whether tringle hs right ngle ientify Pythgoren tri s set of three numers suh tht the sum of the squres of the first two equls the squre of the thir SkillChek Worksheet StrtU ssignment 1 MAT08MGWK Evlute eh exression e f 3 2 þ 4 2 g 5 2 þ 12 2 h Evlute eh squre root. ffiffi ffiffi ffiffi ffiffiffiffiffiffiffiffiffi 196 ffiffiffiffiffiffiffiffiffi e 121 f 49 g 1:21 h 2:25 3 Fin the erimeter of eh she. 4 m 9 m 12 m 27 mm 7 m 11 mm 4 Fin the re of eh she. 10 m 4 m 4 m 5 Write eh numer orret to one eiml le. 10 mm 20 mm Evlute eh exression orret to one eiml le. ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi 3 2 þ 7 2 5:6 2 þ 9:

4 NEW CENTURY MATHS for the Austrlin Curriulum Squre roots n surs ffi The squre root ð Þ of numer is the ositive vlue whih, if squre, will give tht numer. For exmle: 25 ffiffiffiffiffi ¼ 5 euse 5 2 ¼ 25 the squre root of ¼ 7 euse 7 2 ¼ 49 the squre root of 49 Most ffiffiffiffiffi squre roots o not give ext nswers like the ones ove. For exmle, ffi 10 ¼ 3: Suh squre roots re lle surs. A sur is squre root ð Þ, 3 ffi ue root ð Þ, or ny other tye of root whose ext eiml or frtion vlue nnot e foun. As eiml, its igits run enlessly without reeting, so they re neither terminting nor reurring eimls. A sur nnot e written in frtion form so it is lso lle n irrtionl numer. Worksheet Tle of squres n squre roots MAT08MGWK10002 Worksheet Surs MAT08MGWK10003 Exmle 1 Evlute eh exression orret to two eiml les. ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 2 þ ffiffiffiffiffiffiffiffiffiffi 7 2 þ 9 2 ¼ 11: :40 On lultor, enter ( ) = OR: 7 2 þ 9 2 ¼ 130 On lultor, enter = ffiffi 130 ¼ 11: On lultor, enter ANS = 11:40 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 6: :63 OR: On lultor, enter ( ) = ¼ 140 On lultor, enter = ffiffi 140 ¼ 6: On lultor, enter ANS = 6:

5 Chter Exmle 2 Selet the surs from this list of squre roots: 72 ffiffi ffiffi 72 ¼ ¼ ¼ ffiffiffiffiffi ffiffiffiffiffi 28 ¼ so the surs re 72, 90 n ¼ 8 Exerise 1-01 Squre roots n surs See Exmle 1 1 Evlute eh squre root. ffiffi 196 ffiffiffiffiffiffiffi 625 e ffiffi 900 ffiffiffiffiffiffiffi 121 f 64 ffiffiffiffiffiffiffi Evlute eh squre root, orret to two eiml les. ffiffiffiffiffi ffiffi 12 ffiffi ffiffiffiffiffi e 153 ffiffiffiffiffi f 207 ffiffiffiffiffiffiffi g 98 ffiffiffiffiffiffiffi i 24 j 110 k 297 h l ffiffi 325 ffiffiffiffiffiffiffi 888 ffiffiffiffiffiffiffi Evlute eh exression, orret to two eiml les. ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi 12 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 7 ffiffiffiffiffiffiffiffiffiffi e qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 g ð1:5þ 2 þð4:2þ 2 h ð12:5þ 2 ð7:1þ 2 f i ffiffiffiffiffiffiffiffiffiffiffiffi 6 2 þ 11 2 ffiffiffiffiffiffiffiffiffiffi 4 2 þ 9 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð25:7þ 2 þð18:2þ 2 See Exmle 2 4 Whih of the following is not sur? Selet the orret nswer A, B, C or D. A ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi 10:24 B 24 C 1000 D ffiffiffi 3 5 Selet ll the surs from this list of squre roots. ffiffiffi ffiffi 98 ffiffiffiffiffiffiffi 9 ffiffiffiffiffi 225 ffiffiffiffiffi ffiffi 160 ffiffiffiffiffiffiffi ffiffiffiffiffiffiffi ffiffiffiffiffi 81 Just for the reor Pythgors ws mthemtiin who live in nient Greee. The Pythgorens were grou of men who followe Pythgors. This is the symol of the Pythgoren seret soiety. Arently they were so uset out the isovery of surs tht they trie to kee it seret. Hisus, one of the Pythgorens, ws rowne for reveling the seret to outsiers. Pythgors n the Pythgorens Fin the seil nme for this symol involving str insie entgon. This is the symol of the Pythgoren seret soiety

