Diriangolo di vertid $$ A(0 ; 0 ; 3), B(1 ; 4 ; 7) $$ e $$ C $$ ha baricentro $$ G(1 ; 3 ; 3) $$. Determina le coordinate di $$ C $$,

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Consideriamo il triangolo con vertici
$$
A(0,\,0,\,3),\quad B(1,\,4,\,7),\quad C(x,\,y,\,z)
$$
e il baricentro $$ G(1,\,3,\,3) $$. Per il baricentro sappiamo che:
$$
G = \left( \frac{x_A + x_B + x_C}{3},\, \frac{y_A + y_B + y_C}{3},\, \frac{z_A + z_B + z_C}{3} \right)
$$
cioè
$$
\left( \frac{0+1+x}{3},\, \frac{0+4+y}{3},\, \frac{3+7+z}{3} \right) = (1,\, 3,\, 3).
$$

Procediamo col calcolo delle coordinate:

1. Asse $$ x $$:
$$
\frac{0+1+x}{3} = 1 \quad\Longrightarrow\quad \frac{1+x}{3} = 1 \quad\Longrightarrow\quad 1+x = 3 \quad\Longrightarrow\quad x = 2.
$$

2. Asse $$ y $$:
$$
\frac{0+4+y}{3} = 3 \quad\Longrightarrow\quad \frac{4+y}{3} = 3 \quad\Longrightarrow\quad 4+y = 9 \quad\Longrightarrow\quad y = 5.
$$

3. Asse $$ z $$:
$$
\frac{3+7+z}{3} = 3 \quad\Longrightarrow\quad \frac{10+z}{3} = 3 \quad\Longrightarrow\quad 10+z = 9 \quad\Longrightarrow\quad z = -1.
$$

Pertanto, le coordinate del vertice $$ C $$ sono:
$$
C(2,\,5,\,-1).
$$
Le coordinate del vertice $$ C $$ sono $$ (2,\,5,\,-1) $$.

Diriangolo di vertid \( A(0 ; 0 ; 3), B(1 ; 4 ; 7) \) e \( C \) ha baricentro \( G(1 ; 3 ; 3) \). Determina le coordinate di \( C \),

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