ThePythagoreanTheorem
11AProofandsimpleapplications
ThePythagoreanTheorem,whichgivesarelationbetweenthelengthsofthesidesofaright-angledtriangle, isafundamentalresultingeometry.Itstatesthatthesquareofthe hypotenuse (thesideoppositethe right-angle)isequaltothesumofthesquaresoftheothertwosides.Aproofofthistheoremisgivenbelow.
hypotenuse
ThePythagoreanTheoremisnamedaftertheancientGreekphilosopherPythagorasofSamos,1 eventhough itappearsthattheresultwasknowntomathematiciansoftheFirstBabylonianDynastymorethanone thousandyearsearlier.Moreover,thereislittleevidencethatPythagorashimselfwasawareofthetheorem thatbearshisname.
Theorem1(PythagoreanTheorem)
If c isthelengthofthehypotenuseofaright-angledtriangleand a and b arethelengthsoftheother twosides,then
Thismeansthattheareaoftheorangesquareisequaltothesumoftheareasoftheblueandpurple squares.
1 PythagorasofSamos(c.570-c.495BC)wasanancientGreekphilosopherwhoseworkinfluencedPlatoandAristotle. DespitethefactthatthePythagoreanTheorembearshisname,historianscontesttheextentofhiscontributiontomathematics.
Proof
Thetwofiguresbelowhavethesametotalarea.
Therefore, (orangesquare)+ 4(redtriangles) = (bluesquare)+(purplesquare)+ 4(redtriangles) Ifweeliminatethefourtrianglesfromeachfigure,thentheareathatremainswillalsobethesame. Therefore,
Findthevalueof x ineachofthefollowingfigures.
Note. Ineachoftheaboveproblemsweonlyrequiredthepositiverootofeachequation.Thatis,the equation x2 = 25hastwosolutions: x = 5and x = 5,butonlythefirstofthesecanbealength.
Example1
Example2
Givencoordinates A( 3, 1) and B(2, 2),finddistance AB.
Solution
Points A and B areshownontheadjacentsetofaxes. AB2 = 52 + 32 = 25 + 9 = 34 AB = √34.
Therefore, AB = √34.
Example3
Inthegridofsquaresbelow,distance AB = 29.Findthesidelengthofeachsquare.
Solution
Let x bythesidelengthofeachsquare.Then
Example4
Findtheareaoftheisoscelestrianglewithtwosidesoflength3andonesideoflength4.
Solution
Bisectthetriangle.Let h betheheightoftheresultingpairofright-angled triangles.Then
Therefore,theareais
11AExercises
1.Findthevalueof x ineachofthediagramsbelow:
2.Findthevalueof x ineachofthediagramsbelow byrepeatedapplicationsofthePythagorean Theorem:
3.(a)Findthevaluesof x, y and z inthediagram below.
(b)Theabovediagramgivesawayofconstructingsegmentsoflength √2, √3and √4. Showhowyoucouldextendthediagramto constructasegmentoflength √5.
4.Inthediagrambelow,findeachofthefollowing lengths.
11.AmorealgebraicproofofthePythagorean Theoremcanbefoundinthefollowingdiagram.
( 2, 3)
(1, 1)
(2, 1)
( 2, 2)
(a)Findthetotalareaofthelargesquareby expanding (a + b)2 .
(b)Findthetotalareaofthelargesquareby findingthesumoffourtriangularareasand theareaofthesmallersquare.
(e) CD (f)
AB (a) AC (b) AD (c) BC (d)
5.Generalisetheworkyoudidintheprevious questiontofindaformulaforlength AB given coordinates A(x1,y1) and B(x2,y2).
6.Inthesquaregridbelow,distance AB = 34.Find thesidelengthofeachsquare.
(c)Equatethetworesults,anddeducethe PythagoreanTheorem.
12.(a)Findtheareaofthelargestregularhexagon thatfitsinsideacircleofradius1.
(b)Findtheareaofthesmallestregularhexagon thatfitsaroundacircleofradius1.
13.Atrianglehassidesoflength5,6and7.
(a)Findthelengthofitslongestaltitude,as shownbelow.(Notethatanaltitudeisaline segmentdrawnfromavertexofatriangle thatmeetstheopposingsideatarightangle.)
