Question

Triangle ABC is drawn on a coordinate plane with vertices A (-2, - 3)
, B (1, 0), and C (-2, 2) and medians indicating the midpoint of each of the line segments AB, BC', and C'A . Prove that the medians meet at a single point by finding the centroid. (1 point)
O (0, 0)
•（0，5/3 ）
0 (0,-1/3 )
〇（-1，5/3）

Answer 1

Whats the answer based on the given options

Answer 2

The centroid of a triangle can be found by averaging the coordinates of its vertices. For triangle ABC with vertices \( A(-2, -3) \), \( B(1, 0) \), and \( C(-2, 2) \), we can calculate the centroid \( G(x, y) \) as follows:

\[
G_x = \frac{x_A + x_B + x_C}{3}
\]
\[
G_y = \frac{y_A + y_B + y_C}{3}
\]

Calculating the \( x \)-coordinate of the centroid:

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

Now calculating the \( y \)-coordinate of the centroid:

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

Combining these, we find the coordinates of the centroid \( G \):

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

Thus, the centroid of triangle ABC is at the point \( (-1, -\frac{1}{3}) \), and all three medians of the triangle meet at this single point.

Since the options given do not include \( (-1, -\frac{1}{3}) \), the options seem to be incorrect based on the calculation of the centroid. The correct answer is \( (-1, -\frac{1}{3}) \).

Answer 3

Given the options you provided:

1. \( (0, 0) \)
2. \( (0, \frac{5}{3}) \)
3. \( (0, -\frac{1}{3}) \)
4. \( (-1, \frac{5}{3}) \)

From our previous calculation, we found that the centroid of triangle ABC is at \( (-1, -\frac{1}{3}) \).

Upon reviewing the provided options against the calculated centroid, the option that most closely resembles our result is:

**None of the options match \( (-1, -\frac{1}{3}) \) exactly.**

If you must select from the given options, the closest correct interpretation would be option **3** \( (0, -\frac{1}{3}) \) purely for the \( y \)-value, but this does not match the \( x \)-value. The correct centroid is \( (-1, -\frac{1}{3}) \).

If you are tasked strictly with selecting among the choices, there is no exact match to the calculated centroid.