To prove that the medians of triangle XYZ meet at a single point, we will find the coordinates of the centroid of triangle XYZ using the vertices \(X(x_0, 0)\), \(Y(3, 6)\), and \(Z(4, 3)\).
The formula for the centroid \(G\) of a triangle with vertices \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\) is given by:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]
Substituting in our vertices:
- \(X(x_0, 0)\)
- \(Y(3, 6)\)
- \(Z(4, 3)\)
We can now calculate the coordinates of the centroid \(G\):
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Calculate the \(x\)-coordinate of the centroid: \[ G_x = \frac{x_0 + 3 + 4}{3} = \frac{x_0 + 7}{3} \]
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Calculate the \(y\)-coordinate of the centroid: \[ G_y = \frac{0 + 6 + 3}{3} = \frac{9}{3} = 3 \]
Thus, the coordinates of the centroid \(G\) are: \[ G\left(\frac{x_0 + 7}{3}, 3\right) \]
The centroid, being the intersection point of the medians, proves that the medians of triangle XYZ meet at this single point \(G\).
To summarize, we have shown that regardless of the \(x_0\) value (the x-coordinate of vertex X), the centroid \(G\) is determined by the coordinates calculated above. Therefore, the medians of triangle XYZ intersect at the centroid \(G\left(\frac{x_0 + 7}{3}, 3\right)\), confirming that they meet at a single point.
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