PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:

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1 PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for right-ngled tringles 1 07 Pythgoren trids 1 08 Pythgors theorem prolems IN THIS CHAPTER YOU WILL: lulte the squre nd squre root of numer understnd wht surd is understnd nd write Pythgors theorem for right-ngled tringles use Pythgors theorem to find the length of the hypotenuse or shorter side in right-ngled tringle, giving the nswer s surd or rounded deiml use Pythgors theorem to test whether tringle is right-ngled investigte Pythgoren trids solve prolems involving Pythgors theorem Shutterstok.om/isrvut ISBN Chpter 1 Pythgors theorem 1

2 1 01 Squres, squre roots nd surds WORDBANK squred A numer multiplied y itself, for exmple, 5 2 is red 5 squred nd mens 5 5 = 25. squre root The positive vlue whih, if squred, will give tht numer; for exmple, 25 is red the squre root of 25 nd 25 = 5 euse 5 2 = 25. surd A squre root whose nswer is not n ext numer. For exmple, 8 = is surd euse there isn t n ext numer squred tht is equl to 8. As deiml, the digits of 8 run endlessly without ny repeting pttern. EXAMPLE 1 Evlute eh expression ( 13) = 36 Enter 6 = or 6 6 = on the lultor = ( 13) 2 = 169 Enter ( ( ) 13 ) = on the lultor EXAMPLE 2 Evlute eh expression, orret to 2 deiml ples if neessry d = 7 Enter 49 = on the lultor 5 = Enter 5 = on the lultor = 1.6 d 82 = EXAMPLE 3 Whih numers in Exmple 2 re surds? The surds re 5 nd 82 euse they do not simplify to ext deimls. 2 Developmentl Mthemtis Book 3 ISBN

3 exerise Wht is the mening of 8 2? Selet the orret nswer A, B, C or D. A 8 2 B C 8 8 D Evlute eh expression d 14 2 e f 9 2 g h i ( 4) 2 j k ( 8) 2 l Evlute eh expression d 121 e Wht is the most urte nswer for the vlue of 58? Selet A, B, C or D. A 7 B C 8 D Evlute eh expression orret to 2 deiml ples d 150 e Whih one of the numers elow is surd? Selet A, B, C or D. A 9 B 7 2 C 2 3 D 13 7 Is eh sttement true or flse? 10 2 = = = 11 d 169 = 13 2 e ( 9) = 81 8 Find the re of this squre. 5 m 9 If the re of squre is 49 m 2, wht is the length of side of the squre? 10 Sketh squre with n re of 64 m 2, showing the side length of the squre. ISBN Chpter 1 Pythgors theorem 3

4 1 02 Pythgors theorem WORDBANK right-ngled tringle A tringle with one ngle extly 90º. This ngle is lled the right ngle. hypotenuse hypotenuse The longest side of right-ngled tringle, the side opposite the right ngle. right ngle Pythgors theorem The rule or formul 2 = tht reltes the lengths of the sides of right-ngled tringle. Pythgors ws the nient Greek mthemtiin who disovered this rule ( theorem mens rule). PYTHAGORAS THEOREM In ny right-ngled tringle, the squre of the hypotenuse is equl to the sum of the squres of the other two sides. In the digrm, is the length of the hypotenuse (longest side), nd Pythgors theorem is 2 = EXAMPLE 4 Stte Pythgors theorem for eh right-ngled tringle elow. x p r z y q r 2 = p 2 + q 2 z 2 = x 2 + y 2 Rememer, Pythgors theorem egins with (hypotenuse) 2, nd r is the hypotenuse here. EXAMPLE 5 Test Pythgors theorem on eh tringle elow. 12 m 4 m 5 m 5 m 15 m 3 m 4 For Pythgors theorem to e true, hypotenuse 2 = sum of the squres of the other two sides. Does 5 2 = ? 25 = True, so Pythgors theorem is true. This mens tht the tringle is right-ngled. Does 15 2 = ? 225 = Flse, so Pythgors theorem is not true. This mens tht the tringle is not right-ngled. Developmentl Mthemtis Book 3 ISBN

