Triangle ABC

Question

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

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

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

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

Answer 1

To find the centroid of triangle ABC, we can apply the formula for the coordinates of the centroid (G). The centroid of a triangle with vertices \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\) is given by:

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

Given the vertices:

• \(A(-2, -3)\)
• \(B(4, 0)\)
• \(C(-2, 2)\)

We can calculate the x-coordinate of the centroid:

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

Now, we calculate the y-coordinate of the centroid:

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

So the coordinates of the centroid \(G\) are:

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

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

Among the given options, the correct response is:

open paren 0 comma negative 1 third close paren.