To show that the three medians of a triangle meet at a single point, we must demonstrate that they intersect at the centroid of the triangle.
Centroid of a Triangle
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 the formula:
\[ G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]
Given Vertices
Let's use the vertices provided:
- \( A(-5, 1) \)
- \( B(1, 7) \)
- \( C(1, -5) \)
Calculating the Centroid
Now we can calculate the coordinates of the centroid \( G \):
- Calculate the x-coordinate of \( G \):
\[ x_G = \frac{-5 + 1 + 1}{3} = \frac{-5 + 2}{3} = \frac{-3}{3} = -1 \]
- Calculate the y-coordinate of \( G \):
\[ y_G = \frac{1 + 7 - 5}{3} = \frac{3}{3} = 1 \]
Conclusion
The coordinates of the centroid \( G \) are:
\[ G(-1, 1) \]
Answer to Multiple Choice
Among the given options, the correct one corresponding to the centroid is:
\[ O(-1, 1) \]
Thus, we conclude that the three medians of the triangle meet at the point \((-1, 1)\), which is the centroid.