6 NEW CENTURY MATHS for the Austrlin Curriulum Disovering A right-ngle tringle hs one right ngle (90 ) n two smller ngles. The sie oosite the right ngle is the longest sie n is lle the hyotenuse. is rule tht esries the reltionshi etween the hyotenuse n the other two (shorter) sies. Exmle 3 The hyotenuse is lwys oosite the right ngle. Construt right-ngle tringle with the two shorter sies eing of length 3 m n 4 m. Mesure the length of the hyotenuse. Evlute 3 2 þ 4 2. If is the length of the hyotenuse in entimetres, evlute 2. the hyotenuse (the longest sie) Tehnology GeoGer: Pythgors theorem MAT08MGTC00004 Worksheet Pythgors isovery MAT08MGWK10004 Worksheet A ge of right-ngle tringles MAT08MGWK10005 Worksheet Proving Pythgors theorem MAT08MGWK10006 Mesuring with ruler, the hyotenuse is 5 m long. m 3 m 4 m 3 2 þ 4 2 ¼ ¼ 5 2 ¼ 25. Exerise 1-02 Disovering 1 Nme the hyotenuse in eh tringle. q x L M r z y N 2 Drw right-ngle tringle with short sies mesuring 5 m n 12 m. Mesure the length of the hyotenuse. Evlute 5 2 þ See Exmle

7 Chter If is the length of the hyotenuse in entimetres, evlute 2. e Coy n omlete eh sttement. The squre of the hyotenuse ¼ 2 ¼ The sum of the squres of the two shorter sies ¼ 2 þ 2 ¼ 3 Whih eqution is true for the right-ngle tringle shown? Selet the orret nswer A, B, C or D. A 8 þ 15 ¼ 17 B 8 2 þ 15 2 ¼ 17 2 C 8 2 þ 17 2 ¼ 15 2 D 17 2 þ 15 2 ¼ Use GeoGer (or enil, ruler n 5 mm gri er) to rw right-ngle tringle with the shorter sies of length 6 m n 8 m, then mesure the length of the hyotenuse. Clik View Gri n remove Axes efore you egin, then rete the tringle using Intervl Between Two Points. Chek the lengths using Distne or Length. Coy n omlete this tle y: onstruting eh right-ngle tringle with the mesurements given for the two shorter sies n (using GeoGer or enil n er) mesuring the length,, of the hyotenuse evluting the vlues of 2, 2, 2 þ 2, 2 Shorter sies (m) Hyotenuse (m) þ Wht rule or reltionshi i you isover etween the length of the hyotenuse n the lengths of the other two sies? Vieo tutoril 1-03 Fining the hyotenuse MAT08MGVT10001 Tehnology worksheet Summry Exel: Pythgors theorem MAT08MGCT00024 Tehnology Exel: Pythgors theorem MAT08MGCT00009 In right-ngle tringle, the squre of the hyotenuse is equl to the sum of the squres of the other two sies. If is the length of the hyotenuse, n n re the lengths of the other two sies, then: 2 ¼ 2 þ

8 NEW CENTURY MATHS for the Austrlin Curriulum8 A theorem is forml rule or formul. ws isovere y the nient Greek mthemtiin Pythgors. This rule is use to fin the length of one sie of right-ngle tringle when the other two sies re known. Exmle 4 Write for nlnm. L The hyotenuse is LN, the two shorter sies re LM n MN. So is LN 2 ¼ LM 2 þ MN 2 M N Exmle 5 Fin the vlue of in this tringle. 9 m m is the length of the hyotenuse. Using Pythgors theorem: 2 ¼ 2 þ 2 2 ¼ 9 2 þ 40 2 ¼ 1681 ¼ ffiffiffiffiffiffiffiffiffiffi 1681 ¼ 41 Sustitute the lengths of the two shorter sies. Use the squre root to fin. Chek: A hyotenuse of 41 m looks resonle from the igrm: it is the longest sie n little longer thn 40 m. m 40 m Exmle 6 Fin the vlue of x in this tringle: s sur orret to two eiml les. x m 36 m Write. 2 ¼ 2 þ 2 x 2 ¼ 50 2 þ 36 2 ¼ 3796 x ¼ ffiffiffiffiffiffiffiffiffiffi m Sustitute the lengths of the two shorter sies. Beuse the question sks for sur, leve the nswer in squre root ( ) form

