Prove that when the three medians of the triangle are drawn, they meet at a single point. What is the point of the centroid?

Question

(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:

$$G(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})$$

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.