Question

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, 53)
open paren 0 comma 5 thirds close paren

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

(−1, 53)

Answer 1

To find the centroid of triangle ABC, we can use the formula for the centroid (G) of a triangle with vertices A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃):

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

Given the vertices of triangle ABC:
- A(-2, -3)
- B(4, 0)
- C(-2, 2)

Let's substitute these coordinates into the centroid formula.

1. Calculate the x-coordinate:
\[
\frac{-2 + 4 - 2}{3} = \frac{0}{3} = 0
\]

2. Calculate the y-coordinate:
\[
\frac{-3 + 0 + 2}{3} = \frac{-1}{3}
\]

Thus, the coordinates of the centroid G are:
\[
G\left(0, -\frac{1}{3}\right)
\]

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

Among the provided responses, the correct one is:

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