7.Findtheareaofthetrianglesshownbelow.
(a)
(b)
8.Findtheareaofaregularhexagonwhoseside lengthis2.Hint:splitthehexagonintoequilateral triangles.
9.Findthestraightlinedistancefrom A to B inthe diagrambelow.
10.Explainwhythehypotenuseinaright-angled trianglewillalwaysbethelongestside.
5 7 6
(b)Hence,find theareaofthetriangle.
(c)Hence,findthelengthofitsshortestaltitude.
14.Theareaofarectangleis50anditsperimeteris 30.Findthelengthofitsdiagonal.
15.Inaright-angledtriangle,thealtitudedrawnto thehypotenuse c dividesthehypotenuseintotwo linesegmentsoflengths p and q.Ifwedenotethe lengthofthealtitudedrawntothehypotenuseby h,thenprovethat h = √pq. a b p q
11BApplicationsofthePythagoreanTheorem
ThePythagoreanTheoremiscentraltothestudyofgeometryprimarilybecauseofitsconnectiontothenotions ofdistanceandlength.Thisconnectionalsomakesitimmenselyimportantforavarietyofapplications.
Example1
Aropetiedtothetopofaflagpoleis1mlongerthanthepolewhenhangingstraightdown.Whenpulled tightwiththelowerendontheground,itis4mfromthebaseofthepole.Howtallistheflagpole?
Solution
If h mistheheightoftheflagpole,thenthelengthoftheropeis (h + 1) m.So
Theflagpoleis7 5metrestall.
Example2
Townsarelocatedat O(0, 0), A(16, 0),and B(0, 12) (distancesinkilometres).Astraightroadjoins A and B,andanewroadistoconnect O tothisroad.Findtheshortestpossiblelengthofthenewroad.
Solution
Let x bethelengthof AB.Then,
Nowdrawalinethrough O perpendiculartotheexistingroadthough A and B.Thiswillbetheshortestroad.Let h beitslength.Tofind h,we findtheareaof △AOB twodifferentways:
Problemsinvolvingdistanceofteninvolvecircles.A circle isthesetofallpointsintheplanewhosedistance fromafixedpointisequaltoaconstant.Thefixedpointiscalledthe centre ofcircleandtheconstant distanceiscalledthe radius ofthecircle.
Example3
Findthesidelengthofthesmallestsquarethatcontainstwotouchingcirclesofradius1.
Solution
Constructaright-angledtrianglewhosehypotenusespansthe centreofthetwocircles.Let x bethesidelengthofthisisosceles triangle.UsingthePythagoreanTheorem,
Example4
Asquare isinscribedinsideasemicircleofradius5.Findtheareaofthesquare.
Solution
Let2x be thesidelengthofthesquare.Therefore,
thesquareisthen
TheTriangle Inequality
ThePythagoreanTheoremallowsustogiveageometricproofoftheso-called triangleinequality.This characteriseswhenthreepositivenumbersformthesidelengthsofatriangle.Ourproofrequiresonesimple butusefulfact:
Thehypotenuseinaright-angledtriangleislongerthaneitheroftheothertwosides.
Toseewhythisistrue,notethat c2 = a2 + b2 > a2.Therefore, c > a.
Theorem1(TriangleInequality)
Suppose a,b,c > 0.If a, b and c arethesidelengthsofatriangle,then c < a + b
Proof
Theanglebetweenthesidesoflengths a and c iseitheracute,right-angled,orobtuse.Weconsidertwo cases.
Case1.Firstsupposethattheangleisacute.A linesegmentperpendiculartothesideoflength c dividesthissideintotwolinesegmentsoflength d and e.
Case2.Supposethattheangleisrightorobtuse. Alinesegmentperpendiculartothesideoflength c dividesthesideoflength b intolinesegmentsof length d and e.
e
Therefore,
a > d (as a isahypotenuse)
b > e (as b isahypotenuse)
Therefore,
a + b > d + e = c.
Therefore,
d > c (as d isahypotenuse)
Therefore, a + b > b ≥ d > c.
