Strokovno sre canje DMFA Slovenije Marko Razpetmarkor/matematika/Pravilni_petkotnik.pdfPravilni...
Transcript of Strokovno sre canje DMFA Slovenije Marko Razpetmarkor/matematika/Pravilni_petkotnik.pdfPravilni...
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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Marko Razpet Pravilni petkotnik
Pentagram
Pent�grammoc
pènte – petgr�mma – crka, pismenka
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Marko Razpet Pravilni petkotnik
Diagonale ali prekotnice
Diag¸nioc (di�metroc v Evklidovih Elementih)di� – prek, skozigwnÐa – kotmètron – mera, merilo
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Marko Razpet Pravilni petkotnik
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 .
Marko Razpet Pravilni petkotnik
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
Marko Razpet Pravilni petkotnik
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 .
Marko Razpet Pravilni petkotnik
Razmerje med diagonalo in stranico pravilnega petkotnika
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Marko Razpet Pravilni petkotnik
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 .
Marko Razpet Pravilni petkotnik
Zlato razmerje
Iz enakostia2 + ad = d2
sledi relacija d = �a, pri cemer je
� =1 +√
5
2.
Marko Razpet Pravilni petkotnik
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, . . .].
Marko Razpet Pravilni petkotnik
Koti v pravilnem petkotniku
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54∘
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36∘ 36∘
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Marko Razpet Pravilni petkotnik
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.
Marko Razpet Pravilni petkotnik
Petkotnik v naravi – jabolko
Marko Razpet Pravilni petkotnik
Petkotnik v naravi – pasijonka
Marko Razpet Pravilni petkotnik
Petkotnik v naravi – kaktus
Marko Razpet Pravilni petkotnik
Petkotnik v naravi – karambola
Marko Razpet Pravilni petkotnik
Petkotnik v naravi – morska zvezda
Marko Razpet Pravilni petkotnik
Pravilni dodekaeder
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Marko Razpet Pravilni petkotnik
Pravilni ikozaeder
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Marko Razpet Pravilni petkotnik
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∘.
Marko Razpet Pravilni petkotnik
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/� .
Marko Razpet Pravilni petkotnik
Zmaja in zelo dolocata presecisci diagonal
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∙
Marko Razpet Pravilni petkotnik
Zmaj in zelo z GeoGebro
Marko Razpet Pravilni petkotnik
Zlaganje zmajev in zel (Roger Penrose)
Marko Razpet Pravilni petkotnik
Zlaganje zmajev in zel doma in v soli
Marko Razpet Pravilni petkotnik
Marko Razpet Pravilni petkotnik
Hvala za vaso pozornost!
Marko Razpet Pravilni petkotnik
Marko Razpet Pravilni petkotnik