mesurement nd geometry topic 15 Pythgors theorem 15.1 Overview Why lern this? Pythgors ws fmous mthemtiin who lived out 2500 yers go. He is redited with eing the fi rst person to prove tht in ny rightngled tringle there is speil reltionship etween the squres of the three sides. The theorem hs n importnt role to ply in everydy life euse right-ngled tringles our in onstrution, nvigtion, plnning, design nd pkging. Pythgors theorem is one of the gret geometril theorems. Wht do you know? 1 think List wht you know out Pythgors theorem. Use thinking tool suh s onept mp to show your list. 2 Pir Shre wht you know with prtner nd then with smll group. 3 shre As lss, rete thinking tool suh s lrge onept mp tht shows your lss s knowledge of Pythgors theorem. Lerning sequene 15.1 Overview 15.2 Right-ngled tringles 15.3 Finding the hypotenuse 15.4 Finding shorter side 15.5 Working with different units 15.6 Composite shpes 15.7 Pythgoren trids 15.8 Pythgors in 3-D 15.9 Review ONLINE ONLY 530 Mths Quest 8
mesurement nd geometry reflection How will you rememer the mening of the pronumerls, nd in Pythgors theorem? do-11425 15.2 Right-ngled tringles This topi investigtes one of the most importnt ides in geometry relted to right-ngled tringles. Tringles n e lssified ording to the length of their sides. They n e lssified s either equilterl, isoseles or slene. Tringles n e lssified y their ngles s well. A tringle with one right ngle is sid to e right-ngled tringle. Types of tringles Isoseles Equilterl Slene Right-ngled In ny tringle, the longest side is opposite the lrgest ngle; so in right-ngled tringle, the longest side is opposite the right ngle. This side hs speil nme. It is lled the hypotenuse. The theorem, or rule, tht you will lern more out ws nmed fter n nient Greek mthemtiin, lled Pythgors (580 501 BC). It will enle you to solve ll kinds of prtil prolems relted to right-ngled tringles. Eerise 15.2 Right-ngled tringles individul PtHWys PrCtise 1 6, 8, 10 ConsoliDte 1 5, 7, 8, 10, 11 mster 1 5, 7 11 FluenCy 1 For eh of the following tringles, the lengths of the sides re reorded in the tle tht follows. Note tht the hypotenuse is lwys mrked s. Complete the tle. do-11426sample EVALUATION do-11426 Individul pthwy intertivity int-4469 Hypotenuse ONLY 532 Mths Quest 8
mesurement AND geometry d e f g h Lengths (mm) d e f g h 19 24 27 16 31 41 13 24 35 32 30 40 25 19 45 24 40 40 40 43 40 45 47 34 2 2 2 + 2 2 2 Wht do you notie out the results in the tle in question 1? 3 On sheet of pper, refully drw 6 right-ngled tringles of different sizes. You will need to use protrtor, set squre or templte to mke sure the tringles re rightngled. Crefully mesure the sides of eh tringle, nd omplete tle like the one in question 1. Wht do you notie out these results? 4 Now drw some tringles tht re not right-ngled, mesure their sides nd omplete the sme kind of tle. Rememer to lel the longest side s. Wht do you notie this time? 5 Eplin why it is not possile to hve tringle with two right ngles in it. 6 For this tivity you will need grph pper, oloured penils, glue nd sissors. On sheet of grph pper, drw right-ngled tringle with se of 4 m nd height of 3 m. Crefully drw squre on eh of the three sides of the tringle nd mrk grid on eh so tht the squre on the se is divided into 16 smll squres, while the squre on the height is divided into 9 smll squres. Topi 15 Pythgors theorem 533
mesurement AND geometry Colour the squre on the se, nd the squre on the height in different olours s shown t right so tht you n still see the grid lines. d Crefully ut out the two oloured squres from the tringle. e Now stik the lrger of the oloured squres on the unoloured squre of the tringle (the squre on the hypotenuse). f Using the grid lines s guide, see if you n ut the smller squre up nd fit it on the remining spe. The two oloured squres should hve etly overed the third squre. g Wht do you notie out the hypotenuse nd the other two sides of right-ngled tringle? Understnding 7 A right-ngled tringle is speil tringle, ut it hs some properties tht re ommon to ll tringles. One of these ommon properties is tht the sum of the lengths of ny two sides must e greter thn the length of the third side. Using ruler nd ompss, ttempt to onstrut tringle whose side lengths re 7 m, 4 m nd 12 m. Wht do you find? Resoning 8 Wht is the sum of the two smller ngles in right-ngled tringle? Justify your nswer. 9 If the two smller ngles in right-ngled tringle re equl in size, mke sttement out the tringle. Wht size is eh of the smller ngles? Prolem solving 10 Cn you drw n equilterl tringle tht hs right ngle? Eplin. 11 Drw four tringles with 30 ngle like the one shown t right, ensuring tht the length of is different for eh. For eh tringle you hve drwn in prt, find the vlue of: i ii iii Wht do you notie out your nswers to prt? d Repet questions to for nother four right-ngled tringles using n ngle other thn 30. 15.3 Finding the hypotenuse Pythgors ws perhps the first mthemtiin to reognise nd investigte very importnt property of right-ngled tringles. The theorem, or rule, nmed fter him sttes tht: In ny right-ngled tringle, the squre of the hypotenuse is equl to the sum of the squres of the other two sides. The rule is written s 2 = 2 + 2 where nd re the two shorter sides nd is the hypotenuse. 4 m 30 3 m 534 Mths Quest 8
