Explanation:
the point, where all medians intersect (G), is called centroid.
we use the centroid theorem :
the centroid of the triangle is at 2/3 of the distance (median) from the vertex to the opposite mid-point of the sides.
e.g. GE = 2/3 × BE
and that means the second (shorter) part of the median is the remaining 1/3 of the whole median.
1.
BG = 5 = 1/3 × BE
BE = 5×3 = 15
GE = 2/3 × BE = 2/3 × 15 = 10
2.
CG = 16 = 2/3 × CF
CF = 16×3/2 = 8×3 = 24
GF = 1/3 × CF = 24/3 = 8
3.
AD = 30
AG = 2/3 × AD = 2/3 × 30 = 20
GD = 1/3 × AD = 1/3 × 30 = 10
4.
GF = x = 1/3 × CF
CF = 3x
GC = 2/3 × CF = 2/3 × 3x = 2x
5.
AG = 9x = 2/3 × AD
GD = 5x - 1 = 1/3 × AD
therefore,
AG = 2 × GD
9x = 2(5x - 1) = 10x - 2
2 + 9x = 10x
2 = x
GD = 5x - 1 = 5×2 - 1 = 9 = 1/3 × AD
AD = 9×3 = 27