Question

Medians of a Triangle Quick Check 3 of 53 of 5 Items 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 (−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 Skip to navigation page 8 of 8

Answer 1

To find the centroid of triangle ABC with vertices A(-2, -3), B(4, 0), and C(-2, 2), we can use the formula for the centroid \( G(x, y) \):

\[
G(x, y) = \left( \frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3} \right)
\]

Now, substituting the coordinates of the vertices:

- \( x_A = -2 \)
- \( y_A = -3 \)
- \( x_B = 4 \)
- \( y_B = 0 \)
- \( x_C = -2 \)
- \( y_C = 2 \)

Calculating the x-coordinate of the centroid:

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

Calculating the y-coordinate of the centroid:

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

Thus, the coordinates of the centroid \( G \) are:

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

Now, looking at the options provided:

1. (0, -13)
2. (0, -\frac{1}{3})
3. (-1, \frac{5}{3})
4. (0, \frac{5}{3})
5. (0, 0)

The correct response is:

**(0, -\frac{1}{3})**.

This shows that the medians of triangle ABC meet at a single point, which is the centroid.