mesurement nd geometry The hypotenuse is the longest side of right-ngled tringle nd is lwys the side tht is opposite the right ngle. Pythgors theorem gives us wy of finding the length of the third side in tringle, if we know the lengths of the two other sides. We re le to find the length of the hypotenuse when we re given 4 the length of the two shorter sides y sustituting into the formul 2 = 2 + 2 to lulte the hypotenuse. 7 Clultions with Pythgors theorem often result in numer under the root sign tht is not squre numer. In suh se nswers my e left unsimplified, nd this is lled et (surd) form. A lultor my e used to find n pproimte nswer, usully given to speified numer of deiml ples. WorKeD exmple 1 For the tringle t right, find the length of the hypotenuse,, orret to 1 deiml ple. think 1 Copy the digrm nd lel the sides, nd. Rememer to lel the hypotenuse s. Write/DrW = 4 2 Write Pythgors theorem. 2 = 2 + 2 3 Sustitute the vlues of, nd into this rule nd simplify. 2 = 4 2 + 7 2 = 16 + 49 = 65 4 Clulte y tking the squre root of 65. As 65 is not squre numer, lultor n e used to find n pproimte vlue for. Round your nswer orret to 1 deiml ple. The symol mens is pproimtely equl to. An lterntive symol,, n lso e used. In mny ses we re le to use Pythgors theorem to solve prtil prolems. We n model the prolem y drwing digrm, nd use Pythgors theorem to solve the right-ngled tringle. We then use the result to give worded nswer. = 7 = = 65 This is the et (surd) form. 8.1 4 7 Topi 15 Pythgors theorem 535
mesurement nd geometry WorKeD exmple 2 A fire is on the twelfth floor of uilding. A hild needs to e resued from window tht is 20 metres ove ground level. If the resue ldder n e pled no loser thn 6 m from the foot of the uilding, wht is the minimum length ldder needed to mke the resue? Leve your nswer in et form. think 1 Drw digrm nd lel the sides, nd. Rememer to lel the hypotenuse s. Write/DrW 20 m 6 m 2 Write Pythgors theorem. 2 = 2 + 2 3 Sustitute the vlues of, nd into this rule nd simplify. 2 = 20 2 + 6 2 = 400 + 36 = 436 4 Clulte y tking the squre root of 436. = 436 5 Give worded nswer. The ldder needs to e 436 metres long. reflection How do you know whih side is the hypotenuse? do-11427 Eerise 15.3 Finding the hypotenuse individul PtHWys PrCtise 1 5, 9 ConsoliDte 1 6, 9, 10 mster 1 11 FluenCy 1 WE1 For the following tringles, find the length of the hypotenuse,, orret to 1 deiml ple where neessry. 3 5 7 24 4 12 do-11428sample EVALUATION do-11428 d 17.5 12.2 ONLY Individul pthwy intertivity int-4470 e 742 896 f 6.2 1.3 536 Mths Quest 8
mesurement AND geometry 2 For eh of the following tringles, find the length of the hypotenuse. Leve your nswers in et form. 52 63 d 34 125 84 85 e 80 60 UNDERSTANDING 3 A right-ngled tringle hs se of 5 m nd perpendiulr height of 11 m. Find the length of the hypotenuse. Leve your nswers in et form. 4 Find the lengths, orret to 2 deiml ples, of the digonls of squres tht hve side lengths of: 12 m 20 mm 4.9 m 5 Wht is the length, orret to 2 deiml ples, of the digonl of retngle whose sides re: 10 m nd 8 m? 620 m nd 400 m? 17 m nd 3 m? 6 An isoseles tringle hs se of 30 m nd perpendiulr height of 10 m. Find the length of the two equl sides of the isoseles tringle. Give your nswer orret to 2 deiml ples. resoning 7 A right-ngled tringle hs perpendiulr height of 17.2 m, nd se tht is hlf the height. Find the length of the hypotenuse, orret to 2 deiml ples. 8 WE2 A ldder lens ginst vertil wll. The foot of the ldder is 1.2 m from the wll, nd the top of the ldder rehes 4.5 m up the wll. How long is the ldder? Give your nswer orret to 2 deiml ples. 9 Wlly is instlling wtering system in his grden. The pipe is to go ll round the edge of the retngulr grden, nd hve rnh digonlly ross the grden. The grden mesures 5 m y 7.2 m. If the pipe osts $2.40 per metre (or prt thereof), wht will e the totl ost of the pipe? Show your working. prolem solving 10 Clulte the volume of the prism. Find the length of the sloping edge of the ross-setion. 11 A ldder rests ginst vertil wll with the top of the ldder eing 7 m ove the ground. If the ottom of the ldder ws moved 1 m further wy from the foot of the wll, the top of the ldder would rest ginst the foot of the wll. Wht is the length of the ldder? f 13 2 15 8 6 m 7 m 9 m Topi 15 Pythgors theorem 537
mesurement nd geometry 12 A right-ngled isoseles tringle hs n re of 200 m 2. Find the re of the semiirle whih sits on the hypotenuse. 15.4 Finding shorter side Sometimes question will give you the length of the hypotenuse nd sk you to find one of the shorter sides. In suh emples, we need to rerrnge Pythgors formul. Given tht 2 = 2 + 2, we n rewrite this s: 2 = 2 2 or 2 = 2 2. WorKeD exmple 3 Find the length of the unmrked side of the tringle t right. Leve your nswer in et form. think 1 Copy the digrm nd lel the sides, nd. Rememer to lel the hypotenuse s. 2 Write Pythgors theorem for shorter side. 3 Sustitute the vlues of, nd into this rule nd simplify. 4 Find y tking the squre root of 63. Write/DrW = 9 2 = 2 2 = 12 2 = 12 2 9 2 = 144 81 = 63 = 63 m 9 m 12 m Prtil prolems my lso involve you eing required to find the shorter side of right-ngled tringle. 538 Mths Quest 8
