Strokovno sre canje DMFA Slovenije Marko Razpetmarkor/matematika/Pravilni_petkotnik.pdfPravilni petkotnik Strokovno sre canje DMFA Slovenije Marko Razpet Pedago ska fakulteta, Univerza

Pravilni petkotnikStrokovno srecanje DMFA Slovenije

Marko Razpet

Pedagoska fakulteta, Univerza v Ljubljani

Rimske Toplice, 19. oktober 2012

Marko Razpet Pravilni petkotnik

Pentagon

Pent�gwnon

pènte – petgwnÐa – kot

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Ea

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a


Pentagram

Pent�grammoc

pènte – petgr�mma – crka, pismenka

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Diagonale ali prekotnice

Diag¸nioc (di�metroc v Evklidovih Elementih)di� – prek, skozigwnÐa – kotmètron – mera, merilo

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Ptolemajev izrek

PtolemaØoc – Ptolemaj (85–170)

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V vsakem konveksnem tetivnem stirikotniku velja med stranicamia, b, c in d ter diagonalama e in f relacija:

ac + bd = ef .


Dokaz Ptolemajevega izreka

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△ABT ∼ △DBC , △ABD ∼ △TBC

∣AT ∣a

=c

f,∣TC ∣b

=d

f

e = ∣AT ∣+ ∣TC ∣ =ac

f+

bd

fef = ac + bd


Pitagorov izrek

Pravokotnik s stranicama a in b ter diagonalo c je tudi konveksnitetivni stirikotnik.

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Torej velja Pitagorov izrek:

a2 + b2 = c2 .


Razmerje med diagonalo in stranico pravilnega petkotnika

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Uporaba Ptolemajevega izreka v pravilnem petkotniku

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Ce uporabimo pri pravilnem petkotniku ABCDE Ptolemejev izrekna trapezu ABCD, ki je ocitno konveksni tetivni stirikotnik, takojdobimo enakost

a2 + ad = d2 .


Zlato razmerje

Iz enakostia2 + ad = d2

sledi relacija d = �a, pri cemer je

� =1 +√

5

2.


Osnovna lastnost zlatega razmerja

Zlato razmerje ima osnovno lastnost:

�2 = � + 1 .

Posledicno dobimo verizni ulomek:

� = 1 +1

1 +1

1 +1

1 +1

1 +1

1 +1

. . .

.

Tak verizni ulomek spravimo v skatlico:

� = [1; 1, 1, 1, . . .].


Koti v pravilnem petkotniku

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R

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.... 18∘

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54∘

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72∘

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36∘ 36∘

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Nekatere kolicine v pravilnem petkotniku

a – stranica pravilnega petkotnika,R – polmer pravilnemu petkotniku ocrtanega kroga,% – polmer pravilnemu petkotniku vcrtanega kroga,v – visina pravilnega petkotnika,P – ploscina pravilnega petkotnika.S podobnostjo trikotnikov in s Pitagorovim izrekom dobimo:

a =R

2

√10− 2

√5, R =

a

10

√50 + 10

√5,

% =a

10

√25 + 10

√5, v =

a

2

√5 + 2

√5,

P =a2

4

√25 + 10

√5.


Petkotnik v naravi – jabolko


Petkotnik v naravi – pasijonka


Petkotnik v naravi – kaktus


Petkotnik v naravi – karambola


Petkotnik v naravi – morska zvezda


Pravilni dodekaeder

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Pravilni ikozaeder

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Romb v pravilnem petkotniku

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Stranice so dolge a, daljsa diagonala �a, krajsa pa a√

3− � . Ostrakota merita po 72∘, topa pa po 108∘.


Romb razdelimo na zmaja in zelo (kite, dart)

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M∙

Tocka M deli daljso diagonalo v razmerju � . Daljsi odsek meri a,krajsi pa a/� .


Zmaja in zelo dolocata presecisci diagonal

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∙

∙


Zmaj in zelo z GeoGebro


Zlaganje zmajev in zel (Roger Penrose)


Zlaganje zmajev in zel doma in v soli



Hvala za vaso pozornost!



Documents

Strokovno sre canje DMFA Slovenije Marko Razpetmarkor/matematika/Pravilni_petkotnik.pdfPravilni petkotnik Strokovno sre canje DMFA Slovenije Marko Razpet Pedago ska fakulteta, Univerza