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)

Answer 1

To determine the centroid of triangle ABC, we use the formula for the centroid \((G)\) of a triangle with vertices at coordinates \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):

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

Given the vertices:
- \(A(-2, -3)\)
- \(B(4, 0)\)
- \(C(-2, 2)\)

Let's substitute these values into the formula:

**Step 1: Calculate the x-coordinate of the centroid.**

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

**Step 2: Calculate the y-coordinate of the centroid.**

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

Now we have the coordinates of the centroid \(G\):

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

**Final Answer**

The centroid \(G\) of triangle ABC is at the point \((0, -\frac{1}{3})\). Since this point was not among the provided options, please double-check the coordinates in the question or options given. However, the centroid we calculated is indeed \( \left( 0, -\frac{1}{3} \right) \).

Thus, medians intersect at a single point, the centroid.