Centroid Calculation for Triangle Vertices

To find the centroid of a triangle given its vertices, we use the centroid formula. The centroid is the point where the medians of the triangle intersect.

Centroid Formula Application

Let the vertices of the triangle ABC be $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$. The coordinates of the centroid G are given by:

$ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) $

Given vertices are:

• $A = (-16, 13) \implies x_1 = -16, y_1 = 13$
• $B = (-2, -9) \implies x_2 = -2, y_2 = -9$
• $C = (2, 13) \implies x_3 = 2, y_3 = 13$

Calculating Centroid Coordinates

Substitute the coordinates into the centroid formula:

1. Calculate the x-coordinate ($G_x$): $ G_x = \frac{x_1 + x_2 + x_3}{3} = \frac{-16 + (-2) + 2}{3} = \frac{-16 - 2 + 2}{3} = \frac{-16}{3} $
2. Calculate the y-coordinate ($G_y$): $ G_y = \frac{y_1 + y_2 + y_3}{3} = \frac{13 + (-9) + 13}{3} = \frac{13 - 9 + 13}{3} = \frac{4 + 13}{3} = \frac{17}{3} $

Therefore, the coordinates of the centroid are $\left( -\frac{16}{3}, \frac{17}{3} \right)$.

Final Answer Coordinates

The centroid coordinates match option 4.