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, 0) open paren 0 comma 0 close paren (−1, 53) open paren negative 1 comma 5 thirds close paren (0, 53)

Answer 1

To find the centroid of triangle ABC, we can use the formula for the coordinates of the centroid \( G \). The centroid is given by the average of the coordinates of the vertices of the triangle.

The coordinates of the vertices are:
- \( A(-2, -3) \)
- \( B(4, 0) \)
- \( C(-2, 2) \)

The formula for the coordinates of the centroid \( G(x, y) \) is:

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

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

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

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

Thus, the centroid \( G \) is:

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

Hence, among the responses you provided, the correct answer for the centroid of the triangle, indicating the point where the medians meet, is:

\[
(0, -\frac{1}{3}) \quad \text{or} \quad (0, -\frac{1}{3})
\]

Therefore, the centroid is \((0, -\frac{1}{3})\).