Question

Triangle XYZ is drawn on a coordinate plane with vertices X (0, 0), Y (3, 6), and Z (4, 3) and with medians indicating the midpoint of each line XY, YZ, and ZX. Prove that the medians meet at a single point by finding the centroid.

Answer 1

To prove that the medians of triangle XYZ meet at a single point, we find the centroid of the triangle. The centroid of a triangle with vertices at coordinates \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) can be calculated using the formula:

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

For triangle XYZ, the vertices are:
- \( X(0, 0) \)
- \( Y(3, 6) \)
- \( Z(4, 3) \)

Now we will substitute these coordinates into the formula:

1. Calculate the x-coordinate of the centroid:
\[
x_G = \frac{0 + 3 + 4}{3} = \frac{7}{3}
\]

2. Calculate the y-coordinate of the centroid:
\[
y_G = \frac{0 + 6 + 3}{3} = \frac{9}{3} = 3
\]

Thus, the coordinates of the centroid \( G \) are:
\[
G\left(\frac{7}{3}, 3\right)
\]

This point \( G \) is the point where the three medians of the triangle intersect.

### Conclusion:
We've calculated the centroid of triangle XYZ to be \( G\left(\frac{7}{3}, 3\right) \). Since the centroid is the common intersection point of the medians in any triangle, we have proved that the medians of triangle XYZ meet at a single point.