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

Answer 1

To show that the medians of triangle ABC meet at a single point, we will find the centroid of the triangle. The centroid (G) of a triangle with vertices A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃) can be calculated using the following formulas:

\[
G_x = \frac{x_1 + x_2 + x_3}{3}
\]
\[
G_y = \frac{y_1 + y_2 + y_3}{3}
\]

Given the vertices of triangle ABC:
- A(−2,−3) → (x₁, y₁) = (−2, −3)
- B(4, 0) → (x₂, y₂) = (4, 0)
- C(−2, 2) → (x₃, y₃) = (−2, 2)

Now we can substitute the coordinates into the formulas:

1. Calculate \(G_x\):
\[
G_x = \frac{-2 + 4 - 2}{3} = \frac{0}{3} = 0
\]

2. Calculate \(G_y\):
\[
G_y = \frac{-3 + 0 + 2}{3} = \frac{-1}{3}
\]

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

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

Therefore, the medians of triangle ABC indeed meet at a single point, the centroid \((0, -\frac{1}{3})\).

Among the given response options, the correct answer is:
\[
(0, -\frac{1}{3}) \text{ (0, negative 1 third)}
\]