mesurement nd geometry WorKeD exmple 4 A ldder tht is 4.5 m long lens up ginst vertil wll. The foot of the ldder is 1.2 m from the wll. How fr up the wll does the ldder reh? Give your nswer orret to 2 deiml ples. think 1 Drw digrm nd lel the sides, nd. Rememer to lel the hypotenuse s. Write/DrW = 4.5 m = 1.2 m 2 Write Pythgors theorem for shorter side. 2 = 2 2 3 Sustitute the vlues of, nd into this rule nd simplify. 4 Find y tking the squre root of 18.81. Round to 2 deiml ples. 2 = 4.5 2 1.2 2 = 20.25 1.44 = 18.81 = 18.81 4.34 m 5 Give written nswer. The ldder will reh height of 4.34 m up the wll. Some questions will require you to deide whih method is needed to solve the prolem. A digrm will help you deide whether you re finding the hypotenuse or one of the shorter sides. Eerise 15.4 Finding shorter side individul PtHWys PrCtise 1 6, 8, 10, 14 ConsoliDte 1 10, 13, 14 mster 1 15 FluenCy 1 WE3 Find the length of the unmrked side in eh of the following tringles. Leve your nswer in et form. 14 32 10 8 Individul pthwy intertivity int-4471 17 84 reflection Wht should you look out for when using Pythgors theorem to lulte the vlue of missing sides? do-11429 Topi 15 Pythgors theorem 539
mesurement AND geometry d 382 e 1 f 8 457 4 12 2 Find the vlue of the pronumerl, orret to 2 deiml ples. d 0.28 47.2 0.67 d 30.1 2.56 1.98 e 2870 e 1920 UNDERSTANDING For questions 3 to 13, give your nswers orret to 2 deiml ples. 3 The digonl of the retngulr sign t right is 34 m. If the height of this sign is 25 m, find the width. 4 The digonl of retngle is 120 m. One side hs length of 70 m. Find: the length of the other side the perimeter of the retngle the re of the retngle. 5 An equilterl tringle hs sides of length 20 m. Find the height of the tringle. 6 The roundout sign shown t right is in the form of n equilterl tringle. Find the height of the sign nd, hene, find its re. 7 WE4 A ldder tht is 7 metres long lens up ginst vertil wll. The top of the ldder rehes 6.5 m up the wll. How fr from the wll is the foot of the ldder? f 8.4 468 114 17.52 f 76 m 8 A tent pole tht is 1.5 m high is to e supported y ropes tthed to the top. Eh rope is 2 m long. How fr from the se of the pole n eh rope e pegged? 540 Mths Quest 8
mesurement AND geometry 9 Ben s dog M hs wndered onto frozen pond, nd is too frightened to wlk k. Ben estimtes tht the dog is 3.5 m from the edge of the pond. He finds plnk, 4 m long, nd thinks he n use it to resue M. The pond is surrounded y nk tht is 1 m high. Ben uses the plnk to mke rmp for M to wlk up. Will he e le to resue his dog? REASONING 10 A kite is tthed to string 15 m long. Sm holds the end of the string 1 m ove the ground, nd the horizontl distne of the kite from Sm is 8 m s shown t right. How fr ove the ground is the kite? Show your working. 11 Penny, the rpenter, is uilding roof for new house. The roof hs gle end in the form of n isoseles tringle, with se of 6 m nd sloping sides of 7.5 m. She deides to put 5 evenly sped vertil strips of wood s deortion on the gle s shown elow. How mny metres of this deortive wood does she need? Show your working. 7.5 m 6 m 7.5 m 12 The size of retngulr television sreen is given y the length of its digonl. The television elow hs 92-m sreen. Wht is the length of the shorter side? Show your working. 80 m prolem solving 13 An rt student is trying to hng her newest pinting on n eisting hook in wll. She lens 1.2-m ldder ginst the wll so tht the distne etween the foot of the ldder nd the wll is 80 m. Drw sketh showing the ldder lening on the wll. How fr up the wll does the ldder reh? The student lims the ldder to hek whether she n reh the hook from the step t the very top of the ldder. One she etends her rm, the distne from her feet to her fingertips is 1.7 m. If the hook is 2.5 m ove the floor, will the student reh it from the top step? 15 m 1 m 8 m Topi 15 Pythgors theorem 541
mesurement nd geometry do-11431 14 A retngle hs two sides of length 3 m nd digonls of length 5 m. Clulte the length of the other two sides. It is known tht the digonls of this retngle iset eh other t the point of intersetion. Using this ft, hek (using pproprite lultions) whether the digonls re perpendiulr to eh other. 15.5 Working with different units When we use Pythgors theorem, we re usully working with prtil sitution where mesurements hve een given. In ny lultion, it is essentil tht ll of the mesurements re in the sme units (for emple, m). Do you rememer the reltionship etween the units of length? Here is reminder: 10 mm = 1 m To hnge to lrger unit, divide y the onversion ftor. 100 m = 1 m To hnge to smller unit, multiply y the onversion ftor. 1000 m = 1 km For emple, to hnge kilometres to metres, multiply y 1000. To hnge entimetres to metres, divide y 100. The following hrt shows ll the onversions of length tht you re likely to need. kilometres (km) 1000 1000 metres (m) 100 100 entimetres (m) 10 10 millimetres (mm) When using Pythgors theorem, lwys hek the units given for eh mesurement. If neessry, onvert ll mesurements to the sme units efore using the rule. WorKeD exmple 5 Find the length, in mm, of the hypotenuse of right-ngled tringle if the 2 shorter sides re 7 m nd 12 m. Give your nswer orret to 1 deiml ple. think 1 Drw digrm nd lel the sides, nd. Rememer to lel the hypotenuse s. 2 Chek tht ll mesurements re in the sme units. They re. Write/DrW = 12 m 3 Write Pythgors theorem for the hypotenuse. 2 = 2 + 2 4 Sustitute the vlues of nd into this rule nd simplify. 2 = 12 2 + 7 2 = 144 + 49 = 193 = 7 m 542 Mths Quest 8