Note. Theconverseofthetriangleinequalityisalsotrue.Thatis,suppose wehavethreepositivenumbers a,b and c,andthesumofanytwoofthese isgreaterthanthethird.Thenthereisatrianglewhosewhosesideshave lengths a,b and c.Toprovethis,suppose c isthegreatestofthese.Drawa linesegment AB oflength c.Drawacircleofradius b centredat A anda circleofradius a centredat B.Since c < a + b,theinteriorsofthecircles overlap,sothecirclesintersectatsomepoint C.Then ABC istherequired triangle.
a b
11BExercises
1.Aropeistiedtothetopofaflagpole.Whenit hangsstraightdown,theropeis2mlongerthan thepole.Whentheropeispulledtightwiththe lowerendontheground,theropeis3metres fromthepole’sbase.Howtallistheflagpole?
2.Threetownsarelocatedatcoordinates O(0, 0),A(8, 0) and B(0, 6),wherealldistances aremeasuredinkilometres.Thereisastraight roadthatjoinstowns A and B.Athirdstraight roadistojointhetownat O totheexistingroad. Whatistheshortestpossiblelengthofthisroad?
3.Abushwalkerstartsatpoint A andfinishes atpoint B bywalking10kmeast,9kmnorth, 4kmwest,and1kmsouth.Findthestraight-line distancefrom A to B
4.Determinewhichright-angledtriangleshaveside lengthsthatareallconsecutiveintegers.Hint:let n bethesidelengthoftheshortestside.
5.Determinewhichright-angledtriangleshaveside lengthsthatareallconsecutiveevenintegers. Hint:let n bethesidelengthoftheshortestside.
6.Asoccerpitchis100mfromgoaltogoal.Apiece ofropeoflength100metresisattachedtothe baseofeachopposinggoal.Therope’slengthis increasedby(only)20centimetres.Theropeis nowlooseandcanbepickedupinthemiddleto formatriangle.Findtheheightofthetrianglein metres,accuratetothetwodecimalplaces.Can amousefitundertherope?
9.Onesideofsquareliesalongthediameterofa semicircleandtwoverticesofthesquareareon thecircle’scircumference.Iftheradiusofthe semicircleis10,thenfindtheareaofthesquare.
7.Twoantsarelocatedattheorigin (0, 0).One beginswalkingnorthataspeedof2cmper secondatthesametimeasthesecondbegins walkingeastataspeedof3cmpersecond.How manysecondspassbeforethedistancebetween thetwoantsis13cm?
10.(a)Determineaformulaforthestraight-line distance d tothehorizon,intermsofthe radius R ofaplanet,andtheheight h from whichthehorizonisviewed.
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8.Theperimeterofarectangleis16mandthe lengthofitsdiagonalsis6m.Findtheareaof therectangle.
(b)If h istinycomparedto R,thenexplainwhy d ≈ √2Rh
11.Twocylindersofradius r liesidebysideonaflat surface.Athirdcylinderofradius r restsontop ofthetwocylinders.Findtheheightabovethe groundofthecentreofthetopmostcylinder.
12.Therectanglebelowcontainstwotouching semicirclesofradius r.Findtheareaofthe rectangle.
13.Acylindricalglasshasradius4cmandheight 10cm.Anantislocatedat A ontheupperrimof theglass.Thereisadripofmilkontheopposite sideonthebaseoftheglassat M .Theantwalks alongapaththatminimisesthedistanceshemust cover.Howfardoesshewalk?
18† . Threecirclesaredrawntangenttoboththe circumferenceanddiameterofasemicircle,as showninthediagrambelow.Thetwosmaller circlesofradius r aretangenttothelargercircle ofradius R.Findtheratio R ∶ r
14.Acarpenterhasasquarepieceofwoodof dimensions1metreby1metre.Hewantstocreate aregularoctagonbycuttingoffthecorners.What distance x fromtheedgeofthesquaredoesthe carpenterneedtomakehiscuts?
19† . Acircleofradius r andasemicircleofradius R areinscribedinalargerquartercircle,asshown inthediagrambelow.Findtheratio R ∶ r.