5 exerise Whih side of this right-ngled tringle is the hypotenuse? Selet the orret nswer A, B or C. A B C 2 Wht is Pythgors theorem for the tringle in question 1? Selet A, B or C. A 2 = B 2 = C 2 = Write Pythgors theorem for eh right-ngled tringle. n p q m i r d e y f e d z x r t s f 4 Test Pythgors theorem on eh tringle elow. 13 m 5 m 16 m 9 m 12 m 8 m 15 m 17 m 12 m 5 Whih tringles in question 4 re right-ngled? ISBN Chpter 1 Pythgors theorem 5

6 1 03 WORDBANK Finding the hypotenuse ext form When n nswer is written s n ext numer, suh s whole numer, deiml or surd, nd not rounded. To find the length of the hypotenuse in right-ngled tringle: write down Pythgors theorem in the form 2 = where is the length of the hypotenuse solve the eqution hek tht your nswer is the longest side EXAMPLE 6 Find the length of the hypotenuse in eh tringle elow, writing your nswer in ext form. 4 m 5 m 12 m 7 m p 2 = = = p 2 = p is the hypotenuse. = 169 = 65 = 169 p = 65 m This is in ext surd form. = 13 m This is in ext form. 65 does not simplify to n ext deiml From the digrm, hypotenuse of length 13 m looks resonle. It is lso the longest side. EXAMPLE 7 Find d orret to one deiml ple m 8.25 m d 2 = d 2 = = d = = m Rounded to one deiml ple. Shutterstok.om/vihie81 6 Developmentl Mthemtis Book 3 ISBN

7 exerise Drw right-ngled tringle nd mrk the hypotenuse. 2 Copy nd omplete for the tringle shown 2 = m 2 = = = = 8 m 3 Find the length of the hypotenuse in eh tringle elow. Answer in ext form. 10 m 9 m 8 m 24 m m 12 m z 15 m d e y f 16 m 30 m 1 m 2 m 1 m 1 m x 4 Round your nswers to question 3 e nd f to one deiml ple. 5 Find the length of the hypotenuse in eh tringle elow. Answer orret to one deiml ple. 5.2 m 9.6 m 12.4 m 6.3 m 7.8 m 22.6 m Shutterstok.om/JK Photo ISBN Chpter 1 Pythgors theorem 7

8 1 04 Finding shorter side To find the length of shorter side in right-ngled tringle: write down Pythgors theorem in the form 2 = where is the length of the hypotenuse rerrnge the eqution so tht the shorter side is on the LHS (left-hnd side) solve the eqution hek tht your nswer is shorter thn the hypotenuse EXAMPLE 8 Find the length of the unknown side in eh tringle elow. Answer in ext form. 9 m 15 m 22 m 32 m h 15 2 = h is the hypotenuse = h = 15 2 Rerrnging the eqution = 32 2 h 2 = so tht h is on the LHS. 2 = = 144 = 540 h = 144 = 540 m = 12 m From the digrm, length of 12 m looks resonle. It is lso shorter thn the hypotenuse, 15 m. EXAMPLE 9 Find the length of the unknown side in eh tringle elow. Answer orret to one deiml ple. 28 m h 36 m 5.4 m 9.2 m x 36 2 = h = x h = 36 2 x = h 2 = x 2 = = 512 = h = 512 x = = = m Rounded to 1 deiml ple. 7.4 m 8 Developmentl Mthemtis Book 3 ISBN

9 exerise To find the length of the shorter side,, in the tringle elow, whih rule is esier to use? Selet the orret nswer A, B or C A 25 2 = B 25 2 = C 2 = Wht is the length of the hypotenuse in the tringle ove? A 15 B 25 C None of these 3 Copy nd omplete to find = = = = 400 = = 4 Find the vlue of eh pronumerl. Leve your nswers in ext form. 25 m m 26 m 10 m t d 24 m 12 m 25 m x 13 m 5 Find, orret to one deiml ple, the vlue of eh pronumerl. 9 m 18 m 3 m n 12 m d 3.2 m 10.5 m h n 12.2 m 6.4 m ISBN Chpter 1 Pythgors theorem 9

