Pythagoras’TheoremPythagoras’theoremwasdiscoveredover2,000yearsagobyGreekmathematicianandphilosopher,Pythagoras.Thetheoremissimplyanequationthatdescribestherelationshipbetweenthelengthsofthesidesofaright-angledtriangle.Let’stakealookataright-angledtriangle:Thelabellingoftheright-angledtriangleisveryimportant.Thelongestside,c,iscalledtheHypotenuse.Pythagorastheoremsimplystatesthat:

𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐Pythagoras’Theoremhelpstocalculatethelengthofthesidesofthetriangle(a,bandc).Usingthetheoremyoucancalculatethelengthofanysideaslongasyouknowthelengthsoftheothertwosides.Let’sseeWHYthisisthecase:

1. Drawasquareoneachsideofthetriangle.2. Calculatetheareaofeachsquare.3. AddtheareasofsquareAandsquareB.4. Noticeanything?

a c

b

35

4ThisisexactlywhatPythagorasnoticedallthoseyearsago!

Yes!YouwillnoticethattheareaofsquareAplustheareaofsquareBisthesameastheareaofsquareC!

A

B

C

UsingPythagorasTheoremNow,let’sseehowthetheoremworksinquestions.Workedexample1FindthelengthoftheHypotenuse.Giveyouranswerto1decimalplace. Step1:Remember,theHypotenuseisthelongestline,usuallylabelledasc.Therefore,wearecalculatingchere.Step2:Plugyournumbersintotheequation:

𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐𝟓𝟐 + 𝟔𝟐 = 𝒄𝟐𝟐𝟓+ 𝟑𝟔 = 𝒄𝟐𝟔𝟏 = 𝒄𝟐 𝒄 = 𝟔𝟏𝒄 = 𝟕.𝟖

Gotit?Okay,let’smoveontoaharderone…Workedexample2:Findthelengthofa.Giveyouranswerto1decimalplace.Step1:Inthisexample,youarecalculatingthelengthofa,whereaisnottheHypotenuse.Step2:Asyouarecalculatingthelengthofashorterside,youwillneedtorearrangetheequation:

𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐𝒂𝟐 = 𝒄𝟐 − 𝒃𝟐

Step3:Pluginyournumbers:

𝒂𝟐 = 𝟖.𝟓𝟐 − 𝟔.𝟐𝟐𝒂𝟐 = 𝟕𝟐.𝟐𝟓− 𝟑𝟖.𝟒𝟒

𝒂𝟐 = 𝟑𝟑.𝟖𝟏𝒂 = 𝟑𝟑.𝟖𝟏𝒂 = 𝟓.𝟖

5c

c

6

a 8.5

6.2

Pythagoras’Theorem–practicequestions1) FindthelengthoftheHypotenuseinthefollowingtriangles.Giveyouranswerto

1decimalplace.a)

b) 2) Findthelengthofthemissingsides.Giveyouranswerto1decimalplace.

a)

b)

2.2c

3.3

2.6

4.2

c

5.5

7.8

a

3.7

8.0b