mesurement nd geometry 5 Find y tking the squre root. Give the units in the nswer. 6 Chek the units required in the nswer nd onvert if neessry. = 193 13.89 m = 13.89 10 mm = 138.9 mm WorKeD exmple 6 The hypotenuse nd one other side of right-ngled tringle re 450 m nd 3.4 m respetively. Find the length of the third side, in m, orret to the nerest whole numer. think 1 Drw digrm nd lel the sides, nd. Rememer to lel the hypotenuse s. 2 Chek tht ll mesurements re in the sme units. They re different, so onvert 3.4 m into m. Write/DrW = 450 m = 3.4 m 3.4 m = 3.4 100 m = 340 m 3 Write Pythgors theorem for shorter side. 2 = 2 2 4 Sustitute the vlues of nd into this rule nd simplify. 5 Find y tking the squre root of 86 900. Round to the nerest whole numer. 2 = 450 2 340 2 = 202 500 115 600 = 86 900 = 86900 295 6 Give worded nswer. The third side will e pproimtely 295 m long. Eerise 15.5 Working with different units individul PtHWys PrCtise 1 8, 10, 11, 14, 17 ConsoliDte 1, 2 e, 3 e, 4 d, 5 12, 14, 17 Individul pthwy intertivity int-4472 mster 1, 2e h, 3e h, 4 f, 5 18 FluenCy Where pproprite, give nswers orret to 2 deiml ples. 1 WE5 Find the length, in mm, of the hypotenuse of right-ngled tringle, if the two shorter sides re 5 m nd 12 m. 2 Find the length of the hypotenuse of the following right-ngled tringles, giving the nswer in the units speified. Sides 456 mm nd 320 mm, hypotenuse in m. Sides 12.4 mm nd 2.7 m, hypotenuse in mm. reflection How do you rememer whether to multiply or divide when onverting units? do-11430 Topi 15 Pythgors theorem 543
mesurement AND geometry Sides 32 m nd 4750 m, hypotenuse in m. d Sides 2590 mm nd 1.7 m, hypotenuse in mm. e Sides 604 m nd 249 m, hypotenuse in m. f Sides 4.06 km nd 4060 m, hypotenuse in km. g Sides 8364 mm nd 577 m, hypotenuse in m. h Sides 1.5 km nd 2780 m, hypotenuse in km. 3 WE6 The hypotenuse nd one other side of right-ngled tringle re given for eh se elow. Find the length of the third side in the units speified. Sides 46 m nd 25 m, third side in mm. Sides 843 mm nd 1047 mm, third side in m. Sides 4500 m nd 3850 m, third side in km. d Sides 20.3 m nd 123 mm, third side in m. e Sides 6420 mm nd 8.4 m, third side in m. f Sides 0.358 km nd 2640 m, third side in m. g Sides 491 mm nd 10.8 m, third side in mm. h Sides 379 000 m nd 82 700 m, third side in km. 4 Two sides of right-ngled tringle re given. Find the third side in the units speified. The digrm shows how eh tringle is to e lelled. Rememer: is lwys the hypotenuse. = 37 m, = 180 m, find in m. = 856 mm, = 1200 mm, find in m. = 4950 m, = 5.6 km, find in km. d = 125 600 mm, = 450 m, find in m. e = 0.0641 km, = 0.153 km, find in m. f = 639 700 m, = 2.34 km, find in m. 5 MC Note: There my e more thn one orret nswer. 4 m Wht is the length of the hypotenuse in this tringle? A 25 m B 50 m 3 m C 50 mm D 500 mm Wht is the length of the third side in this tringle? A 0.698 m B 69.8 m C 6.98 m D 69.8 mm The most urte mesure for the length of the third side in the tringle t right is: A 4.83 m B 23.3 m C 2330 mm D 4826 mm 43 m 82 m 5.6 m 2840 mm d Wht is the length of the third side in this tringle? A 34.71 m B 5.89 m C 1722 m D 4.4 m 4380 mm 6 A retngle mesures 35 mm y 4.2 m. Find the length of the digonl in mm. 394 m 544 Mths Quest 8
mesurement AND geometry 7 A sheet of A4 pper mesures 210 mm y 297 mm. Find the length of the digonl in entimetres. 8 A retngulr envelope hs length of 21 m nd digonl mesuring 350 mm. Find: the width of the envelope the re of the envelope. UNDERSTANDING 9 A right-ngled tringle hs hypotenuse of 47.3 m nd one other side of 0.308 m. Find the re of the tringle. 10 A horse re is 1.2 km. The trk is stright, nd 35 m wide. How muh further thn 1.2 km will horse run if it strts on the outside nd finishes on the inside s shown? 35 m Strting gte 1.2 km Finishing post 11 A rmp is 9 metres long, nd rises to height of 250 m. Wht is the horizontl distne, in metres, etween the ottom nd the top of the rmp? 12 Srh is mking gte, whih hs to e 1200 mm wide. It must e red with digonl strut mde of different type of timer. She hs only 2 m of this kind of timer ville. Wht is the mimum height of the gte tht she n mke? REASONING 13 A retngulr prk is 260 m y 480 m. Dnny usully trins y running round the edge of the prk. After hevy rin, two djent sides re too muddy to run long, so he runs tringulr pth long the other two sides nd the digonl. Dnny does 5 iruits of this pth for trining. How fr does he run? Give your nswer in km. 14 A swimming pool is 50 m y 25 m. Peter is ored y his usul trining routine, nd deides to swim the digonl of the pool. How mny digonls must he swim to omplete his norml distne of 1.2 km? 15 A hiker wlks 4.5 km west, then 3.8 km south. How fr in metres is she from her strting point? 16 A squre hs digonl of 10 m. Wht is the length of eh side in mm? Prolem solving 17 The tringulr sides of whet field mesure 2.4 km, 2500 m nd 700 000 mm. Is the tringle right-ngled? Show your working. 18 The sides of retngle mesure m y 5 mm. The digonl mesures m. Find the reltionship etween nd. 10 15.6 Composite shpes In ll of the eerises so fr, we hve een working with only one right-ngled tringle. Mny situtions involve more omple digrms, where the right-ngled tringle is not s ovious. A net digrm is essentil for these questions. Topi 15 Pythgors theorem 545
