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Numere pitagorice

 
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Marcelina Popa
Bernoulli


Joined: 05 Mar 2008
Posts: 208
Location: Tulcea

PostPosted: Fri Oct 24, 2008 12:19 am    Post subject: Numere pitagorice Reply with quote

Trei numere naturale a, b si c se numesc numere pitagorice daca a^2+b^2=c^2. De exemplu, numerele 3, 4 si 5 sunt pitagorice

a). Aratati ca daca inmultim trei numere pitagorice cu acelasi numar, obtinem tot numere pitagorice.

b). Dati inca doua exemple de numere pitagorice (altele decat 3, 4 si 5).

c). Daca ati terminat cu bine punctele a) si b), aruncati o privire AICI. Ce credeti ca reprezinta numerele din coloana din dreapta?


NOTA. Numerele pitagorice sunt legate de celebra teorema a lui Pitagora. Aceasta suna cam asa: "Daca un triunghi are un unghi drept (triunghi dreptunghic), atunci patratul celei mai mari laturi este egal cu suma patratelor celorlalte doua laturi". Cu alte cuvinte, numerele pitagorice pot fi lungimile laturilor unui triunghi dreptunghic.


P.S. Intrucat acesta este cel de-al 50-lea mesaj al meu, tocmai am capatat titlul de Pitagora Smile.
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miruna.lazar
Bernoulli


Joined: 08 Oct 2008
Posts: 224
Location: Tulcea

PostPosted: Fri Oct 24, 2008 5:29 pm    Post subject: Reply with quote

A. (ya)^2 +  (yb)^2 =  (yc)^2 -> y ( a^2 +  b^2) =  (yc)^2 / :y ->  a^2 +  b^2 = c^2 -> Indiferent cu ce nr natural inmultim fiecare nr pitagoric , relatia  a^2 + b^2=c^2

Last edited by miruna.lazar on Wed Oct 29, 2008 7:36 pm; edited 5 times in total
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Marcelina Popa
Bernoulli


Joined: 05 Mar 2008
Posts: 208
Location: Tulcea

PostPosted: Fri Oct 24, 2008 8:21 pm    Post subject: Reply with quote

Este aproape bine. Hai sa sistematizam un pic lucrurile: cu notatiile tale, problema este urmatoarea:

Se stie ca numerele a, b si c sunt pitagorice.
Le inmultim pe toate cu acelasi numar, pe care-l notam cu y. Obtinem numerele ay, by si cy.
__________________________/___________________________

Trebuie sa demonstram ca si numerele ay, by si cy sunt pitagorice
_________________________//___________________________


Altfel spus, stim ca a^2+b^2=c^2 si trebuie sa demonstram ca (ay)^2+(by)^2= (cy)^2 .

(ay)^2 + (by)^2 = a^2 \cdot y^2 + b^2\cdot y^2 = y^2 \cdot (a^2 +b^2 ) = y^2 \cdot c^2 = (cy)^2,

deci (ay)^2+(by)^2= (cy)^2.
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miruna.lazar
Bernoulli


Joined: 08 Oct 2008
Posts: 224
Location: Tulcea

PostPosted: Wed Oct 29, 2008 7:35 pm    Post subject: Reply with quote

B. 5 , 10 , 15 , si 12 , 16 , 20
C. Eu cred ca reprezinta numerele pitagorice de baza cu ajutorul carora s-au format alte numere pitagorice. ( cele din stanga )
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