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.

A. (−1, 53)
B. (0,−13)
C. (0, 0)
D. (0, 53)

Answer 1

To find the centroid of triangle ABC with vertices A(−2,−3), B(4, 0), and C(−2, 2), we use the formula for the centroid \( G \), which is given by:

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

where \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are the coordinates of the vertices of the triangle.

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

- \( x_1 = -2 \)
- \( y_1 = -3 \)
- \( x_2 = 4 \)
- \( y_2 = 0 \)
- \( x_3 = -2 \)
- \( y_3 = 2 \)

Now we calculate the centroid:

1. Calculate \( x \) coordinate of \( G \):
\[
x = \frac{-2 + 4 - 2}{3} = \frac{0}{3} = 0
\]

2. Calculate \( y \) coordinate of \( G \):
\[
y = \frac{-3 + 0 + 2}{3} = \frac{-1}{3} = -\frac{1}{3}
\]

So the coordinates of the centroid \( G \) are:

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

Examining the options given:

A. (−1, 53)

B. (0,−13)

C. (0, 0)

D. (0, 53)

None of the options provided match the computed centroid coordinate \( G(0, -\frac{1}{3}) \). Therefore, we can conclude that while the centroid calculation confirms that the medians meet at a single point, none of the provided options represent that point accurately.