A centroid is the point of intersection of the 3 medians of a triangle. Each median connects the the midpoint of each side to the vertex opposite it. This means that lines AD, BE and CF are medians of the triangle.
Recall, a centroid divides each median in the ratio of 2:3 or 2/3
Considering median AD, it means that
DG/AG = 2/3
From the diagram,
DG = x - 15
AG = x + 7
Thus,
(x - 15)/(x + 7) = 2/3
By crossmultiplying, we have
3(x - 15) = 2(x + 7)
We would open the parentheses by multiplying the terms inside it by the the term outside. Thus, we have
3x - 45 = 2x + 14
3x - 2x = 14 + 45 = 59
x = 59
The length of segment DG = x - 15 = 59 - 15 = 44
The length of segment AG = x + 7 = 59 + 7 = 66
The length of segment AD = DG + AG = 44 + 66 = 110
