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)

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

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

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

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

(0,−13)

GPT-4o mini · Answer

To find the centroid of triangle ABC, we can use the formula for the centroid \((G)\) of a triangle with vertices \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\):

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

Given the vertices:

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

Now we can substitute the coordinates into the formula.

Find \(x_G\):

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

Find \(y_G\):

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

So the centroid \(G\) is:

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

Given the responses provided, we can see that:

\((0, -\frac{1}{3})\) does not match any of the provided options, but based on the given points and considering a mistake in the responses, please double-check the given choices to ensure they are correctly reflected or provided. However, the computed centroid represents that all three medians of triangle ABC intersect at the point \(G(0, -\frac{1}{3})\).