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

Answer 1

To find the centroid of triangle ABC, we use the formula for the centroid (G) of a triangle given by its vertices A(x1, y1), B(x2, y2), and C(x3, y3):

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

Given the vertices of triangle ABC:

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

We 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:

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

Thus, the centroid \( G \) of triangle ABC is given by:

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

This means the medians of triangle ABC meet at the point:

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

From the options provided, the correct response is:

\(\left( \frac{1}{3}, 0 \right)\).