Constructible polygons

Constructible polygons

Season 3

Episode 07 Time frame 2 periods

Prerequisites :

^{Rulerandompassrulesandmethods}

Objectives :

•

^{study the} onstrutibility of regularpolygons.

Materials :

•

^Ruler

•

^Compas

•

^{Task sheet}

•

^{Hints for the pentagon, 15-gon and 16-gon.}

•

^Beamer

1 – The easy ones 20 mins

Students workinpairs. They havetond the minimalnumberof ationsneeded todraw

an equilateraltriangle,a square, a regularhexagona, and aregular otogon. This part is

marked over10.

2 – The regular pentagon 20 mins

The seondtaskistoonstrutaregularpentagon. Thispartis markedover10.Progres-

sive hints are available on demand. Every hint asked by a group takes one point of the

nal mark.

3 – The regular pentadecagon 15 mins

The third taskis to onstrut a regular 15-gon. One hint may be given : use apentagon

and an equilateraltriangle.This par isnot marked

4 – The regular heptadecagon 20 mins

The third task is toonstrut a regular17-gon. A onstrution protoolis given to eah

student and has tobe arried out. The end result isalso marked over 10.

5 – Constructible polygons ^{20 mins}

Eahgrouphas tolistallonstrutiblepolygonswithanumberof sideslessthan orequal

to

20

^.

6 – Lecture : Some results about constructible polygons 15 mins

A quik history of the problem of onstrutible polygons, inluding Eulid's methods,

Document Answer sheet

Part 1 – Construct some easy regular polygons

Equilateral triangle Square

Regular Octogon Regular hexagon

Construction of the pentagon

Trytondawaytoonstrutaregularpentagon.Ifyoudon'tmanage,gototheteaher's

desk and ask for ahint. The hintsare progressiveand ost 1pointover 10eah.

Construction of the pentadecagon

Trytond a way toonstrut a regularpentadeagon (with15 equalsides).If you don't

manage, gotothe teaher's desk and ask for ahint. Thereis onlyone available hintand

The heptadecagon : a construction protocol

Followthe following instrutions toonstrut a regularheptadeagon.

1

^{. Given anarbitrary point}

O

,draw airle entered on

O

and a horizontaldiameter drawn through

O

.

2

^{. Callthe right end of the diameter dividingthe irle intoa semiirle}

P ₁

.

3

^{. Construt the diameter} perpendiular to the original diameter by nding the per- pendiular bisetor

OB

, with

B

atthe top of the irle.

4

^{. Construt}

J

a quarter of the way up

OB

.

5

^{. Join}

J P ₁

and nd

E

on linesegment

OP ₁

sothat

∠ OJ E

isa quarter of

∠ OJ P ₁

.

6

^{. Find}

F

online

OP ₁

, but on the other side of

O

, sothat

∠ EJ F

is 45degrees.

7

^{. Construt the semiirlewithdiameter}

F P ₁

, onthe same sideas

J

. Thissemiirle uts

OB

at

K

.

8

^{. Draw a semiirle with enter}

E

and radius

EK

, on the same side as

B

and with both endpoints on

OP ₁

.This uts the linesegment

OP ₁

at

N ₄

.

9

^{. Construt a line} perpendiular to

OP ₁

through

N ₄

. This line meets the original semiirleat

P ₄

.

10

^{. You now have points}

P ₁

and

P ₄

of a heptadeagon. Use

P ₁

and

P ₄

to get the remaining15pointsof the heptadeagon aroundthe original irle by onstruting

P ₁

,

P ₄

,

P ₇

,

P ₁₀

,

P ₁₃

,

P ₁₆

,

P ₂

and so on.

11

^{. Connet the adjaent points}

P _i

for

i = 1

^to

17

^{, forming the} heptadeagon.

The constructible polygons

1

^{. Listallregularpolygonswith}

20

^{orless sides thatyouthinkare} onstrutiblewithe ruler and ompass. Explain ina few words eahonstrution.

2

^{. AFermatprimeisaprimenumberoftheform}

F _n = 2 ^{2 n} +1

^,where

n

isanonnegative integer. Compute the rst ve Fermat primes.

3

^{. There is a onnetion between the} onstrutible polygons and the Fermat prime.

Document 1

^{Hints for the onstrution of the regularpentagon}

1

^{. Draw a irle in whih to insribe the pentagon and mark}

the enter point

O

.

2

^{. Choose a point}

A

on the irle that will serve as one vertex of the pentagon. Draw a line through

O

and

A

.

3

^{. Construt aline}perpendiularto theline

OA

passingthrough

point

O

. Mark its intersetion with one side of the irle as the point

B

.

4

^{. Construt the point}

C

as the midpoint of

O

and

B

.

5

^{. Draw a irle entered at}

C

through the point

A

. Mark its

intersetion with the line

OB

(inside the original irle) as the point

D

.

6

^{. Draw a irle entered at}

A

through the point

D

. Mark its intersetions with the original irle as the points

E

and

F

.

7

^{. Draw a irle entered at}

E

through the point

A

. Mark its other intersetion with the original irle as the point

G

.

8

^{. Draw a irle entered at}

F

through the point

A

. Mark its other intersetion with the original irle as the point

H

.