mesurement nd geometry WorKeD exmple 7 Find the length of the side,. Give your nswer orret to 2 deiml ples. 8 m 4 m 10 m think 1 Copy the digrm. On the digrm, rete right-ngled tringle nd use the given mesurements to work out the lengths of 2 sides. 2 Lel the sides of your right-ngled tringle s, nd. Rememer to lel the hypotenuse s. 3 Chek tht ll mesurements re in the sme units. They re the sme. 4 Write Pythgors theorem for the hypotenuse. 5 Sustitute the vlues of, nd into this rule nd simplify. 6 Find y tking the squre root of 20. Round your nswer orret to 2 deiml ples. Write/DrW 2 8 8 2 = 2 + 2 2 = 4 2 + 2 2 = 16 + 4 = 20 = 20 4.47 m Some situtions will involve shpes tht ontin more thn one tringle. In this se it is good ide to split the digrm into seprte right-ngled tringles first. WorKeD exmple 8 For the digrm t right, find the length of the sides mrked nd y to 2 deiml ples. think 1 Copy the digrm. 2 Find nd drw ny right-ngled tringles ontined in the digrm. Lel their sides. 3 To find n unknown side in right-ngled tringle, we need to know 2 sides, so find first. Write/DrW 15 y 15 4 y 6 2 6 2 546 Mths Quest 8
mesurement nd geometry 4 For the tringle ontining, write down Pythgors theorem for shorter side. Use it to find. 5 We now know 2 sides for the other tringle euse we n sustitute = 5.66. 6 For the tringle ontining y, write down Pythgors theorem for shorter side nd use it to find y. Eerise 15.6 Composite shpes individul PtHWys PrCtise 1 8, 13 ConsoliDte 1 10, 13, 14 Where pproprite, give nswers orret to 2 deiml ples. FluenCy 1 WE7 Find the length of the side in the figure t right. 2 = 2 2 2 = 6 2 2 2 = 36 4 = 32 = 32 5.66 2 = 2 2 y 2 = 15 2 5.66 2 = 225 32 = 193 y = 193 13.89 mster 1 15 2 For the following digrms, find the length of the sides mrked. 7 12 15 200 100 3 WE8 For eh of the following digrms, find the length of the sides mrked nd y. 3 5 10 y Individul pthwy intertivity int-4473 120 y 12 4 8 8 m 12 m 37 52 7 m 20 reflection Wht do you look for when dividing omposite shpes? Topi 15 Pythgors theorem 547
mesurement AND geometry 5 y d 18 12 20 10 2 5 4 MC The length of the digonl of the retngle t right is: A 9.7 B 19 C 13.9 D 12.2 The re of the retngle t right is: A 12.1 B 84.9 C 98 D 169.7 The vlue of in this shpe is: A 24 B 26 C 38.4 D 10 d Wht is the vlue of in this figure? A 5.4 B 8.6 C 10.1 D 10.3 UNDERSTANDING 5 A grden ed is in the shpe of right-ngled trpezium with the sloping edge 2.0 m, nd prllel sides of 3.2 m nd 4.8 m. Find the width of the grden nd, hene, the re. 6 Two uildings, 10 nd 18 m high, re diretly opposite eh other on either side of Europen street tht is 6 m wide (s shown in the following figure). Wht is the distne etween the top of the two uildings? 18 m 10 m 5 7 24 12 20 30 14 2 7 7 6 m 548 Mths Quest 8
mesurement AND geometry 7 Jess pddles noe 1700 m to the west, then 450 m south, nd then 900 m to the est. She then stops for rest. How fr is she from her strting point? 8 A yht re strts nd finishes t A nd onsists of 6 legs; AB, BC, CA, AE, EC, CA, in tht order s shown in the figure t right. If AB = 4 km, BC = 3 km nd CE = 3 km, find: AC AE the totl length of the re. B 3 km C 4 km 3 km 9 A pinter uses trestle to stnd on in order to pint eiling. It onsists of 2 stepldders onneted y 4 m long plnk. The inner feet of the 2 stepldders re 3 m prt, nd eh ldder hs sloping sides of 2.5 m. How high off the ground is the plnk? 4 m 2.5 m 2.5 m 2.5 m 3 m 10 A feture wll in grden is in the shpe of trpezium, with prllel sides of 6.5 m nd 4.7 m. The wll is 3.2 m high. It is to hve firy lights round the perimeter (eept for the se). How mny metres of lighting re required? REASONING 11 A retngulr gte is 3.2 m long nd 1.6 m high, nd onsists of three horizontl ems, nd five vertil ems s shown in the digrm t right. Eh setion is red with digonls. How muh timer is needed for the gte? 12 Find the distne AB in the following pln of the pddok. Distnes re in metres. A 2.5 m 4.7 m 6.5 m 3.2 m 10 E 3.2 m 1.6 m 10 A 17 B Topi 15 Pythgors theorem 549
mesurement nd geometry Prolem solving 13 The front door shown is 1 m wide nd 2.2 m high nd hs four identil glss pnels, eh 76 m long 15 m nd 15 m wide. Clulte the totl re of the glss pnels. The door is to e pinted inside nd outside. Clulte the totl re to e pinted. Two ots of pint re needed on eh side of the door. If the pint is sold in 1-L tins t $24.95 per litre nd eh litre overs 8 m 2 of the surfe, lulte the totl ost of pinting the door. 14 The ledlight pnel shown depits sunrise over the mountins. The mountin is represented y green tringle 45 m high. The yellow sun is represented y setion of irle with n 18-m rdius. There re 10 yellow sunrys in the shpe of isoseles tringles with se of 3 m nd height of 12 m. The sky is lue. Clulte the re of the ledlight pnel mde of: green glss yellow glss lue glss 15 On eh of the four sides of squre n equilterl tringle is drwn, the finl figure resemling four pointed str. If the sides of the squre re m long, wht is the re of the omplete figure? 15.7 Pythgoren trids Pythgoren trids, or trids for short, re 3 integers (whole numers) tht stisfy Pythgors theorem. This mens tht the 3 numers ould e the sides of right-ngled tringle. We test Pythgoren trids y first lulting 2, then lulting 2 + 2. If oth lultions yield the sme result, the originl 3 numers form Pythgoren trid. The numers 3, 4 nd 5 form Pythgoren trid euse 3 2 + 4 2 = 25 nd 5 2 = 25. If we multiply eh of these 3 numers y nother numer, the resulting numers will lso form Pythgoren trid. For emple, if we multiply 3, 4 nd 5 y 5, we get 15, 20 nd 25. These 3 numers stisfy Pythgors theorem. Chek: 2 = 25 2 = 625 2 + 2 = 15 2 + 20 2 = 625 so 2 = 2 + 2 Alger n e used to find sets of numers tht re Pythgoren trids. There re two wys in whih this n e done. Method 1 Strt with ny odd numer, nd mke this the shortest length of the tringle. 1 m 76 m 90 m 2.2 m 100 m 550 Mths Quest 8