9 Chter x ¼ ffiffiffiffiffiffiffiffiffiffi 3796 ¼ 61: From rt. Chek: From the igrm, this nswer looks resonle. Note: The sur nswer is more ext thn the eiml nswer euse it is not roune. Exerise 1-03 Fining the hyotenuse See Exmle 4 1 Wht is for this tringle? Selet the orret nswer A, B, C or D. A z 2 ¼ x 2 þ y 2 B ZY 2 ¼ ZX 2 þ XY 2 C XY 2 ¼ ZY 2 þ ZX 2 D y 2 ¼ x 2 þ z 2 Z y x X z Y 2 Write for eh right-ngle tringle. m r s n t L J K e 10 f x 3 3 See Exmle 5 3 Fin the vlue of eh ronumerl. m 9 m h m 6 m 15 m 20 m x m 12 m 8 m See Exmle 6 4 Fin the vlue of eh ronumerl s sur. m km 17 km 20 mm 24 km m m 4 m 13 mm x mm

10 NEW CENTURY MATHS for the Austrlin Curriulum8 1 m e 3.4 m 1 m m m 2.8 m 5 For eh tringle in question 4, fin orret to one eiml le the vlue of eh ronumerl. 6 In this tringle, wht is the vlue of? Selet the orret nswer A, B, C or D. A 22.8 B 12.9 C 34.5 D Worke solutions Exerise 1-03 MAT08MGWS Fin the vlue of eh ronumerl, orret to two eiml les h z 8 Coy n omlete eh eqution þ 32 2 ¼ þ 15 2 ¼ þ 40 2 ¼ þ 2 ¼ 35 2 e 60 2 þ 2 ¼ 61 2 f 2 þ 15 2 ¼ 17 2 g 2 þ 24 2 ¼ 51 2 h 2 þ 2 ¼ 10 2 i 2 þ 2 ¼ 15 2 Tehnology Fining the hyotenuse In this tivity, we will rete sresheet to lulte the length of the hyotenuse,, of right-ngle tringle, given the lengths of the other two sies ( n )

11 Chter Enter the heings n vlues shown elow. Rememer: ^2 þ ^2 mens 2 þ 2. 2 Enter into ell C2 the formul =A2^2þB2^2, n use Fill Down to oy this formul into ells C3 to C5. 3 To lulte the length of eh hyotenuse in olumn D, we nee to fin the squre root of the vlues in olumn C, so in ell D2, enter =sqrt(c2). Then Fill Down to oy this formul into ells D3 to D5. 4 Use your sresheet to hek your nswers to questions 3 to 6 of Exerise Fining shorter sie Vieo tutoril MAT08MGVT10001 n lso e use to fin the length of one of the shorter sies in rightngle tringle if the hyotenuse n the other sie re given. In this se, we sutrt the length of the known shorter sie from the length of the hyotenuse. Tehnology Exel: Pythgors theorem MAT08MGCT00009 Summry To fin the length of shorter sie,, in rightngle tringle with hyotenuse n other shorter sie, use in the form: 2 ¼ 2 2 Exmle 7 Fin the vlue of in this tringle. 6 mm mm 10 mm

12 NEW CENTURY MATHS for the Austrlin Curriulum8 We wnt to fin the length of one of the shorter sies. Use. 2 ¼ ¼ Sustitute the hyotenuse n the other sie. ¼ 64 ¼ ffiffiffiffiffi 64 Tke the squre root to fin. ¼ 8 Chek: From the igrm, the nswer ¼ 8 looks resonle. Exmle 8 Fin the vlue of y in this tringle: s sur orret to two eiml les. 2 ¼ 2 2 y 2 ¼ ¼ 405 y ¼ ffiffiffiffiffiffiffi 405 y ¼ ffiffiffiffiffiffiffi 405 ¼ 20: (Hyotenuse) 2 (other sie) 2 Sur nswer from rt y m 43 m 38 m Chek: From the igrm, this nswer ers to e resonle. Exerise 1-04 Fining shorter sie 1 Fin the vlue of eh ronumerl. 5 mm See Exmle 7 y m 34 m x mm 13 mm 30 m 17 m 15 m y m mm 25 mm 20 mm e 24 m 30 m m f x mm 9 mm 15 mm

