Triangle XYZ

Question

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.(1 point)
centroid=
(
,
)

Answer 1

To find the centroid of triangle XYZ defined by the vertices \(X(0,0)\), \(Y(3,6)\), and \(Z(4,3)\), we can use the formula for the centroid \((G_x, G_y)\) of a triangle given its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):

\[ G_x = \frac{x_1 + x_2 + x_3}{3} \] \[ G_y = \frac{y_1 + y_2 + y_3}{3} \]

Substituting in the coordinates of the vertices:

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

Now we can calculate \(G_x\):

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

Next, we calculate \(G_y\):

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

Therefore, the centroid \(G\) of triangle XYZ is:

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

Thus, the centroid is:

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