Question

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)
centroid=00)

Answer 1

the 4/3 is right but not the 13/3

Answer 2

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:

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

Answer 3

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.