If the coordinates of the vertices of a trian | vedclass

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(C) The coordinates of the vertices are $(1, a)$,$(2, b)$,and $(c^2, 3)$.The centroid $(G)$ of a triangle with vertices $(x_1, y_1)$,$(x_2, y_2)$,and

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Cannot lie on the $y$-axis

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(C) The coordinates of the vertices are $(1, a)$,$(2, b)$,and $(c^2, 3)$.The centroid $(G)$ of a triangle with vertices $(x_1, y_1)$,$(x_2, y_2)$,and $(x_3, y_3)$ is given by $G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)$.Substituting the given values,the centroid is $\left( \frac{1 + 2 + c^2}{3}, \frac{a + b + 3}{3} \right) = \left( \frac{3 + c^2}{3}, \frac{a + b + 3}{3} \right)$.For the centroid to lie on the $y$-axis,its $x$-coordinate must be zero.$\frac{3 + c^2}{3} = 0 \implies 3 + c^2 = 0 \implies c^2 = -3$.Since $c^2$ cannot be negative for any real value of $c$,the $x$-coordinate can never be zero.Therefore,the centroid cannot lie on the $y$-axis.

If the coordinates of the vertices of a triangle are $(1, a)$,$(2, b)$,and $(c^2, 3)$,then the centroid of the triangle:

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