Question

Triangle ABC is drawn on a coordinate plane with vertices A (-3,-3), B (0,6), and C (4,-3) and medians indicating with midpoint of each of the lines AB, BC, and CA. Prove the medians meet at a single point by finding the centroid.

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, which is the average of the coordinates of the vertices. The vertices of triangle ABC are given as:

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

The formula for the centroid (G) of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is:

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

Substituting the coordinates of A, B, and C into the formula:

\[
x_1 = -3, \quad y_1 = -3
\]
\[
x_2 = 0, \quad y_2 = 6
\]
\[
x_3 = 4, \quad y_3 = -3
\]

Calculating the x-coordinate of the centroid:

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

Calculating the y-coordinate of the centroid:

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

Thus, the coordinates of the centroid G are:

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

This matches option (d):

**d. (1/3, 0)**

Therefore, the medians of triangle ABC meet at a single point, which is the centroid at \((\frac{1}{3}, 0)\).