13 Chter See Exmle 8 2 Fin the vlue of eh ronumerl, giving your nswer in sur form. 1 m y m m km 48 km x m 3 m 10 m 4 m 125 km e 110 m f 4 m y m 7 m 326 m 12 m x m x m 6 m Worke solutions Exerise 1-04 MAT08MGWS Fin the vlue of eh ronumerl in question 2, orret to one eiml le. 4 Wht is the vlue of? Selet the orret nswer A, B, C or D. A B C D Fin the vlue of eh ronumerl, orret to two eiml les. k q km km 30 km A B 6 Wht oul e the lengths of the ffiffi other two sies of right-ngle tringle if: one of the shorter sies is 7 m long? one of the shorter sies is 7 m long? the hyotenuse is 20 m long? the hyotenuse is 20 m long? C

14 NEW CENTURY MATHS for the Austrlin Curriulum8 Tehnology Fining shorter sie In this tivity, we will rete sresheet to lulte the length of shorter sie,, of rightngle tringle, given the lengths of the hyotenuse () n the other sie (). 1 Enter the heings n vlues shown elow. 2 Enter formul into ell C2 to lulte 2 2, whih is 225, n use Fill Down to oy this formul into ells C3 to C5. 3 To lulte the length of the shorter sie () in olumn D, we nee to fin the squre root of 2 2. Enter formul into ell D2 to lulte the squre root of ell C2, whih is 15, n use Fill Down to oy this formul into ells D3 to D5. 4 Use your sresheet to hek your nswers to Exerise Mentl skills 1 Mths without lultors Squring numer ening in 5 or 1 The squre of numer ening in 5 lwys ens in 25. For exmle, 35 2 ¼ 1225 n ¼ A mentl lultion trik requires three esy stes: elete the 5 from the numer multily the remining numer y the next onseutive numer write 25 t the en of the rout. 1 Stuy eh exmle. To lulte 35 2 : eleting the 5 from 35 leves 3 multily 3 y the next onseutive numer: ¼ 12 write 25 t the en: ¼

15 Chter To lulte : eleting the 5 from 105 leves 10 multily 10 y the next onseutive numer: ¼ 110 write 25 t the en: ¼ Now lulte eh squre numer e f g 95 2 h i j 65 2 k l The squre of numer ening in 1 lwys ens in 1. For exmle, 41 2 ¼ 1681 n 71 2 ¼ A mentl lultion trik requires three stes: roun the numer own to the nerest 10 (y sutrting 1) to mke new numer squre the new numer to your nswer, the new numer n its next onseutive numer 3 Stuy eh exmle. To lulte 41 2 : roun 41 own to 40 squre 40: 40 2 ¼ n 41: 1600 þ 40 þ 41 ¼ ¼ 1681 To lulte 71 2 : roun 71 own to 70 squre 70: 70 2 ¼ þ 70 þ 71 ¼ ¼ Now lulte eh squre numer e f 81 2 g 61 2 h i j Worksheet Fining n unknown sie MAT08MGWK10007 Skillsheet MAT08MGSS Mixe rolems To fin the length of n unknown sie in right-ngle tringle, follow these stes. 1 Deie whether the unknown sie is the hyotenuse or one of the shorter sies. 2 To fin the hyotenuse, use 2 ¼ 2 þ 2 n. To fin shorter sie, use 2 ¼ 2 2 n sutrt. ffi 3 If sur nswer is require, leve the nswer in squre root ( ) form. Otherwise, lulte the vlue orret to the require numer of eiml les

16 NEW CENTURY MATHS for the Austrlin Curriulum8 Exmle 9 Fin the vlue of x s sur. x is the hyotenuse. 2 ¼ 2 þ 2 x 2 ¼ 7 2 þ 8 2 ¼ 113 x ¼ ffiffiffiffiffiffiffi m x m 7 m As sur. Puzzle sheet Pythgors 1 MAT09MGPS00039 Puzzle sheet Pythgors 2 MAT09MGPS00040 Exmle 10 Fin the vlue of n orret to one eiml le. 1.9 km n is shorter sie. 2 ¼ 2 2 n 2 ¼ 4:3 2 1:9 2 ¼ 14:88 n ¼ ffiffiffiffiffiffiffiffiffiffiffi 14:88 ¼ 3: :9 (Hyotenuse) 2 (other sie) 2 Fin the squre root. 4.3 km n km Exerise 1-05 Mixe rolems 1 Fin the vlue of eh ronumerl. 77 m 36 m f m 10.4 m 9.6 m h m 44 mm 33 mm s mm r m e 20 mm 21 mm 5.1 m 2.4 m l mm

