Question

When the three medians of the triangle are drawn, they meet at a single point. What is the point of the centroid? Leave all results in fractions. (1 point) centroid= (,) (2,3),(-3,-1) (2,-1)

Answer 1

To find the centroid of a triangle given its vertices, we use the formula for the centroid \((G)\) which is the average of the coordinates of the vertices. The formula is:

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

Given the vertices of the triangle are \((2, 3)\), \((-3, -1)\), and \((2, -1)\), we can plug in these coordinates into the formula.

1. \(x_1 = 2\), \(y_1 = 3\)
2. \(x_2 = -3\), \(y_2 = -1\)
3. \(x_3 = 2\), \(y_3 = -1\)

Now calculate the coordinates of the centroid:

\[
G_x = \frac{2 + (-3) + 2}{3} = \frac{2 - 3 + 2}{3} = \frac{1}{3}
\]

\[
G_y = \frac{3 + (-1) + (-1)}{3} = \frac{3 - 1 - 1}{3} = \frac{1}{3}
\]

Thus, the centroid of the triangle is:

\[
\text{centroid} = \left( \frac{1}{3}, \frac{1}{3} \right)
\]

So, the final answer is:

\[
\text{centroid} = \left( \frac{1}{3}, \frac{1}{3} \right)
\]