15† . (a)Aright-angledtrianglehasintegerside lengths.Ifitsperimeterequals30,thenfind thetriangle’slengths.
(b)Aright-angledtrianglehasintegerside lengths.Ifitsperimeterequals40,thenfind thetriangle’slengths.
(c)Showthattherearetwodifferentright-angled triangleswithintegersidelengthswhose perimeterequals60.
(d)Showthatthereisnoright-angledtriangle withintegersidelengthswhoseperimeter equals20.
16† . Findthelengthofthediagonalofthesmallest squarethatcontainstwo tangent circles (touchingatexactlyonepoint),oneofradius r andtheotherofradius R
17† Twosemicirclesofradius R aredrawnonthe diameterofasemicircleofradius2R.Acircleof radius r isdrawntangenttoallthreesemicircles, asshowninthediagrambelow.Findtheratio R
20† Twosemicirclesofradius R aredrawntangent toacircleofradius r,asshowninthediagram below.Eachsemicirclepassesthroughthecentre oftheothersemicircle.Findtheratio R
r
21† Threecirclesofradii r1 < r2 < r3 aremutually tangentandalsotangenttoahorizontalline,as showninthediagrambelow.Findanexpression for r1 intermsof r2 and r3
22† . Let E beapointinsiderectangle ABCD.Let AE = a, BE = b, CE = c and DE = d.Provethat a2 + c2 = d2 + b2 .
(a)WritedowntheFareysequences F4, F5, F6, and F7
a b c d
A
B C D E
23† Consideraright-angledtrianglewithsides a,b and c.Drawanaltitudefromtheright-angleto thehypotenuse,andletitslengthbe h a b c h
(a)Showthat c = ab h
(b)The InversePythagoreanTheorem statesthat 1 a2 + 1 b2 = 1 h2 .
Provethisresult.
24† . Findalloftheright-angledtriangleswithinteger sidelengthssuchthattheareaandperimeterare equal.
25‡ Forthisquestionallourfractionswillbeassumed tobewrittenwithnon-negativenumeratorsand denominators.
The nth Fareysequence Fn isthelist,written inincreasingorder,ofallfractionsbetween0and 1whichhavedenominatorslessthanorequalto n (whenexpressedasirreduciblefractions).For example,
Wehaverepresented F4 onthenumberlinewith neighbouringfractionsconnectedbycirculararcs. Thisiscalleda Fareydiagram
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1 = { 0 1 , 1 1 } , F2 = { 0 1 , 1 2 , 1 1 } ,
Wesaythat a b and c d kiss if ad bc = ±1.
(b)Provethatifthefractions a b and c d kiss,then theymustbothbeirreducible.
Infact, a b and c d kissifandonlyiftheyare neighboursinaFareysequence,thoughwewill notprovethishere.
Wedefinethe mediant of a b and c d tobe a+c b+d
(c)Provethatif a b < c d ,thentheirmediant satisfies:
(d)Provethatif a b and c d kiss,thentheirmediant a+c b+d kissesboth a b and c d
(e)Provethatif a b and c d areneighboursina Fareysequence,thentheirmediantisthe uniquefractionthatoccursinbetween a b and c d thatkissesthemboth.Inparticular, themediantisthefirstfractionthatoccurs between a b and c d inalaterFareysequence.
Toanyfraction a b wedefineitsassociated Ford circle tobethecirclewithradius 1 2b2 thatlies abovethenumberline,andtouchesitatthepoint a b
(f)Provethat a b and c d kissifandonlyiftheir associatedFordcirclesaretangent.
Beginningwiththefractions 0 1 and 1 1 ,wecan takemediantsandsketchtheassociatedFord circles.Wecaniteratethisproceduretogenerate
infinitelymanycircles,andhavedrawnthefirst fewiterationsbelow.Eachcirclerepresentsa rationalnumber,andstartingwiththeintegers n 1 wecangenerateallrationalnumbersinthis way,thoughwewillnotprovethishere.Notethat thediagramisparticularlypleasingwhenoverlaid ontotheFareydiagram,asshownadjacent.
(g)ProvethatthecirculararcsintheFarey diagramadjacentmeettheFordcirclesat rightangles.
