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

centroid = (_,_)

1 answer

To find the centroid of triangle \( XYZ \) with vertices \( X(0,0) \), \( Y(3,6) \), and \( Z(4,3) \), we can use the formula for the centroid \( G \) of a triangle, which is given by the coordinates:

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

where \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are the coordinates of the vertices \( X \), \( Y \), and \( Z \).

Substituting the coordinates of points \( X \), \( Y \), and \( Z \):

  • \( (x_1, y_1) = (0, 0) \)
  • \( (x_2, y_2) = (3, 6) \)
  • \( (x_3, y_3) = (4, 3) \)

We calculate the \( x \)-coordinate of the centroid:

\[ x = \frac{0 + 3 + 4}{3} = \frac{7}{3} \]

Next, we calculate the \( y \)-coordinate of the centroid:

\[ y = \frac{0 + 6 + 3}{3} = \frac{9}{3} = 3 \]

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

\[ G\left( \frac{7}{3}, 3 \right) \]

So the centroid of triangle \( XYZ \) is:

\[ \text{centroid} = \left( \frac{7}{3}, 3 \right) \]