To find the coordinates of the centroid of a triangle with given vertices, we can use the formula for the centroid (also known as the center of gravity or the barycenter) of a triangle. The centroid is located at the average of the x-coordinates and the average of the y-coordinates of the vertices of the triangle.
Given the vertices \(\( A(-7, 2) \)\), \(\( B(9, 3) \)\), and \(\( C(-5, -5) \)\), we can determine the coordinates of the centroid using the following steps:
1. Calculate the average of the x-coordinates:
\(\[
\text{centroid}_x = \frac{x_A + x_B + x_C}{3}
\]\)
Substituting the given x-coordinates:
\(\[
\text{centroid}_x = \frac{-7 + 9 - 5}{3}
\]\)
Simplifying the expression:
\(\[
\text{centroid}_x = \frac{-3}{3} = -1.0
\]\)
2. Calculate the average of the y-coordinates:
\(\[
\text{centroid}_y = \frac{y_A + y_B + y_C}{3}
\]\)
Substituting the given y-coordinates:
\(\[
\text{centroid}_y = \frac{2 + 3 - 5}{3}
\]\)
Simplifying the expression:
\(\[
\text{centroid}_y = \frac{0}{3} = 0.0
\]\)
Therefore, the coordinates of the centroid of the triangle with vertices \(\( A(-7, 2) \)\), \(\( B(9, 3) \)\), and \(\( C(-5, -5) \)\) are \(\((-1.0, 0.0)\)\).
