Question

a razão entre o numero de lados de dois polígonos convexos é 1/2 e a razão entre o numero de diagonais desses polígonos é 1/10. A razão entre as somas dos ângulos internos desses polígonos é

Answer 1

$\mathrm{\dfrac{n_1}{n_2}=\dfrac{1}{2}\ \to\ n_2=2.n_1\ \ \bigg\| \ \ \dfrac{d_1}{d_2}=\dfrac{1}{10}}\\\\\\ \mathrm{\dfrac{d_1}{d_2}=\dfrac{\frac{n_1(n_1-3)}{2}}{\frac{n_2(n_2-3)}{2}}\ \to\ \dfrac{1}{10}=\dfrac{n_1(n_1-3)}{2}.\dfrac{2}{n_2(n_2-3)}}\\\\ \mathrm{\dfrac{1}{10}=\dfrac{n_1(n_1-3)}{n_2(n_2-3)}\ \to\ \dfrac{1}{10}=\dfrac{1}{2}.\dfrac{(n_1-3)}{(n_2-3)}}\\\\ \mathrm{\dfrac{1}{5}=\dfrac{(n_1-3)}{(2.n_1-3)}\ \to\ 2.n_1-3=5(n_1-3)}\\\\ \mathrm{2.n_1-3=5.n_1-15\ \to\ -3+15=5n_1-2.n_1}$

$\mathrm{3.n_1=12\ \to\ n_1=\dfrac{12}{3}\ \to\ n_1=4}\\\\ \mathrm{n_2=2.n_1\ \to\ n_2=2.4\ \to\ n_2=8}\\\\\\ \mathrm{\dfrac{S_{i(1)}}{S_{i(2)}}=\dfrac{(n_1-2)180\º}{(n_2-2)180\º}\ \to\ \dfrac{S_{i(1)}}{S_{i(2)}}=\dfrac{(4-2)}{(8-2)}}\\\\ \mathrm{\dfrac{S_{i(1)}}{S_{i(2)}}=\dfrac{2}{6}}\ \to\ \mathbf{\dfrac{S_{i(1)}}{S_{i(2)}}=\dfrac{1}{3}}$