Triangle ABC is drawn on a coordinate plane with vertices A(−2,−3) , B(4, 0) , and C(−2, 2) and medians indicating the midpoint of each of the line segments AB¯¯¯¯¯¯¯¯ , BC¯¯¯¯¯¯¯¯ , and CA¯¯¯¯¯¯¯¯ . Prove that the medians meet at a single point by finding the centroid. (1 point) Responses (0, 53) open paren 0 comma 5 thirds close paren (0,−13) open paren 0 comma negative 1 third close paren (0, 0) open paren 0 comma 0 close paren (−1, 53)

Answer 1

To find the centroid of triangle \( ABC \) with vertices \( A(-2, -3) \), \( B(4, 0) \), and \( C(-2, 2) \), we use the formula for the centroid \( G \) of a triangle, which is given by:

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

Here, the coordinates of the vertices are:

• \( A(x_1, y_1) = (-2, -3) \)
• \( B(x_2, y_2) = (4, 0) \)
• \( C(x_3, y_3) = (-2, 2) \)

Now we can substitute these values into the centroid formula:

\[ G_x = \frac{-2 + 4 - 2}{3} = \frac{0}{3} = 0 \]

\[ G_y = \frac{-3 + 0 + 2}{3} = \frac{-1}{3} \]

Thus, the coordinates of the centroid \( G \) are:

\[ G\left(0, -\frac{1}{3}\right) \]

Therefore, the medians of triangle \( ABC \) meet at the centroid \( G(0, -\frac{1}{3}) \), proving that they intersect at a single point.

In terms of the given choices, the answer is: \[ (0, -\frac{1}{3}) \]

Hence, the correct response is: (0, −1/3)