Question

Triangle ABC is drawn on a coordinate plane with vertices A(−3,−3) , B(0,6) , and C(4,−3) and medians indicating the midpoint of each of the lines AB , BC , and CA . Prove the medians meet at a single point by finding the centroid.(1 point) Responses (13,0) left parenthesis Start Fraction 1 over 3 End Fraction comma 0 right parenthesis (13,1) left parenthesis Start Fraction 1 over 3 End Fraction comma 1 right parenthesis (0,13) left parenthesis 0 comma Start Fraction 1 over 3 End Fraction right parenthesis (16,3) left parenthesis Start Fraction 1 over 6 End Fraction comma 3 right parenthesis

Answer 1

To find the centroid of triangle ABC, we use the formula for the centroid \( G \) of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \):

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

Given the vertices:
- \( A(-3, -3) \)
- \( B(0, 6) \)
- \( C(4, -3) \)

Substituting in the coordinates:

\[
G\left( \frac{-3 + 0 + 4}{3}, \frac{-3 + 6 - 3}{3} \right)
\]

Calculating the x-coordinate of the centroid:

\[
\frac{-3 + 0 + 4}{3} = \frac{1}{3}
\]

Calculating the y-coordinate of the centroid:

\[
\frac{-3 + 6 - 3}{3} = \frac{0}{3} = 0
\]

Thus, the coordinates of the centroid are:

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

This shows that all three medians intersect at the centroid, confirming that they meet at a single point.

**Final Answer:** \(\left( \frac{1}{3}, 0 \right)\)