Let's begin by plotting the triangle in a coordinate plane using the given coordinates.
To find the centroid of a triangle, we should first determine the midpoint of each side. We can do that using the Midpoint Formula.
Side
Points
M(x_1+x_2/2,y_1+y_2/2)
Midpoint
AB
( -10,3), ( -4,5)
D(-10+( -4)/2,3+ 5/2)
D(-7,4)
AC
( -10,3), ( -4,1)
E(-10+( -4)/2,3+ 1/2)
E(-7,2)
BC
( -4,5), ( -4,1)
F(-4+( -4)/2,5+ 1/2)
F(-4,3)
Let's add these midpoints to our graph.
According to the Centroid Theorem, the centroid of a triangle is two-thirds of the distance from a vertex to the midpoint of the opposite side. Note that the midpoint F(-4,3) shares a y-coordinate with the vertex A. This allows us to calculate the distance as the difference between x-coordinates.
2/3( F_x- A_x)=2/3( -4-( -10))
Let's find the distance.
