Question

triangle ABC is drawn on a coordinate plane with vertices A (1,3), B (6,6) and C (3,1) and medians indicating the midpoint of each lines AB, BC, and CA. Prove that the medians meet at a single point by finding the centroid. Express all result in fractions.

centroid = ( _, _)

Answer 1

To find the centroid of triangle ABC with vertices A(1, 3), B(6, 6), and C(3, 1), we can use the formula for the centroid \( G \) of a triangle given its vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \). The coordinates of the centroid are calculated as follows:

\[
G_x = \frac{x_1 + x_2 + x_3}{3}
\]
\[
G_y = \frac{y_1 + y_2 + y_3}{3}
\]

Plugging in the coordinates of the vertices:

- \( A(1, 3) \) so \( x_1 = 1 \) and \( y_1 = 3 \)
- \( B(6, 6) \) so \( x_2 = 6 \) and \( y_2 = 6 \)
- \( C(3, 1) \) so \( x_3 = 3 \) and \( y_3 = 1 \)

Now, calculate \( G_x \):

\[
G_x = \frac{1 + 6 + 3}{3} = \frac{10}{3}
\]

Next, calculate \( G_y \):

\[
G_y = \frac{3 + 6 + 1}{3} = \frac{10}{3}
\]

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

\[
G = \left( \frac{10}{3}, \frac{10}{3} \right)
\]

This shows that the medians of triangle ABC meet at the single point, which is the centroid. Therefore, the final answer is:

\[
\text{centroid} = \left( \frac{10}{3}, \frac{10}{3} \right)
\]