Question

Prove that when the three medians of the triangle are drawn, they meet at a single point. What is the point of the centroid?
(1 point)
O (-1,1)
( -1/2 , 1/2)
( -1/3 1/3)
• (-3,1)
The triangle points are
A -5 1
B 1,7
C 1,-5

Answer 1

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 \( G \):

1. Calculate the x-coordinate of \( G \):

\[
x_G = \frac{-5 + 1 + 1}{3} = \frac{-5 + 2}{3} = \frac{-3}{3} = -1
\]

2. 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.