mesurement nd geometry Use the formul M = s2 1 where S = shortest side nd M = middle side, to lulte 2 the middle side of the tringle. You now hve two sides of the tringle nd n use 2 = 2 + 2 to lulte the third side (hypotenuse, ). WorKeD exmple 9 If the smllest numer of Pythgoren trid is 7, find the middle numer nd, hene, find the third numer. think Write 1 Write down the given informtion, nd the S = 7 formul to find the middle side, M. M = S2 1 2 2 Sustitute S = 7 into the formul nd evlute. = 72 1 2 = 49 1 2 = 48 2 = 24 The middle numer is 24. 3 Use Pythgors theorem to find the third numer. 2 = 2 + 2 = 7 2 + 24 2 = 49 + 576 = 625 = 625 = 25 The third numer is 25. 4 Stte the solution. The Pythgoren trid required is 7, 24, 25. Why do you think this rule works only when the smllest side is n odd numer? Method 2 This method tkes ny 2 numers, nd pplies set of rules to them, in order to generte Pythgoren trids. Selet 2 numers, nd y, with the following rules: Rule 1 The numer hosen for must e lrger thn y; tht is > y. Rule 2 One numer must e odd nd the other even, to find the first trid. Rule 3 The numers hosen for nd y must hve no ommon ftors. For emple, we ould not hoose 6 nd 9 sine they hve ommon ftor of 3. The Pythgoren trid is given y 2y, 2 y 2, 2 + y 2. Additionl trids n e found y multiplying eh of these side lengths y ommon multiple. Topi 15 Pythgors theorem 551
mesurement nd geometry reflection Wht strtegies ould you use to memorise s mny ommon trids s possile? int-2765 WorKeD exmple 10 Find Pythgoren trid using the vlues = 7 nd y = 2. think Write 1 Write down the vlues given in the question. = 7, y = 2 2 Sustitute these vlues into the epression for eh term. 3 Chek tht the numers otined stisfy Pythgors theorem. Does 2 = 2 + 2? 2y = 2 7 2 = 28 2 y 2 = 7 2 2 2 = 49 4 = 45 2 + y 2 = 7 2 + 2 2 = 49 + 4 = 53 2 = 53 2 = 2809 2 + 2 = 28 2 + 45 2 = 784 + 2025 = 2809 Therefore, 2 = 2 + 2. 4 Stte the solution. The Pythgoren trid required is 28, 45, 53. Eerise 15.7 Pythgoren trids individul PtHWys PrCtise 1 9 ConsoliDte 1 10, 12 Individul pthwy intertivity int-4474 mster 1 13 FluenCy 1 Use Pythgors theorem to deide whih of these tringles re right-ngled. 6, 8, 10 5, 12, 13 4, 5, 6 d 24, 7, 25 e 16, 20, 12 f 14, 16, 30 2 Use the trids elow to rete three other trids, nd hek tht they stisfy Pythgors theorem. 3, 4, 5 5, 12, 13 3 WE9 The smllest numers of four Pythgoren trids re given elow. Find the middle numer nd, hene, find the third numer. 9 11 13 d 29 4 Wht do you notie out the trids in question 3? 5 WE10 For eh of the following, find Pythgoren trid using the given vlues of nd y. (In eh se hek tht the 3 numers found stisfy Pythgors theorem.) = 6 nd y = 1 = 7 nd y = 4 = 8 nd y = 3 d = 11 nd y = 6 e = 14 nd y = 9 f = 15 nd y = 2 552 Mths Quest 8
mesurement nd geometry understnding 6 Are the following tringles right-ngled? Use Pythgors theorem to hek. 10 12 7 11 14 7 Use Pythgors theorem to prove tht the following tringle is not right-ngled. 6.5 7 8 Does the method used in question 5 work if oth nd y re even? Does it work if oth nd y re odd? Try some emples of your own. resoning 9 If 32,, 68 is Pythgoren trid, wht is the vlue of? Show your working. 10 Find Pythgoren trid in whih the smllest numer is 33. Eplin how you rehed the solution. Prolem solving 11 Prove tht ( 2 2 ), 2, ( 2 + 2 ) form Pythgoren trid. 12 The lengths of the sides of prtiulr tringle re m, 2 m nd 3 m. Prove tht the trid formed is not Pythgoren trid. 13 If (p q), p, (p + q) form Pythgoren trid, find reltionship etween p nd q. If p = 8, find the Pythgoren trid. 15.8 Pythgors in 3-D B Mny rel-life situtions involve 3-dimensionl (3-D) shpes: shpes with length, width nd height. Some ommon 3-D shpes used in this setion inlude oes nd pyrmids. The importnt thing out 3-D shpes is tht in digrm, right ngles my not look like right ngles, so it is importnt to redrw setions of the digrm in 2 dimensions, where the right ngles n e seen urtely. A 8 C 13 Bo Pyrmid do-11433 Topi 15 Pythgors theorem 553
mesurement nd geometry WorKeD exmple 11 Find the length AG in this o. A B D E C 5 6 F think Write/DrW 1 Drw the digrm in 3-D. A B 2 Drw, in 2-D, right-ngled tringle tht ontins AG nd lel the sides. Only 1 side is known, so we need to find nother right-ngled tringle to use. 3 Drw EFGH in 2-D nd show the digonl EG. Lel the side EG s. 4 Use Pythgors theorem to find EG. 2 = 2 + 2 2 = 5 2 + 10 2 = 25 + 100 = 125 = 125 11.18 5 Ple this informtion on tringle AEG. Lel the side AG s y. D H 6 E E 5 H A E A 6 E 10 10 y 11.18 6 Solve this tringle for AG. 2 = 2 + 2 y 2 = 6 2 + 11.18 2 = 36 + 125 = 161 y = 161 12.69 7 Write your nswer. The length of AG is 12.69. H C 10 G 5 G 6 F G F 5 G G 554 Mths Quest 8
mesurement nd geometry Eerise 15.8 Pythgors in 3-D individul PtHWys PrCtise 1 6, 10 ConsoliDte 1 8, 10 Where pproprite, give nswers orret to 2 deiml ples. FluenCy 1 WE11 Find i length EG nd hene ii length AG. A B A B D H E 10 C G Individul pthwy intertivity int-4475 10 10 F 2 Find BD nd, hene, the height of the pyrmid t right. 3 The sloping side of one is 10 m nd the height is 8 m. Find the rdius, r, of the se. 4 An ie-rem one hs dimeter ross the top of 6 m, nd sloping side of 13 m. How deep is the one? 5 Jodie trvels to Bolivi, tking with her suitse s shown in the photogrph. She uys rved wlking stik 1.2 m long. Will she e le to fit it in her suitse for the flight home? D H E 5 C G 5 10 F 6 m mster 1 11 A B D H E 8.2 13 m D C G 8 7.3 A 10.4 V F C 8 8 8 m 10 m d r 30 m B reflection Wht re the key fetures to look for when solving 3-D prolem involving tringles? 65 m 90 m Topi 15 Pythgors theorem 555
