The value of x is 7/6.
To find the value of x, we can utilize the given information about the lengths of the medians and segments within triangle ABC.
Using medians CE and GC:
Since G is the centroid of triangle ABC, it divides each median into two segments in a ratio of 2:1. Therefore, we can set up two equations based on the given lengths:
CE = 2/3 * GC
6x = 2/3 * (3x + 7)
Solving for x, we get:
6x = 2x + 14/3
4x = 14/3
x = 7/6
Using medians BG and FG:
Similarly, we can set up two equations based on the given lengths of medians BG and FG:
FG = 2/3 * BG
x + 8 = 2/3 * (5x - 1)
Solving for x, we get:
x + 8 = 10x/3 - 2/3
8/3 = 9x/3
x = 8/9
Using medians DG and AG:
Following the same approach, we can set up two equations based on the given lengths of medians DG and AG:
AG = 2/3 * DG
9x - 6 = 2/3 * (4x - 5)
Solving for x, we get:
9x - 6 = 8x/3 - 10/3
x - 6 = 8x/3 - 10/3
-5 = 5x/3
x = -3
Comparing the values obtained from each set of equations, we find that x = 7/6 is consistent across all three sets. Therefore, the value of x is 7/6.