Question

considerando-se um polígono regular de n lados, n  4, e tomando-se ao acaso uma das diagonais do polígono, a probabilidade de que ela passe pelo centro é:

Answer 1

$\mathbf{Se\ n\ for\ par:}\\\\ \mathrm{*\ N\º\ de\ diagonais\ \to\ d=\dfrac{n(n-3)}{2}}\\\\ \mathrm{*\ N\º\ de\ diagonais\ que\ passam\ pelo\ centro\ \to\ d_{pc}=\dfrac{n}{2}}\\\\ \mathrm{P=\dfrac{d_{pc}}{d}=\dfrac{\frac{n}{2}}{\frac{n(n-3)}{2}}=\dfrac{n}{2}.\dfrac{2}{n(n-3)}=\dfrac{1}{n-3}\ \to\ \boxed{\mathbf{P=\dfrac{1}{n-3}}}}$

$\mathbf{Se\ n\ for\ \'impar:}}\\\\ \mathrm{*\ N\º\ de\ diagonais\ \to\ d=\dfrac{n(n-3)}{2}}\\\\ \mathrm{*\ N\º\ de\ diagonais\ que\ passam\ pelo\ centro\ \to\ d_{pc}=0}\\\\ \mathrm{P=\dfrac{d_{pc}}{d}=\dfrac{0}{\frac{n(n-3)}{2}}=0\ \to\ \boxed{\mathbf{P=0}}}$