mesurement AND geometry UNDERSTANDING 6 A desk tidy is shped like ylinder, height 18 m nd dimeter 10 m s shown. A penil tht is 24 m long rests inside. Wht length of the penil is ove the top of the ylinder? 18 m 7 A 10 m high flgpole is in the orner of retngulr prk 10 m tht mesures 240 m y 150 m. 240 m Find: i the length of the digonl of the prk B ii the distne from A to the top of the pole iii the distne from B to the top of the pole. A ird flies from the top of the pole to the entre of the prk. How fr does it fly? REASONING 8 The totl height of the shpe t right is 15 m. Find the length of the sloping side of the pyrmid. 9 A tent is in the shpe of tringulr prism, with height of 120 m s shown t right. The width ross the se of the door is 1 m, nd the tent is 2.3 m long. Find the length of eh sloping side, in metres. Then find the re of fri used in the onstrution of the sloping retngles whih form the sides. Prolem solving 10 A wooden plnk of the gretest possile length is pled inside grden shed. Use the digrm to lulte: the length of the digonl of the floor of the shed 1.8 m the length of the plnk of wood to 1 deiml ple. 11 A squre-sed pyrmid hs height of h m nd se side length of m. Find the length of the slnt height in terms of h nd. 120 m 14 m 10 m 14 m 2.4 m If the volume of pyrmid is 1 of the volume of the prism into whih it snugly fits, 3 find the volume of the pyrmid in terms of h nd. 1 m A 150 m 15 m 6 m 2.3 m R 90 m 556 Mths Quest 8
MEASUREMENT AND GEOMETRY ONLINE ONLY 15.9 int-3385 int-3386 int-3194 Review The Mths Quest Review is ville in ustomisle formt for students to demonstrte their knowledge of this topi. The Review ontins: Flueny questions llowing students to demonstrte the skills they hve developed to effiiently nswer questions using the most pproprite methods Prolem Solving questions llowing students to demonstrte their ility to mke smrt hoies, to model nd investigte prolems, nd to ommunite solutions effetively. A summry of the key points overed nd onept mp summry of this topi re ville s digitl douments. Lnguge 3 dimensions omposite shpe hypotenuse Link to ssesson for questions to test your rediness FOR lerning, your progress AS you lern nd your levels OF hievement. ssesson provides sets of questions for every topi in your ourse, s well s giving instnt feedk nd worked solutions to help improve your mthemtil skills. www.ssesson.om.u Pythgors theorem Pythgoren trids right ngle www.jplus.om.u Review questions Downlod the Review questions doument from the links found in your ebookplus. right-ngled tringle solve units Link to SpyClss, n eiting online gme omining omi ook style story with prolem-sed lerning in n immersive environment. Join Jesse, Toy nd Dn nd help them to tkle some of the world s most dngerous riminls y using the knowledge you ve gined through your study of mthemtis. www.spylss.om.u Topi 15 Pythgors theorem 557
<investigtion> For rich tsk or <mesurement nd geometry> For PuZZle investigtion rich tsk SA M PL E EV AL U AT IO N O N LY Are these wlls t right ngles? 558 Mths Quest 8 15PythgorsTheorem.indd 558 09/05/16 1:34 PM
mesurement nd geometry We will investigte the use of the uilder s squre. Cut two thin strips of pper to represent the rms nd s shown in the digrm ove. Join these two strips t one point with pin to mke the shpe of the uilder s squre. The length of n e otined y mesuring the distne from the end of to the end of. Complete questions 1 to 4, nd reord your results in the tle on the following pge. 1 Open the rms so tht they mke n ngle of 90. Use protrtor to mesure the ngle. Crefully mesure the strips to otin the vlues for, nd in millimetres. Reord your results in the fi rst row of the tle. 2 Open the rms so tht they mke n ngle less thn 90. Use protrtor to mesure the ngle. Crefully mesure the strips to otin the vlues for, nd in millimetres. Complete the seond row of the tle. 3 Repet question 2 with the rms opened up to n ngle greter thn 90. Complete the third row of the tle. 4 Chnge the lengths of nd y onstruting new uilder s squre. Repet questions 1 to 3 nd omplete the lst three rows of the tle. 5 Consider the lst three olumns of the tle. Using your results, wht onlusions n you drw? 6 Construt new uilder s squre nd open it to n ngle tht you estimte to e 90. Tke mesurements of, nd nd reord them elow. Wht onlusions would you drw regrding your ngle estimte? Chek y mesuring your ngle with protrtor. 7 From your investigtions, write prgrph outlining how the mesurements of the lengths of, nd on uilder s squre n e used y the uilder to determine whether the ngle etween djent wlls would e equl to, greter thn or less thn right ngle. Topi 15 Pythgors theorem 559
<investigtion> mesurement nd For geometry rich tsk or <mesurement nd geometry> For PuZZle CoDe PuZZle N LY Wht is Bgheer in The Jungle Book? O Clulte the lengths of the lettered sides in the tringles. Ple the lengths with their letters ove them, in sending order in the oes elow. The letters will spell out the puzzle s nswer. AT IO N Give lengths urte to 2 deiml ples. 6m R 130 m 9m A 155 m SA 1.2 m E L 1.5 m 520 m 170 m 6.31 m A 2m B T M PL E 180 m 7m EV 6m AL U 1m N 7.2 m C 4m 6.2 m 5.8 m 13 m 12.98 m H 3m K A 4m 3m 578 m 621 m 567 m 560 P Mths Quest 8 15PythgorsTheorem.indd 560 09/05/16 1:35 PM
