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, 0)
open paren 0 comma 0 close paren

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

(0, 53)
open paren 0 comma 5 thirds 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 can use the formula for the coordinates of the centroid \( G \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \):

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

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

- \( x_1 = -2 \)
- \( y_1 = -3 \)
- \( x_2 = 4 \)
- \( y_2 = 0 \)
- \( x_3 = -2 \)
- \( y_3 = 2 \)

Now, calculate the coordinates of the centroid:

1. Calculate the x-coordinate of the centroid:

\[
x = \frac{-2 + 4 + (-2)}{3} = \frac{0}{3} = 0
\]

2. Calculate the y-coordinate of the centroid:

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

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

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

This matches the option:

**(0, -\( \frac{1}{3} \))**

Thus, the medians of triangle ABC meet at a single point, which is the centroid \( G(0, -\frac{1}{3}) \).