Question

Determine as coordenadas do baricentro do triângulo indicado na figura

Answer 1

$\boxed{\begin{array}{l}\sf Vamos~encontrar~a~equac_{\!\!,}\tilde ao~da~reta~AB:\\\sf passa~pelos~~pontos~M(-4,0)~e~B(3,5)\\\sf a=\dfrac{5-0}{3-(-4)}=\dfrac{5}{7}\\\sf y=y_0+m(x-x_0)\\\sf y=0+\dfrac{5}{7}\cdot(x-[-4])\\\sf y=\dfrac{5}{7}(x+4)\\\sf y=\dfrac{5}{7}x+\dfrac{20}{7}\\\sf o~ponto~A~tem~coordenadas(-x,-2)\\\sf substituindo~temos:\\\sf -2=\dfrac{5}{7}x+\dfrac{20}{7}\cdot(7)\\\sf -14=5x+20\\\sf 5x=20+14\\\sf 5x=34\\\sf x=\dfrac{34}{5}\implies -x=-\dfrac{34}{5}\end{array}}$

$\boxed{\begin{array}{l}\sf As~coordenadas~do~ponto~A~s\tilde ao\\\sf A\bigg(-\dfrac{34}{5},-2\bigg)\end{array}}$

$\boxed{\begin{array}{l}\sf Sejam~A(x_A,y_A),B(x_B,y_B~e~C(x_C,y_C)\\\sf os~v\acute ertices~de~um~tri\hat angulo~no~plano~cartesiano.\\\sf O~baricentro~deste~tri\hat angulo~\acute e~o~ponto~G(x_G,y_G)~tal~que\\\sf x_G=\dfrac{x_A+x_B+x_C}{3}~e~y_G=\dfrac{y_A+y_B+y_C}{3}\end{array}}$

$\boxed{\begin{array}{l}\sf x_G=\dfrac{1}{3}\cdot\bigg(-\dfrac{34}{5}+3+1\bigg)\\\sf x_G=\dfrac{1}{3}\cdot\bigg(\dfrac{-34+15+5}{5}\bigg)\\\sf x_G=\dfrac{1}{3}\cdot(\bigg(-\dfrac{14}{5}\bigg)=-\dfrac{14}{15}\\\sf y_G=\dfrac{1}{3}(-2+5-3)=0\end{array}}$

$\boxed{\begin{array}{l}\sf Portanto~o~baricentro~\acute e~G\bigg(-\dfrac{14}{15},0\bigg)\end{array}}$