17 Chter See Exmle 9 2 Fin the vlue of eh ronumerl s sur. 9 m x mm g m 7 m 22 mm 18 mm t mm 17 mm 32 mm See Exmle 10 3 Fin the vlue of eh ronumerl orret to one eiml le. q m 90 m m 27 m 4 km km 17 m 2.4 km 65 m 4 Wht is the vlue of y? Selet the orret nswer A, B, C or D. A B C D y km 28 km 5 Fin the vlue of eh ronumerl orret to two eiml les. 12 km 19.5 m i m 8.9 m 34.1 m y m 46 mm v mm 8.9 m 29 mm Worke solutions 6 Fin the vlue of eh ronumerl s sur. Exerise 1-05 MAT08MGWS m 5 m m 14 mm 16 mm x mm 12 m x m 12 m

18 NEW CENTURY MATHS for the Austrlin Curriulum8 e 10 km f 6 m 11 m x km 27 m m m g 2.4 m 3.3 m 8 km 13 m h 32 m i 21.5 km x m 20 m x m y km 18 km 1-06 Testing for right-ngle tringles n e use to test whether tringle is right-ngle. Summry Tehnology worksheet tester MAT08MGCT10011 If the sies of tringle hve lengths, n, where is lrgest, n they follow the formul 2 ¼ 2 þ 2, then the tringle must e right-ngle. The right ngle is oosite the longest sie,. Vieo tutoril Testing for right-ngle tringles MAT08MGVT10002 Worksheet Pythgoren tris MAT08MGWK00056 This is lle the onverse of. The wor onverse mens to turn roun or reverse. So the onverse of theorem is the theorem written k-to-front. Exmle 11 Test whether eh tringle is right-ngle. 15 m 27 m 45 m 14 m 36 m 6 m

19 Chter ¼ 2025 Squring the longest sie þ 36 2 ¼ 2025 Squring the two shorter sies, n ing. [ 45 2 ¼ 27 2 þ 36 2 The three sies follow 2 ¼ 2 þ 2. [ The tringle is right-ngle [ is the symol for therefore (with the right ngle oosite the 45 m sie) ¼ 225 Squring the longest sie. 6 2 þ 14 2 ¼ 232 Squring the two shorter sies, n ing. [ ¼ 6 2 þ 14 2 The three sies o not follow 2 ¼ 2 þ 2. [ The tringle is not right-ngle. Exerise 1-06 Testing for right-ngle tringles See Exmle 11 Worke solutions Exerise 1-06 MAT08MGWS Coy eh tringle, test whether it is right-ngle, then mrk the right ngle on the tringle e f g h i j k l

20 NEW CENTURY MATHS for the Austrlin Curriulum Pythgoren tris A Pythgoren tri or Pythgoren trile is ny grou of three numers tht follow Pythgors theorem, for exmle, (3, 4, 5) or (2.5, 6, 6.5). The wor tri mens grou of three relte items ( tri- mens 3). Summry (,, ) ispythgoren tri if 2 ¼ 2 þ 2. Any multile of (,, ) is lso Pythgoren tri. Exmle 12 Test whether (5, 12, 13) is Pythgoren tri. Worksheet Pythgoren tris MAT08MGWK10008 Tehnology worksheet Exel: Pythgoren triles MAT08MGCT00025 Tehnology Exel: Pythgoren triles MAT08MGCT ¼ 169 Squring the lrgest numer. 5 2 þ 12 2 ¼ 169 Squring the two smller numers, n ing. [ 13 2 ¼ 5 2 þ 12 2 These three numers follow. [ (5, 12, 13) is Pythgoren tri. Exmle 13 (3, 4, 5) is Pythgoren tri. Crete other Pythgoren tris y multilying (3, 4, 5) y: (3, 4, 5) ¼ (6, 8, 10) Cheking: 10 2 ¼ þ 8 2 ¼ 100 [ 10 2 ¼ 6 2 þ 8 2 [ (6, 8, 10) is Pythgoren tri. 9 3 (3, 4, 5) ¼ (27, 36, 45) Cheking: 45 2 ¼ þ 36 2 ¼ 2025 [ 45 2 ¼ 27 2 þ 36 2 [ (27, 36, 45) is Pythgoren tri

