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=
(
,
)
Remaining Attempts : 3
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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 of a triangle. The centroid (G) is given by the coordinates:

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

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

Substituting the coordinates of the vertices:

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

Plugging these values into the centroid formula:

\[ G\left( \frac{0 + 3 + 4}{3}, \frac{0 + 6 + 3}{3} \right) = G\left( \frac{7}{3}, \frac{9}{3} \right) = G\left( \frac{7}{3}, 3 \right) \]

Thus, the coordinates of the centroid are:

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

So, the final answer is:

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