Question

Determine os valores de x e y sabendo que A (2,4) B (x,5) e C (5,y) são vértices de um triângulo cujo baricentro é o ponto G (2,3).

Answer 1

G=(Xa+Xb+Xc)/3
2=(2+Xb+5)/3
3.2=Xb+7
6=Xb+7
Xb=6-7
Xb=-1

G=(Ya+Yb+Yc)/3
3=(4+5+Yc)/3
3.3=Yc+9
9=Yc+9
Yc=9-9
Yc=0

B (-1,5) e C (5,0)

Answer 2

Se $G(x_{G}},\,y_{G}})$ é baricentro do triângulo formado pelos pontos

$A(x_{_A},\,y_{_A}),~B(x_{_B},\,y_{_B})~\text{ e }~C(x_{_C},\,y_{_C});$


então as coodenadas de $G$ são dadas por

$\left\{ \!\begin{array}{l} x_{G}=\dfrac{x_{_A}+x_{_B}+x_{_C}}{3}\\\\ y_{G}=\dfrac{y_{_A}+y_{_B}+y_{_C}}{3} \end{array} \right.$

_________________


Substituindo os valores que conhecemos:

$\bullet\;\;x_G=2\\\\\\ 2=\dfrac{2+x+5}{3}\\\\\\ 2+x+5=6\\\\ x+7=6\\\\ x=6-7\\\\ \therefore~~\boxed{\begin{array}{c}x=-1 \end{array}}$


$\bullet\;\;y_G=3\\\\\\ 3=\dfrac{4+5+y}{3}\\\\\\ 4+5+y=9\\\\ 9+y=9\\\\ y=9-9\\\\ \therefore~~\boxed{\begin{array}{c}y=0 \end{array}}$


Os vértices do triângulo são os pontos

$A(2,\,4),~B(-1,\,5)~\text{ e }~C(5,\,0).$


Bons estudos! :-)