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

centroid = (_,_)

Answer 1

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:

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