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 (16,3) left parenthesis Start Fraction 1 over 6 End Fraction comma 3 right parenthesis (13,1) left parenthesis Start Fraction 1 over 3 End Fraction comma 1 right parenthesis (13,0) left parenthesis Start Fraction 1 over 3 End Fraction comma 0 right parenthesis (0,13)

Answer 1

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

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

Given the vertices:

• \( A(-3, -3) \)
• \( B(0, 6) \)
• \( C(4, -3) \)

We can substitute the coordinates into the formula:

1. Calculate the x-coordinate of the centroid:

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

2. Calculate the y-coordinate of the centroid:

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

So, the centroid \( G \) of triangle ABC is:

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

Thus, the centroid is given by the coordinates:

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

Therefore, the correct response corresponds to:

\((\frac{1}{3}, 0)\)