MEASUREMENT AND GEOMETRY Ativities 15.2 Right-ngled tringles Digitl dos SkillSHEET (do-11425) Mesuring lengths nd ngles SkillSHEET (do-11426) Finding the squre of numer Intertivity IP intertivity 15.2 (int-4469) Right-ngled tringles 15.3 Finding the hypotenuse Digitl dos SkillSHEET (do-11427) Finding the squre root of numer SkillSHEET (do-11428) Rounding to given numer of deiml ples Intertivity IP intertivity 15.3 (int-4470) Finding the hypotenuse 15.4 Finding shorter side Digitl dos SkillSHEET (do-11429) Rerrnging formuls WorkSHEET 15.1 (do-11431) Intertivity IP intertivity 15.4 (int-4471) Finding shorter side 15.5 Working with different units Digitl do SkillSHEET (do-11430) Converting units of length To ess ebookplus tivities, log on to Intertivity IP intertivity 15.5 (int-4472) Working with different units 15.6 Composite shpes Intertivity IP intertivity 15.6 (int-4473) Composite shpes 15.7 Pythgoren trids Intertivities Pythgoren triples (int-2765) IP intertivity 15.6 (int-4474) Skething liner grphs 15.8 Pythgors in 3-D Intertivity IP intertivity 15.8 (int-4475) Pythgors in 3-D Digitl do WorkSHEET 15.2 (do-11433) 15.9 Review Intertivities Word serh (int-3385) Crossword (int-3386) Sudoku (int-3194) Digitl dos Topi summry (do-10766) Conept mp (do-10779) Topi review (Word do-14950, PDF do-14951) www.jplus.om.u Topi 15 Pythgors theorem 561
mesurement AND geometry Answers topi 15 Pythgors theorem 15.2 Right-ngled tringles 1 d e f g h 19 24 27 16 31 41 13 24 35 32 30 40 25 19 45 24 40 40 40 43 40 45 47 34 2 361 576 729 256 961 1681 169 576 2 1225 1024 900 1600 625 361 2025 576 2 + 2 1586 1600 1629 1856 1586 2042 2194 1152 2 1600 1600 1600 1849 1600 2025 2209 1156 2 2 + 2 2 3 2 + 2 = 2 4 2 + 2 2 5 The ngle sum of tringle is 180. If tringle ontined two right ngles, the third ngle would e 0, whih would men it ws stright line rther thn tringle. 6 g The squre on the hypotenuse is equl to the sum of the squres on the two other sides. 7 It is not possile tringle euse 7 + 4 < 12. 8 90. The right ngle is 90 nd the ngle sum of tringle is 180. Therefore the sum of the other two ngles must equl 90. 9 The tringle is isoseles; the smller ngles re 45. 10 It is not possile s ll ngles re 60 in n equilterl tringle. 11 Answers will vry. i 0.577 ii 0.5 iii 0.866 The vlues re the sme for ll the tringles. 15.3 Finding the hypotenuse 1 = 5 = 13 = 25 d = 21.3 e = 1163.3 f = 6.3 2 9760 11194 394 d 16 781 e 100 f 68 3 146 m 4 16.97 m 28.28 mm 6.93 m 5 12.81 m 737.83 m 17.26 m 6 18.03 m 7 19.23 m 8 4.66 m 9 $81.60 10 189 m 3 6.95 m 11 25 m 12 314.16 m 2 Chllenge 15.1 Side length 2.83 m; re = 8 m 2 15.4 Finding shorter side 1 6 93 6032 d 62 925 e 15 f 80 2 36.36 1.62 15.37 d 0.61 e 2133.19 f 453.90 3 23.04 m 4 97.47 m 334.94 m 6822.90 m 2 5 17.32 m 6 65.82 m; 2501.16 m 2 7 2.60 m 8 1.32 m 9 Yes 10 13.69 m 11 20.61 m 12 45.43 m 13 1.2 m 80 m 89.44 m Yes, she will reh the hook from the top step. 14 4 m The digonls re perpendiulr to eh other euse they rete right-ngled tringles. 15.5 Working with different units 1 130 mm 2 55.71 m 29.71 mm 57.27 m d 3098.08 mm e 6.53 m f 5.74 km g 10.16 m h 3.16 km 3 386.13 mm 62.09 m 2.33 km d 16.15 m e 541.70 m f 2615.61 m g 478.97 mm h 369.87 km 4 176.16 m 147.40 m 2.62 km d 432.12 m e 165.88 m f 6811.55 m 5 C A, B D d B 6 54.67 mm 7 36.37 m 8 28 m 588 m 2 9 552.86 m 2 10 0.51 m 11 8.65 m 12 1600 mm 13 6.43 km 14 21.46 digonls, so would need to omplete 22 15 5889.82 m 16 70.71 mm 17 Yes 18 2 = 80 2 15.6 Composite shpes 1 8.06 m 2 = 7.62 = 60 = 20.87 3 = 4, y = 9.17 = 6.93, y = 5.80 = 13, y = 15.20 d = 19.55 4 C B B d D 5 1.2 m; 4.8 m 2 6 10 m 7 917.88 m 8 5 km 4 km 24 m 9 2.45 m 10 11.35 m 11 26.65 m 12 18.44 m 13 4560 m 2 or 0.456 m 2 34 880 m 2 or 3.488 m 2 The pint jo would need 1 tin of pint, osting $24.95. The ost of the pint used would e $21.76. 14 2025 m 2 943.41 m 2 6031.59 m 2 15 (1 + 3) 2 2 15.7 Pythgoren trids 1 Yes:,, d, e No:, f 2 Answers will vry. Possile nswers: 6, 8, 10; 9, 12, 15; 12, 16, 20 10, 24, 26; 15, 36, 39; 20, 48, 52 3 40, 41 60, 61 84, 85 d 420, 421 562 Mths Quest 8
mesurement AND geometry 4 The two lrger numers re onseutive numers. 5 12, 35, 37 33, 56, 65 48, 55, 73 d 85, 132, 157 e 115, 252, 277 f 60, 221, 229 6 10 2 + 12 2 14 2 No 7 2 + 11 2 13 2 No 7 6.5 2 + 7 2 = 91.25 8 2 = 64 Therefore, the tringle is not right-ngled. 8 Yes Yes 9 60 10 The following Pythgoren trids n e found: 33, 44, 55; 33, 56, 65; 33, 180, 183; 33, 544, 545. Eplntions will vry. 11, 12 Chek with your teher. 13 p = 4q 6, 8, 10 Chllenge 15.2 36 15.8 Pythgors in 3-D 1 i 14.14 ii 17.32 i 7.07 ii 12.25 i 10.98 ii 15.12 2 11.31, 5.66 3 6 m 4 12.65 m 5 No: mimum stik n only e 115 m long. 6 3.41 m 7 i 283.02 m ii 240.21 m iii 150.33 m 141.86 m 8 13.38 m 9 1.3 m, 5.98 m 2 10 2.563 m 3.1 m 11 h 2 + 2 4 1 3 2 h Investigtion Rih tsk 1 to 4 Answers will vry. Teher to hek. 5 When the ngle is equl to 90, 2 = 2 + 2. For n ngle greter thn 90, 2 > 2 + 2. For n ngle smller thn 90, 2 < 2 + 2. 6 If 2 = 2 + 2, then the ngle is equl to 90. Chek y mesuring the ngle. 7 Prgrph eplntion of the results in question 5; teher to hek. Code puzzle A lk pnther Topi 15 Pythgors theorem 563