11CThePythagoreanTheoreminthreedimensions
A facediagonal ofarectangularprismisaline connectingoppositeverticesofthesameface.A spacediagonal ofarectangularprismisaline connectingtwoverticesthatarenotonthesame face.Wecanfindthelengthofaspacediagonalby repeatedapplicationofthePythagoreanTheorem.
Example1
Findthelengthofthespacediagonalinaboxwhosesidesare2cm,3cmand5cm.
Solution
Wefirstconsiderthefacediagonaloflength x cm,aspictured.Usingthe PythagoreanTheorem,wefindthat,
Wenowfindthelength d cmofthespacediagonal.Applyingthe PythagoreanTheoremonceagain,weseethat,
Theorem1(PythagoreanTheoreminThreeDimensions)
Let a,b and c bethesidelengthsofarectangularprism.If d isthelengthofthespacediagonal,then
Proof
Wefirstconsiderthefacediagonaloflength x,aspictured.Usingthe PythagoreanTheorem,wefindthat,
Wenowfindthelength d ofthespacediagonal.ApplyingthePythagorean Theoremonceagain,weseethat,
Theaboveformulareducesatwo-stepproblemtoaone-stepproblem.
Example2
Findthelengthofthespacediagonalofarectangularprismwhosesidesare8,4and1unitsinlength.
Solution
ByanapplicationofthePythagoreanTheoreminthreedimensions,
Example3
Theface diagonalsofarectangularprismaremeasuredtobe5, 6and7units.Findthelengthofthe spacediagonal.
Solution
Letthesidesbeoflength a,b and c.If d isthelengthofthespacediagonal, then d2 = a2 + b2 + c2.BythreeapplicationsofthePythagoreanTheorem,
Addingthesethreeequationstogether,thendividingbytwo,gives,
1.Findthelengthofthespacediagonalforeachof theserectangularprisms.
5.Foreachofthesepyramids,findtheperpendicular distancefromtheapextothebase,asshown:
2.Findthelengthofthelongestthin,inflexiblestick thatcanbeplacedinsideacubeofsidelength3 metres.
3.Thespacediagonalofacubeis12cmlong.Find itssidelength.
4.Findthevalueof x inthefigurebelow:
6.Thefacediagonalsofarectangularprismare measuredtobe7, 8and9units.Findthelength ofthespacediagonal.
7.Thespacediagonalofarectangularprismis1 cmlongerthanitsfirstside,4cmlongerthanits secondside,and5cmlongerthanitsthirdside. Findthelengthofthespacediagonal.
8.Thespacediagonalofarectangularprismis1 cmlongerthanitsfirstside,5cmlongerthanits secondside,and8cmlongerthanitsthirdside. Findthelengthofthespacediagonal.
11DTheconverseofthePythagoreanTheorem
WearenowacquaintedwiththePythagoreanTheorem.Takeatrianglewithsidelengths a,b and c
Statement: Iftheanglebetween a and b isaright-angle,then a2 + b2 = c2
Ifweswitchthehypothesisandtheconclusioninthisstatement,thenweobtainthe converse statement:
Converse: If a2 + b2 = c2,thentheanglebetween a and b isaright-angle.
Wewillnowprovethattheconverseholds.
Theorem1(ConverseofthePythagoreanTheorem)
Foranytrianglewithsidelengths a, b and c,if a2 +b2 = c2,thentheanglebetween a and b isaright-angle.
Proof
Wearegiventhat a2 + b2 = c2.Firstsupposetheanglebetweenthesides oflength a and b exceeds 90○.Thenwecanextendonesidetoformthe right-angledtriangleshownadjacent.UsingthePythagoreanTheorem
Therefore, c2 = c2 + 2ae > c2,whichisimpossible.Wecanfindasimilar contradictioniftheanglebetweenthetwosidesislessthan90○
Determineifthesetrianglesareright-angled.
Example1
Solution
(a)
Ifthisisaright-angledtriangle,thenthe longestsidewillbethehypotenuse.Therefore,
Ifthisisaright-angledtriangle,thenthe longestsidewillbethehypotenuse.Therefore,
11DExercises
1.Determinewhichofthesetrianglesarerightangled.
trianglehavelengths2and3,respectively.Find thelengthofthethirdside.
