traingle 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.

Question

a. (0,1/3)
b. (1/6,3)
c.(1/3,1
d(1/3,0)

Answer 1

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

The coordinates of the centroid G are given by:

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

Given the vertices of triangle ABC:

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

We can plug the coordinates of the vertices into the centroid formula.

1. Calculate \( x \)-coordinate of G:

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

2. Calculate \( y \)-coordinate of G:

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

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

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

The centroid \( G \) is at the point \( \left( \frac{1}{3}, 0 \right) \).

From the given options, the correct answer is:

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

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