Question

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,−13)
open paren 0 comma negative 1 third close paren

(0, 0)
open paren 0 comma 0 close paren

(−1, 53)
open paren negative 1 comma 5 thirds close paren

(0, 53)
open paren 0 comma 5 thirds close paren

Answer 1

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

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

Substituting in the coordinates of the vertices:

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

Now, we calculate:

1. \(x\)-coordinate of the centroid:
\[
\frac{-2 + 4 - 2}{3} = \frac{0}{3} = 0
\]

2. \(y\)-coordinate of the centroid:
\[
\frac{-3 + 0 + 2}{3} = \frac{-1}{3}
\]

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

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

Therefore, the centroid of the triangle \(ABC\) is at the point \((0, -\frac{1}{3})\).

Among the provided responses, the correct one is:

**(0, −1/3)**