Solve for centroid

The centroid is the point where the three medians intersect.

 

If you find the point of intersection of two medians, the third median will also pass through this point.

 

A median of a triangle is the line that passes through one vertex and the middle point of the opposite side.

 

We will need to find the equations of two of the medians.

First, let's find the equation of the median that passes through point C and the middle of line segment DE.

The midpoint of DE is  ( [6 + -4]/2 , [2 + -2] / 2 )  =  (1, 0).

The slope of the line that passes through C(-2,4) and (1,0) is:  (0 - 4) / (1 - -2)  =  -4/3.

The equation of the line is:  y - 4  =  (-4/3)(x - -2)     --->     y - 4  =  (-4/3)(x + 2)

     --->     3y - 12  =  -4x - 8     --->     4x + 3y  =  4

 

Now, let's find the equation of the midian that passes through point E and the middle of line segment CD.

The midpoint of CD is  ( [-2 + 6]/2, [4 + 2]/2 )  =  (2,3).

The slope of the line that passes through E(-4,-2) and (2,3) is:  (3 - -2)/(2 - -4)  =  5/6.

The equation of the line is:  y - -2  =  (5/6)(x - -4)     --->     y + 2  =  (5/6)(x + 4)

     --->     6y + 12  =  5x + 20     --->     -5x + 6y  =  8     --->     5x - 6y  =  -8

 

Finally, we need to find the point where these two lines intersect:

4x + 3y  =  4         --->  x 2   --->     8x + 6y  =  8

5x - 6y  =  -8                                   5x -  6y  =  -8

Adding down:                                        13x  =  0     --->     x  =  0

If  x = 0:    4(0) + 3y  =  4     --->     0 + 3y  =  4     --->     y  =  4/3

 

So, the coordinates of the centroid are:  (0, 4/3)