Question

Obeserve o ABC em um plano cartesiano ​

Answer 1

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$\Large\boxed{\begin{array}{l}\sf sejam~A(x_A,y_A),B(x_B,y_B)~e~C(x_C,y_C)~as~coordenadas\\\sf de~um ~tri\hat angulo~no~plano~cartesiano.\\\sf o~baricentro~do~tri\hat angulo~a~qual\\\sf representa-se~por~G(x_G,y_G)~\acute e~dado~por~\\\sf x_G=\dfrac{x_A+x_B+x_C}{3}~e~y_G=\dfrac{y_A+y_B+y_C}{3}\end{array}}$

$\Large\boxed{\begin{array}{l}\tt a)~\sf A(-4,-1)~B(-2,4)~e~C(2,3)\\\sf x_G=\dfrac{x_A+x_B+x_C}{3}\\\sf x_G=\dfrac{-4-\diagdown\!\!\!2+\diagdown\!\!\!2}{3}=-\dfrac{4}{3}\\\sf y_G=\dfrac{y_A+y_B+y_C}{3}\\\sf y_G=\dfrac{-1+4+3}{3}\\\sf y_G=\dfrac{6}{3}\\\sf y_G=2\\\boxed{\boxed{\boxed{\boxed{\sf G\bigg(\!-\dfrac{4}{3},2\bigg)}}}}\end{array}}$

$\large\boxed{\begin{array}{l}\tt b)~\sf (x_C-x_G)^2=\bigg(2-\bigg[-\dfrac{4}{3}\bigg]\bigg)^2\\\sf=\bigg(2+\dfrac{4}{3}\bigg)^2=\bigg(\dfrac{10}{3}\bigg)^2=\dfrac{100}{9}\\\sf (y_C-y_G)^2=(3-2)^2=1^2=1\\\sf d_{G,C}=\sqrt{(x_C-x_G)^2+(y_C-y_G)^2}\\\sf d_{G,C}=\sqrt{\dfrac{100}{9}+1}\\\sf d_{G,C}=\sqrt{\dfrac{100+9}{9}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf d_{G,C}=\dfrac{\sqrt{109}}{3}}}}}\end{array}}$