5.(a)Showthatatrianglewhosesideshavelengths √2, √3and √5hasarightangle.
(b)Let a,b > 0.Showthatatrianglewithsides √a, √b and √a + b hasarightangle.
6.Aright-angledtrianglehasonesideoflength 5andanothersideoflength6.Therearetwo possiblesidelengthsforthethirdsideofthis triangle.Whatarethey?
7.Consider △ABC shownbelow.
2.Atrianglehassideswhoselengthsare7,24and 25.Determineifthistrianglehasarightangle.
3.Atrianglehassideswhoselengthsare √2, √7and 3.Determineifthistrianglehasarightangle.
4.Onesideandthehypotenuseofaright-angled
(a)Findeachofthelengths:
(b)Showthat △ABC hasarightangle,andthen finditsarea.
8.Let r > 1.Supposearight-angledtrianglehas sidesoflength1, r and r2.Findthevalueof r. Hint:let x = r2
9† . Let m > 0.Considerthetwolinessketchedbelow. Theequationsoftheselinesare y = mx and y = 1 m x.Point A islocatedonthefirstline,and point B islocatedonthesecondline.
(a)Findeachofthefollowinglengths: OA (a) OB (b) AB (c)
(b)Hence,usingtheconverseofthePythagorean Theorem,showthatthesetwolinescrossat right-angles.
10† Arjunmeasurestwosidesofatriangleandthese sumto31.Blakemeasurestwosidesofthesame triangleandthesesumto32.Cadenmeasurestwo sidesofthetriangleandthesesumto49.Without measuringanything,Divyadeclaresthistobea right-angledtriangle.HowcanDivyaknowthis forsure?
11EPythagoreantriples
A Pythagoreantriple consistsofthreepositiveintegers a, b and c forwhich a2 + b2 = c2.Suchatripleis oftenwrittenas (a,b,c).Well-knownPythagoreantriplesare (3, 4, 5) and (5, 12, 13)
TheconverseofthePythagoreanTheoremensuresthatifthesidelengthsofatriangleformaPythagorean triple,thenthistrianglewillhavearightangle.
DeterminewhichofthesearePythagoreantriples:
8, 15, 17) (a)
289. As a2 + b2 = c2,thisisaPythagoreantriple.
7, 11, 13)
49 + 121
170 c 2 = 132
169. As a2 +b2 ≠ c2,thisisnotaPythagoreantriple.
Example1
Example2
Provethatif (a,b,c) isaPythagoreantripleand k isapositiveinteger,then (ka,kb,kc) isalsoa Pythagoreantriple.
Solution
Wearegiventhat a2 + b2 = c2.Toshowthat (ka,kb,kc) isaPythagoreantriple,wecalculate
Therefore, (ka,kb,kc) isalsoaPythagoreantriple.
ConsiderthePythagoreantriple (a,b,c).If a, b and c donotshareanyfactorsotherthan1,thenthisiscalled a primitivePythagoreantriple.EachPythagoreantriplecanbeassociatedwithaprimitivePythagorean triplebydividingtheintegersbytheirgreatestcommondivisor.Astheaboveexampleshows,anyprimitive PythagoreantriplealsogivesrisetoaninfinitefamilyofPythagoreantriples.
Example3
ShowthatthereisexactlyonePythagoreantripleforwhich3istheshortestside.
Solution
Since (3,b,c) isaPythagoreantriple,
As c + b > c b wemusthave c b = 1and c + b = 9.Addingthesetwoequationsgives2c = 10.Therefore, c = 5and b = 4.
11EExercises
1.Determinewhichofthefollowingare Pythagoreantriples:
(3, 4, 5) (a)
(5, 6, 7) (b)
(6, 8, 10) (c) (14, 22, 26) (d)
(8, 15, 17) (e) (5, 12, 13) (f)
2.ShowthatthereisexactlyonePythagoreantriple oftheform (5,b,c).
3.ShowthatthereareexactlytwoPythagorean triplesoftheform (9,b,c).
4.ShowthatthereareexactlytwoPythagorean triplesoftheform (8,b,c).
