Question

Num triângulo de baricentro G(0,1/2), dois dos vértices são A(1,1) e B(-2,2/3). Obter o outro vértice?

​

Answer 1

O baricentro é dado por:

$(x_G~,~y_G)~=~\left(\frac{x_A+x_B+x_C}{3}~,~\frac{y_A+y_B+y_C}{3}\right)\\\\\\Substituindo~os~dados~fornecidos, ~temos:\\\\\\\left(0~,~\frac{1}{2}\right)~=~\left(\frac{1+(-2)+x_C}{3}~,~\frac{1+\frac{2}{3}+y_C}{3}\right)\\\\\\\left(0~.~3~,~\frac{1}{2}~.~3\right)~=~\left(1-2+x_C~,~1+\frac{2}{3}+y_C\right)\\\\\\\left(0~,~\frac{3}{2}\right)~=~\left(-1+x_C~,~\frac{5}{3}+y_C\right)\\\\\\Igualando~coordenada~a~coordenada, temos:$

$\left\{\begin{array}{ccc}~\,0~=\,-1+x_C\\\frac{3}{2}~=~\frac{5}{3}+y_C\end{array}\right\\\\\\Da~1^a~equacao,~temos:\\\\\boxed{x_C~=~1}\\\\\\Da~1^a~equacao,~temos:\\\\\frac{3}{2}~=~\frac{5}{3}+y_C\\\\y_C~=~\frac{3}{2}-\frac{5}{3}\\\\\boxed{y_C~=~-\frac{1}{6}}$

O terceiro vértice é o ponto:

$(x_C~,~y_C)~=~\left(1~,\,-\frac{1}{6}\right)$