10 1 05 Mixed prolems For this right-ngled tringle: to find the length of the hypotenuse, use 2 = where is the hypotenuse to find the length of one of the shorter sides, use the shortut 2 = 2 2 to find side or 2 = 2 2 to find side. EXAMPLE 10 Find the length of the unknown side in eh tringle elow. Leve your nswer in ext form. 7 m 15 m 8 m 17 m x 24 m t 32 m d d is shorter side Squre nd sutrt x is the hypotenuse Squre nd dd t is shorter side Squre nd sutrt d 2 = = = 225 d = 225 = 15 m the lrgest side (hypotenuse) goes first x 2 = = 625 x = 625 = 25 m t 2 = = 799 t = 799 m Shutterstok.om/Angelin Dimitrov 10 Developmentl Mthemtis Book 3 ISBN

11 exerise To find the length of the hypotenuse, h, in this tringle, whih rule is esier to use? Selet the orret nswer A, B, C or D. A h 2 = i 2 + j 2 h i B h 2 = i 2 j 2 C j 2 = h 2 i 2 j D i 2 = h 2 + j 2 2 To find the length of the shorter side, p, in this tringle, whih rule is orret? A p 2 = q 2 + r 2 B p 2 = r 2 q 2 p r C q 2 = r 2 p 2 D r 2 = p 2 + q 2 q 3 Find the length of the unknown side in eh tringle. Leve your nswers in ext form. 16 m 6 m 10 m 14 m 22 m 9 m y x m d e f n 9 m 42 m 17 m 40 m 52 m 18 m 4 Find, orret to one deiml ple, the vlue of eh pronumerl. m p 8.2 m 12.6 m 15.9 m 4.9 m 11.5 m 17.3 m n x 5 A ldder is pled ginst uilding to reh window on the seond floor. Wht is the nme for the side of the tringle where the ldder is positioned? Window Ldder ISBN Chpter 1 Pythgors theorem 11

12 1 06 Testing for right-ngled tringles If the sides of tringle follow the rule 2 = 2 + 2, then the tringle must e right-ngled. This is lled the onverse of Pythgors theorem, the theorem used in reverse. To prove tht tringle is right-ngled: sustitute the lengths of its sides into the rule 2 = if it is true, then the tringle is right-ngled if it is flse, then the tringle is not right-ngled EXAMPLE 11 Test whether eh tringle is right-ngled Sustitute the lengths of the sides into the rule 2 = Does 25 2 = ? 625 = = 625 Yes This tringle is right-ngled. The right ngle is opposite the hypotenuse, 25, etween the sides mrked 15 nd 20. Does 15 2 = ? 225 = No This tringle is not right-ngled. Almy/Mike Pwley 12 Developmentl Mthemtis Book 3 ISBN

13 exerise Explin in words how you n prove tht tringle is right-ngled. 2 In the tringle elow, whih ngle is the right ngle? Selet the orret nswer A, B or C. A A B B C C A 3 5 C B 4 3 Wht is Pythgors theorem for the tringle ove? A 4 2 = B 3 2 = C 5 2 = Copy nd omplete to test if this tringle is right-ngled. Does 17 2 = ? 289 = = So the tringle right-ngled. 5 Test whether eh tringle is right-ngled. 10 m P R P 20 m 17 R m 40 m 25 m 15 m Q Q P d P 2 m R 15 m 22 m 1.6 m 1.2 m Q Q 16 m R e 6.1 m R 1.1 m f P 8 m Q 74 m 22 m R P 6 m Q 6 For eh right-ngled tringle in question 5, stte whih ngle is the right ngle. ISBN Chpter 1 Pythgors theorem 13

