C 1 with entre A and radius r.

(1)

Épreuve de setion européenne

Doubling theirle

The etymologyof thewordirle is from theGreek, kirkos,from the base ker-whih means

toturn. Theoriginofthewordirus isloselyrelatedaswell.

The irle was known before the beginning of reorded history. Natural irles would have

beenobserved,suh astheMoon,theSun,and ashort plantstalkblowingin thewindonsand,

whihforms airle shapein thesand. Theirle isthebasisforthewheel,whih,with related

inventionssuhasgears,makesmuhofmodernivilizationpossible. Inmathematis,thestudyof

theirlehashelpedinspirethedevelopmentofgeometry,astronomy,andalulus. Earlysiene,

partiularlygeometryandastronomy,wasonnetedtothedivineformostmedievalsholars,and

manybelievedthatthere wassomethingintrinsiallydivine orperfet thatouldbefoundin

irles.

Let'sonsider airle

C ₁

^with ^entre

A

^and ^radius

r

^.

Howanweonstrutairle

C ₂

^with^entre

A

ândânârea

equaltothedoubleof theareaof

C ₁

^? ^One^way^to^do ^this

istouseaset-square. Plaetheset-squareasinthepiture

ontheright,withthetwoinnersidestangenttoirle

C ₁

^.

Then,rotateitaround

C ₁

^. ^A^pen^plaed^at^theright-angle ofthesetsquare(point

M

^on^the^gure)^will^then^trae^the

requiredirle

C ₂

^.

A +

M N S

Q

P X

AdaptedfromWikipedia,thefreeenylopedia.

Questions

1. Aordingtothetext,quoteafewnaturalirles.

2. Ifwedrawairlewithradiusthedoubleof

r

^,^do^we^obtain^the^required^irle

C ₂

^? ^Explain

youranswer.

3. Provethatwearelookingforairleof radius

√ 2 r

^.

4. Provethattheonstrutionwiththeset-squaregivestherequiredirle

C ₂

^.

5. What should bethe width

e = SQ = P X

ôf^the ^set-square^so^that â^pen^plaedât ôrner

N

^will^drawâîrle^withârea^three^times^theâreaôf

C ₁

^?

C 1 with entre A and radius r.

C 1

A

r

C 2

A

C 1

C 1

C 1

M

C 2

A +

M N S

Q

P X

r

C 2

√ 2 r

C 2

e = SQ = P X

N

C 1

C ₁

C ₂

C ₁

C ₁

C ₁

C ₂

C ₂

C ₂

C ₁