Triangle ABC is drawn on a coordinate plane with vertices A (1, 4), B (9,1), 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. (1 point)

To find the centroid of triangle ABC with vertices \( A(1, 4) \), \( B(9, 1) \), and \( C(3, -1) \), we can use the formula for the centroid \( G \) of a triangle given its vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \):

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

Here, the coordinates of the vertices are:

• \( A(1, 4) \): \( x_1 = 1, y_1 = 4 \)
• \( B(9, 1) \): \( x_2 = 9, y_2 = 1 \)
• \( C(3, -1) \): \( x_3 = 3, y_3 = -1 \)

Now let's calculate the coordinates of the centroid \( G \):

1. Calculate \( x \)-coordinate of \( G \): \[ x_G = \frac{1 + 9 + 3}{3} = \frac{13}{3} \]

2. Calculate \( y \)-coordinate of \( G \): \[ y_G = \frac{4 + 1 - 1}{3} = \frac{4}{3} \]

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

\[ G\left(\frac{13}{3}, \frac{4}{3}\right) \]

This point represents the meeting point of the medians of triangle ABC, and it shows that the medians do indeed meet at a single point.

Finally, we conclude that:

\[ \text{centroid} = \left( \frac{13}{3}, \frac{4}{3} \right) \]

Let’s recalculate the \( x \)-coordinate of the centroid \( G \) using the vertices of triangle ABC, which are:

• \( A(1, 4) \)
• \( B(9, 1) \)
• \( C(3, -1) \)

The formula for the coordinates of the centroid \( G \) is:

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

We can find the \( x \)-coordinate:

1. Calculate \( x_G \):

\[ x_G = \frac{x_1 + x_2 + x_3}{3} = \frac{1 + 9 + 3}{3} = \frac{13}{3} \]

Next, let’s check the \( y \)-coordinate to confirm its correctness:

2. Calculate \( y_G \):

\[ y_G = \frac{y_1 + y_2 + y_3}{3} = \frac{4 + 1 - 1}{3} = \frac{4}{3} \]

All calculations are correct.

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

\[ G\left( \frac{13}{3}, \frac{4}{3} \right) \]

If you believe \( \frac{13}{3} \) is incorrect, please provide more context or clarify where the error may be originating from, as the calculation follows the standard procedure.