14 1 07 Pythgoren trids WORDBANK Pythgoren trid A set of 3 numers tht follows Pythgors theorem, 2 = To prove tht group of 3 numers is Pythgoren trid: sustitute the numers into the rule 2 = where is the lrgest numer if it is true, then the numers form Pythgoren trid if it is flse, then the numers do not form Pythgoren trid EXAMPLE 12 Test whether eh set of numers form Pythgoren trid. {15, 20, 25} {8, 12, 13} Does 25 2 = ? Does 13 2 = ? Alwys sustitute the lrgest numer for. 625 = = = 625 true flse So {15, 20, 25} is Pythgoren trid So {8, 12, 13} is not Pythgoren trid. If the three numers in Pythgoren trid re multiplied y the sme vlue, then the new numers lso form Pythgoren trid. EXAMPLE 13 By multiplying eh numer of the Pythgoren trid {7, 24, 25} y numer of your hoie, find two more trids. {7 2, 24 2, 25 2} = {14, 48, 50} Multiplying {7, 24, 25} y 2. Chek tht 50 2 = {7 5, 24 5, 25 5} = {35, 120, 125} Multiplying {7, 24, 25} y Shutterstok.om/gui jun peng Developmentl Mthemtis Book 3 ISBN

15 exerise Copy nd omplete: A Pythgoren trid is set of numers whih follow theorem. 2 To prove tht set of numers form Pythgoren trid, whih vrile in the formul 2 = should e sustituted y the lrgest numer? Selet the orret nswer A, B or C. A B C 3 Whih set of numers is Pythgoren trid? Selet A, B, C or D. A {3, 4, 4} B {3, 4, 5} C {3, 4, 6} D {3, 4, 7} 4 Copy nd omplete: 13 2 = = = So {5, 12, 13} Pythgoren trid. 5 Test whether eh set of numers form Pythgoren trid. {9, 40, 41} {7, 20, 25} {12, 16, 20} d {11, 50, 52} e {5, 6, 7} f {7, 24, 25} g { 8, 15, 17} h {4, 8, 12} 6 If {5, 12, 13} is Pythgoren trid, find two more Pythgoren trids using multiples of {5, 12, 13}. 7 Prove tht {16, 30, 34} is Pythgoren trid. Whih Pythgoren trid is this multiple of? 8 Write down 2 more Pythgoren trids using {7, 24, 25}. 9 Is eh sttement true or flse? {30, 40, 50} is Pythgoren trid. {9, 12, 15} is multiple of {3, 4, 5}. There re only 3 Pythgoren trids. 10 Drw right-ngled tringle with sides 6 m, 8 m nd 10 m. Is {6, 8, 10} Pythgoren trid? Almy/Dinodi Photos ISBN Chpter 1 Pythgors theorem 15

16 1 08 Pythgors theorem prolems To solve prolem using Pythgors theorem: drw digrm if it is not given nd drw right-ngled tringle identify the unknown vlue use 2 = to solve n eqution nswer the prolem in words. EXAMPLE 14 Toy is flying kite t height of 32 m ove the ground. He is stnding on the ground 24 m wy from the ground level of the kite. How long is the piee of string tht he is using to fly the kite? 32 m 2 = = = 1600 = 1600 = 40 The length of the string is 40 m. 24 m is the length of the string nd is the hypotenuse of the tringle. Sustitute in the numers given in the question. EXAMPLE 15 A window in uilding is 6 m ove the ground. A ldder is pled 9 m from the se of the uilding so tht it rehes the window. How long is the ldder (orret to 1 deiml ple)? Let the length of the ldder e x m. x 2 = = 117 x = 117 = m The ldder is 10.8 m long. 6 m 9 m x exerise Gemm lens ldder ginst 6 m high wll so tht it rehes the top of it. She ples the ldder 2.5 m from the se of the wll. Whih is the orret digrm for the prolem elow? Selet A, B or C. A B C 6 m 2.5 m 6 m 2.5 m 6 m 2.5 m 16 Developmentl Mthemtis Book 3 ISBN

