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

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

Transcript text: $\triangle A B C$ is drawn on a coordinate plane with vertices $A(1,3), B(6,6)$, and $C(3,1)$ and with medians indicating the midpoint of each of the lines $A B, B C$, and $C A$. Prove that the medians meet at a single point by finding the centroid. Express all results in fractions.

Solution

Solution Steps

To find the centroid of a triangle given its vertices, we can use the formula for the centroid, which is the average of the x-coordinates and the y-coordinates of the vertices. The centroid $ G $ of a triangle with vertices $ A(x_1, y_1) $, $ B(x_2, y_2) $, and $ C(x_3, y_3) $ is given by:

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

Step 1: Calculate the Centroid Coordinates

To find the centroid $ G $ of triangle $ ABC $ with vertices $ A(1, 3) $, $ B(6, 6) $, and $ C(3, 1) $, we use the formula:

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

Substituting the coordinates of the vertices:

\[ G_x = \frac{1 + 6 + 3}{3} = \frac{10}{3} \quad \text{and} \quad G_y = \frac{3 + 6 + 1}{3} = \frac{10}{3} \]

Step 2: Express the Centroid

Thus, the coordinates of the centroid $ G $ are:

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