21 Chter (3, 4, 5) ¼ (1.5, 2, 2.5) 2 Cheking: ¼ þ 2 2 ¼ 6.25 [ ¼ þ 2 2 [ (1.5, 2, 2.5) is Pythgoren tri. Exerise 1-07 Pythgoren tris See Exmle 12 1 Test whether eh tri is Pythgoren tri. (8, 15, 17) (10, 24, 26) (30, 40, 50) (5, 7, 9) e (9, 40, 41) f (4, 5, 9) g (11, 60, 61) h (7, 24, 25) i (15, 114, 115) 2 Whih of the following is Pythgoren tri? Selet the orret nswer A, B, C or D. A (4, 6, 8) B (5, 10, 12) C (6, 7, 10) D (20, 48, 52) See Exmle 13 Worke solutions Exerise 1-07 MAT08MGWS Use the sresheet you rete in Tehnology: tester to hek your nswers to questions 1 n 2. 4 For eh Pythgoren tri, rete nother Pythgoren tri y multilying eh numer in the tri y: i whole numer ii frtion iii eiml. (5, 12, 13) (8, 15, 17) (30, 40, 50) (7, 24, 25) Chek tht eh nswer follows. 5 Pythgors eveloe formul for fining Pythgoren tris (,, ). If one numer in the tri is, the formuls for the other two numers re ¼ 1 2 ð 2 1Þ n ¼ 1 2 ð 2 þ 1Þ. If ¼ 5, use the formuls to fin the vlues of n. Hene show tht (,, ) is Pythgoren tri. 6 Use the formuls to fin Pythgoren tris for eh vlue of. ¼ 7 ¼ 11 ¼ 15 ¼ 4 e ¼ 9 f ¼ 19 g ¼ 10 h ¼ 51 7 There re mny other formuls for reting Pythgoren tris. Use the Internet to serh for some of them rolems Worksheet Alitions of hs mny rtil litions: from fining the igonl length of television sreen to lulting the istne shi hs sile. MAT08MGWK

22 NEW CENTURY MATHS for the Austrlin Curriulum8 Exmle 14 The size of TV sreen is esrie y the length of its igonl. If flt sreen TV is 58 m wie n 32 m high, wht is the size of its sreen? Answer to the nerest entimetre. Let the igonl length e m. 58 m 32 m Worksheet in 2D n 3D MAT08MGWK10010 Worksheet TV sreens MAT08MGWK ¼ 58 2 þ 32 2 ¼ 4388 ¼ ffiffiffiffiffiffiffiffiffiffi 4388 ¼ 66: The size of the sreen is 66 m. is the hyotenuse. Exmle 15 A tower is suorte y wire tht is 20 m long n tthe to the groun 10 m from the se of the tower. How high oes the wire reh u the tower? Answer to the nerest 0.1 m. The igrm 20 m wire Let h m e how high the wire rehes u the tower. h 2 ¼ ¼ 300 h ¼ ffiffiffiffiffiffiffi 300 ¼ 17: :3 The wire rehes 17.3 m u the tower. h is shorter sie. 10 m Exerise 1-08 rolems 1 Fin the length of this lygroun slie orret to two eiml les. See Exmle 14 2 m m 23

23 Chter A shi sils 60 kilometres south n then 70 kilometres est. How fr is it from its strting oint, orret to one eiml le? 60 km 3 Wht is the length of the le use to stilise flgole tht is 12 metres high, if the le is seure to the groun 4 m from the se of the flgole? Selet the orret nswer A, B, C or D. A 11.3 m B 12.6 m C 16 m D 80 m le 70 km 12 m See Exmle 15 4 Fin the height of n LCD TV sreen with 51 m igonl if its length is 41 m. Give your nswer orret to one eiml le. 5 Fin the length of the igonl of squre with sies of length 6 metres. Give your nswer s sur. 6 m 4 m 6 A firefighter les ler on winow sill 4.5 m ove the groun. If the foot of the ler is 1.6 m from the wll, how long is the ler, orret to one eiml le? 4.5 m Worke solutions Exerise 1-08 MAT08MGWS m 7 By first using to fin the length of the unknown sie, fin the erimeter of eh she elow, orret to two eiml les, where neessry. 9 m 65 m 16 m 13 m 12 m 25 m