5.(a)ShowthatthereisonlyonePythagorean triplewhosehypotenuseis5.
(b)Provethatthismethodalwaysresultsina Pythagoreantriple.
8† DetermineallPythagoreantriplesforwhichthe hypotenuseisoneunitlongerthanoneofitssides.
9.Let m,n ∈ N where m > n. Euclid’sformula statesthatif
a = m 2 n 2,b = 2mn,c = m 2 + n 2 ,
then (a,b,c) isaPythagoreantriple.Infact,it canbeshownthateveryprimitivePythagorean tripleisofthisformforsomechoiceof m and n ofoppositeparity,satisfyingGCD(m,n) = 1.
(a)FindthePythagoreantriplecorresponding to:
m = 3and n = 4(i)
(b)ShowthatthereisnoPythagoreantriple whosehypotenuseis6.
m = 4and n = 7(ii)
m = 5and n = 8(iii)
(c)Somenumberscanappearasthehypotenuse ofmorethanonetriple.Showthat (15, 20, 25) and (7, 24, 25) arebothPythagoreantriples.
6.(a)Showthat (1,b,c) and (2,b,c) cannotbe Pythagoreantriples.
(b)ShowthatthereisaPythagoreantripleofthe form (n,b,c) forall n ≥ 3.Hint:considerthe casewhere n isevenandthenthecasewhere n isodd.
7.Inthisquestionwegiveaclevermethodforeasily generatingPythagoreantriples.
• Pickanytwooddnumbersthatdifferbytwo, suchas3and5.
• Addtheirreciprocals.Here, 1 3 + 1 5 = 8 15 .
• Thenumeratorandthedenominatorofthe resultingfractionwillbethefirsttwoterms ofaPythagoreantriple.Here,thetripleis (8, 15, 17),since82 + 152 = 172 .
(a)Findthetripleobtainedifyoubeginwiththe numbers:
(a)
(b)Findthevaluesof m and n thatyieldthese Pythagoreantriples:
(3, 4, 5) (i)
(5, 12, 13) (ii)
(7, 24, 25) (iii)
(c)Provethatif a,b and c aregivenbyEuclid’s formula,then (a,b,c) isaPythagoreantriple.
10† Suppose (a,b,c) isaprimitivePythagoreantriple.
(a)Showthatpreciselyoneof a and b isdivisible by2.
(b)Showthatpreciselyoneof a and b isdivisible by3.
(c)Showthatpreciselyoneof a and b isdivisible by4.
(d)Showthatpreciselyoneof a, b and c is divisibleby5.
11† Suppose (a,b,c) isaprimitivePythagoreantriple. Provethat (c a)(c b) 2 isalwaysaperfectsquare.
12† . Isitpossiblefortwodifferentright-angled triangleswithintegersidelengthstohave thesameperimeter?Hint:Considerthetriples (a,b,c) and (d,e,f ) where
and
Byequatingtheirperimeters,showthat
(m + n) = p(p + q)
Canyoufindvaluesof m,n,p and q thatsatisfy thisequation?
13† Usingasquaresheetofpaper,wecaneasilyfolda trianglesimilartothe (3, 4, 5) triangle,asfollows.
• Startwithasquaresheetofpaper.
• Findthemidpointsofthesides.
• Foldalinethroughthetopmid-pointtothe bottomleftcorner,andalinethroughthetop mid-pointtothebottomrightcorner.
• Foldalinethroughtherightmid-pointtothe bottomleftcorner.
Usethediagrambelow,andsimilartriangles, toprovethatthetriangularregionformedin themiddleissimilartothe (3, 4, 5) triangle. Hint:Firstshowthattheanglesindicatedinthe diagramareequal.
14† . Supposethat (a,b,c) isaPythagoreantriple. Provethat (a + 1,b + 1,c + 1) isnotaPythagorean triple.
15† Withalittlemorework,wecangeneralisethe previousquestion.Supposethat (a,b,c) isa Pythagoreantriple.Let d ∈ N.Provethat (a + d,b + d,c + d) isnotaPythagoreantriple.Hint:In anytrianglewithsides a, b and c,weknowthat a + b > c