AABC 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.

Question

GPT-4o mini · Answer

To find the centroid of triangle $AABC$, we can use the formula for the centroid, which is given by $$ G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) $$ where $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$ are the vertices of the triangle. Given the vertices: - $A(1, 3)$ - $B(6, 6)$ - $C(3, 1)$ We can identify the coordinates of the points: - $x_1 = 1$, $y_1 = 3$ - $x_2 = 6$, $y_2 = 6$ - $x_3 = 3$, $y_3 = 1$ Now, we calculate the coordinates of the centroid $G$: 1. **Calculating $G_x$**: $$ G_x = \frac{x_1 + x_2 + x_3}{3} = \frac{1 + 6 + 3}{3} = \frac{10}{3} $$ 2. **Calculating $G_y$**: $$ G_y = \frac{y_1 + y_2 + y_3}{3} = \frac{3 + 6 + 1}{3} = \frac{10}{3} $$ Therefore, the coordinates of the centroid $G$ are: $$ G\left(\frac{10}{3}, \frac{10}{3}\right) $$ ### Conclusion The centroid $G$ of triangle $AABC$ is $\left(\frac{10}{3}, \frac{10}{3}\right)$. Since the centroid is the point where all three medians of the triangle intersect, we have thus shown that the medians of triangle $AABC$ meet at the single point $G$ with coordinates $\left(\frac{10}{3}, \frac{10}{3}\right)$. This confirms that the medians indeed meet at a single point.