Use the image to answer the question A(2,3) B(-3,-1) C(2,-1). When the three medians of the triangle are drawn, they meet at a single point. What is the point of the centroid? Leave all results in fractions. (1 point) centroid= ( ) ( )

Answer 1

To find the centroid of a triangle with vertices A(2, 3), B(-3, -1), and C(2, -1), we use the formula for the centroid (G) which is given by:

\[ G\left(x, y\right) = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]

where \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are the coordinates of the vertices of the triangle.

Substituting the coordinates of points A, B, and C:

\[ x_1 = 2, \quad y_1 = 3 \] \[ x_2 = -3, \quad y_2 = -1 \] \[ x_3 = 2, \quad y_3 = -1 \]

Now, substituting these values into the centroid formula:

\[ G\left(x, y\right) = \left(\frac{2 + (-3) + 2}{3}, \frac{3 + (-1) + (-1)}{3}\right) \]

Calculating the x-coordinate:

\[ x = \frac{2 - 3 + 2}{3} = \frac{1}{3} \]

Calculating the y-coordinate:

\[ y = \frac{3 - 1 - 1}{3} = \frac{1}{3} \]

Therefore, the coordinates of the centroid G are:

\[ \boxed{\left(\frac{1}{3}, \frac{1}{3}\right)} \]