17 exerise Find the length of the ldder for the prolem desried in question 1. 3 Find the length of the digonl in this retngle. 8 m 4 In this retngle, the digonl is 13 m long nd one side mesures 12 m. Wht is the length of the other side? 13 m 15 m 12 m 5 Wht length of wood is needed to mke the ross-r of this grden gte? 1.5 m 2 m 6 A retngulr plyground is 95 m from one orner, ross the plyground, to the opposite orner. Its longest side mesures 73 m. Drw sketh to represent the plyground. Find the length of the shorter side (orret to the nerest metre). 7 A 3-metre ldder lens ginst wll. The foot of the ldder is 1.5 metres from the wll. How high up the wll does the ldder reh? Answer orret to one deiml ple.? 3 m 1.5 m 8 An empty lok of lnd mesures 12 m y 20 m. Wht is the shortest distne from A to C, to the nerest metre? A 12 m B 9 For the lok of lnd in question 8, how muh frther is it to wlk from A to B then B to C, rther thn from A to C? D 20 m C 10 Tim ws strnded on n islnd 3.5 km est of lighthouse. He knew there ws town on the minlnd 12.8 km south of the lighthouse. How fr (orret to one deiml ple) would he hve to swim to reh the town if he swm in stright line from where he ws? Minlnd Lighthouse Tim Islnd Town ISBN Chpter 1 Pythgors theorem 17

18 LANGUAGE ACTIVITY FIND-A-WORD PUZZLE Mke opy of this pge, then find ll the words listed elow in this grid of letters. A O J S R O O T W Y P R T G B K J E Z G W A B B A W J M X E B X M D A I H S S T F P H V Q R Q M T X D J A F Y U N U V A C B D R U E O C O N V E R S E H X N D G A F Z U N M X G B W G Q Y H D S V E G L K F F I Q C B U A S V L G R E U F T I V A E L R D Y Y D X H M A S Y Q G L O U I W T P V Q Q P L A K T Y A D E Q O P O S N G W B B E K E N I D Y R Q V N V Y R T W C M U S Z L Q J E R G P H O I X H P Y K Q Q L O G S B A D A T Z J R Z U T L F C C Q E N Y Q A E S W F Y T P U B Y P S O R S A U V Q M T G O K D X H E K S R Z F G I D Y U O M S M R L J N D E W K O J B R V D K F Q Q V F Q D G I F V O I B U T Q Z C Z S U E L G N A S A G W L R L Y O I H B J A Q I C E P L Z U T J D G E C A E L D R G P Y H C K Z L E R Q W E M M D Q O E F C X J B U L C A N K D D N P V B G A W L E I W U I U G U A R C A I B I ANGLE CONVERSE DRAW HYPOTENUSE PROBLEM PROVE PYTHAGORAS ROOT SIDE SQUARE SURD THEOREM TRIAD TRIANGLE 18 Developmentl Mthemtis Book 3 ISBN

19 PRACTICE TEST 1 Prt A Generl topis Clultors re not llowed. 1 Write 2108 in 12-hour time. 2 Complete: 60.8 m = mm 3 Evlute Evlute Expnd 2(x 4). 7 Find 2 of $ Find the perimeter of this shpe. 12 m 10 m 4 m 8 Find the men of 1, 2, 8, 3, 6. 9 How mny fes hs tringulr prism? 10 Kthy pys groery ill of $83.45 with $100 note. Clulte the hnge. Prt B Pythgors theorem Clultors re llowed Squres, squre roots nd surds 11 Evlute eh expression, orret to two deiml ples Selet ll the surds from this list of squre roots Pythgors theorem 13 Nme the hypotenuse nd write Pythgors theorem for this tringle. h j k 1 03 Finding the hypotenuse 14 Find the vlue of eh pronumerl, giving your nswer s surd. 13 m 5 m 9 m r 6 m ISBN Chpter 1 Pythgors theorem 19

20 PRACTICE TEST Finding shorter side 15 Find the vlue of eh pronumerl, giving your nswer orret to two deiml ples. n 4 m 7 m 11.3 m 4.6 m 1 05 Mixed prolems x 16 Find the vlue of eh pronumerl, orret to one deiml ple. 8 km 12.4 m 6.2 m 112 m 208 m 22 km p s w 1 06 Testing for right-ngled tringles 17 Test whether eh tringle is right-ngled Pythgoren trids Test whether eh set of numers is Pythgoren trid. {1.8, 2.4, 3.0} {7, 24, 26} 1 08 Pythgors theorem prolems 19 Find, orret to one deiml ple, the length of the longest umrell tht n fit inside suitse mesuring 60 m long nd 46 m wide. 46 m 60 m 20 Find the perimeter of this trpezium. 8 m 12 m 24 m 20 Developmentl Mthemtis Book 3 ISBN