Question

Triangle 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.

Answer 1

To find the centroid of triangle ABC with vertices A(-3, -3), B(0, 6), and C(4, -3), 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 \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are the coordinates of the vertices of the triangle.

Let's identify the coordinates of each vertex:
- \(A = (-3, -3)\)
- \(B = (0, 6)\)
- \(C = (4, -3)\)

Next, we substitute the coordinates into the centroid formula:

1. Calculate the \(x\)-coordinate of the centroid:
\[
x_G = \frac{-3 + 0 + 4}{3} = \frac{1}{3}
\]

2. Calculate the \(y\)-coordinate of the centroid:
\[
y_G = \frac{-3 + 6 - 3}{3} = \frac{0}{3} = 0
\]

Therefore, the coordinates of the centroid \(G\) of triangle ABC are:
\[
G\left( \frac{1}{3}, 0 \right)
\]

Now we have shown that the centroid (the point where the medians meet) is at the coordinates \(\left( \frac{1}{3}, 0 \right)\). This proves that the medians of triangle ABC meet at a single point, which is the centroid \(G\).