The solutions for the given triangle are:
GE = 10/3, BE = 5/3
GF = 32/3, CF = 16/3
AG = 10, GD = 20
GC = 3X/2, CF = X/2
X = 7, AD = 43
If BG=5, find GE and BE:
Since G is the centroid of the triangle, it divides each median (BG, CG, and AG) in a 2:1 ratio.
Therefore, GE = 2/3 * BG = 2/3 * 5 = 10/3 and
BE = BG - GE = 5 - 10/3 = 5/3.
If CG=16, find GF and CF:
Similarly, GF = 2/3 * CG = 2/3 * 16 = 32/3 and
CF = CG - GF = 16 - 32/3 = 16/3.
If AD=30, find AG and GD:
AG = 1/3 * AD = 1/3 * 30 = 10 and
GD = AD - AG = 30 - 10 = 20.
If GF=X, find GC and CF:
Since GF = 2/3 * GC, X = 2/3 * GC.
Solving for GC, we get GC = 3X/2.
Therefore, CF = CG - GF = 3X/2 - X = X/2.
If AG=9 =X, GD=5X-1, find X and AD:
We are given that AG = 9 and GD = 5X-1.
We also know that AG + GD = AD.
Substituting the first two equations, we get 9 + 5X-1 = AD.
Simplifying the equation, we get AD = 5X + 8.
Since AG = 1/3 * AD, we can also substitute the first equation to get 9 = 1/3 * (5X + 8).
Solving for X, we get X = 7.
Therefore, AD = 5 * 7 + 8 = 43.