24 NEW CENTURY MATHS for the Austrlin Curriulum8 3 m e 8 m 4.5 m 2.5 m 12 m 4 m 8 An equilterl tringle hs sies of 12 m. Fin its ereniulr height, h, orret to two eiml les. 12 m h m 12 m 9 A roe is tie to the to of the 6 metre wll of tent n tie to eg in the groun. The eg is 2 m from the ottom of the tent. How long is the roe, orret to two eiml les? 10 Olg hols kite string 1.2 m ove the groun. How high is the kite ove the groun, to the nerest metre? 30 m 26 m 1.2 m 11 Fin the re of eh she elow, orret to one eiml le where neessry. 16 m 20 m 41 mm 25 mm 24 mm 4 m 5 m e 11 m 9 mm 18 m 7 m 11 m 12 This sell imon is squre she of length 27 m. Wht is the istne from the home lte to seon se, to the nerest metre? 27 m

25 Chter The slnt height of one is 27 m n its vertil height is 14 m. Fin the rius of the se irle, orret to the nerest entimetre. 14 m 27 m 14 A ler 5 m long lens ginst wll, with its se 2 m from the ottom of the wll. How fr oes the ler reh u the wll, orret to nerest entimetre? r m Power lus 1 Fin the vlue of x in eh tringle, orret to one eiml le. x 2x 15 x x x x 3x 17 x e 2x 23 x f x 9 3x 2 Fin, orret to one eiml le, the length of: HD DE A 3 For this ue of sie length 15 m fin, orret to one eiml le, the length of: QS, the igonl of the se QT, the igonl of the ue B H E 15 m M C D U R F 6 m 5 m G T N S Q 15 m P 4 For this squre yrmi, fin the slnt height EF orret to one eiml le. E 18 m Chter 1 review A D 8 m B F C

26 Chter 1 review n Lnguge of mths re onverse igonl formul hyotenuse irrtionl erimeter Pythgors right-ngle shorter sie squre root 1 Who ws Pythgors n wht ountry i he ome from? 2 Desrie the hyotenuse of right-ngle tringle in two wys. 3 Wht is nother wor for theorem? 4 For wht tye of tringle is use? 5 Wht is sur? sur theorem tri unknown 6 Wht is the nme given to set of three numers tht follows? n Toi overview How relevnt o you think is to our worl? Give resons for your nswer. Give three exmles of jos tht woul use. Wht i you fin eseilly interesting out this toi? Is there ny setion of this toi tht you foun iffiult? Disuss ny rolems with your teher or frien. Puzzle sheet fin--wor MAT08MGPS10001 Worksheet Min m: Pythgors theorem MAT08MGWK10012 Coy n omlete this min m of the toi, ing etil to its rnhes n using itures, symols n olour where neee. Ask your teher to hek your work. PYTHAGORAS THEOREM Tris 3, _, 5 5, 12, _ Surs = Testing for right-ngle tringles Hy Prolems

27 Chter 1 revision See Exerise 1-01 See Exerise 1-01 See Exerise Evlute eh exression, orret to two eiml les. ffiffi ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi þ Selet ll the surs from this list of squre roots. ffiffi 25 ffiffiffiffiffiffiffi 104 ffiffiffiffiffiffiffi 96 ffiffiffiffiffi ffiffi 256 ffiffiffiffiffiffiffi 729 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3:4 2 þ 5: ffiffiffiffiffi 88 3 For eh right-ngle tringle, nme the hyotenuse n write. A B e h s C f k See Exerise Fin the vlue of eh ronumerl, giving your nswer s sur x 5 See Exerise Fin the vlue of eh ronumerl, giving your nswer orret to two eiml les. m A 12 f B C See Exerise Fin the vlue of eh ronumerl, orret to one eiml le. 6.4 m z km 100 km 9.5 m y m 33 mm v mm 105 km 27 mm

28 Chter 1 revision 7 Test whether eh tringle is right-ngle. See Exerise Test whether eh tri is Pythgoren tri. (15, 20, 25) (11, 14, 20) (20, 21, 29) (2.5, 6, 6.5) 9 Fin the length of the longest enil tht n fit insie this enil se. Answer to the nerest mm. See Exerise 1-07 See Exerise mm 180 mm 10 Fin the erimeter of eh she, orret to one eiml le. 1.1 m 20 m See Exerise m 40 m 45 m