Răspuns:
[tex]13. \ \boldsymbol{ \red{14}}; \ 14. \ \boldsymbol{ \red{-1}}; \ 15. \ \boldsymbol{ \red{6}}[/tex]
Explicație pas cu pas:
13. O este centrul cercului circumscris triunghiului ABC
[tex](x_A-x_O)^2 + (y_A-y_O)^2 = (x_B - x_O)^2 + (y_B-y_O)^2 = (x_C-x_O)^2 + (y_C-y_O)^2 = R^2\\[/tex]
[tex](1 - 0)^2 + (8 - 0)^2 = (b - 0)^2 + (4 - 0)^2 = (c - 0)^2 + (4 - 0)^2\\[/tex]
[tex]\begin{cases}b^2 + 16 = 1 + 64 \Rightarrow b^2 = 49 \Rightarrow |b| = 7 \\ c^2 + 16 = 1 + 64 \Rightarrow c^2 = 49 \Rightarrow |c| = 7 \end{cases}[/tex]
[tex]\Rightarrow \bf |b| + |c| = 14[/tex]
[tex]\boldsymbol{ \red{\star \star \star}}[/tex]
14. Abcisa centrului de greutate:
[tex]x_O = \dfrac{x_A+x_B+x_C}{3} \Rightarrow \dfrac{1 + b + c}{3} = 0 \\[/tex]
[tex]\Rightarrow \bf b + c = -1[/tex]
[tex]\boldsymbol{ \red{\star \star \star}}[/tex]
[tex]15. \ \left|\begin{array}{ccc}1&8&1\\b&-4&1\\c&-4&1\end{array}\right| = -4+8c-4b+4c-8b+4 = 12(c-b)[/tex]
[tex]\mathcal{A}_{\Delta ABC} = \dfrac{1}{2}|\Delta| \Rightarrow \dfrac{1}{2} \cdot |12(c-b)| = 36 \Rightarrow 12|c-b| = 72\\[/tex]
[tex]\Rightarrow \boldsymbol { |b - c| = 6}[/tex]