Determine the lenght

Question

1. 

Answer 1

The centroid is the intersection of the medians, meaning Point G is the centroid. The Centroid Theorem says the centroid is located 2/3 of the distance from each vertex. So BG is 2/3 of the length of BE and GE is 1/3 of the length of BE or, in other words BG is twice as long as GE. So:

2(GE) = BG

2(31 + 2x) = (-4x - 2)

62 + 4x = -4x -2

8x = -84

x = -8

Since DF = (-3x + 6) and x = -8 then DF = -3(-8) + 6 = 24 + 6 = 30

Then DG is 2/3(30) = 20

Answer 2

The centroid at G is 2/3 of the distance from the vertex point B to the midpoint of CD at point E.

BG is 2/3 of the length of BE, and is thus twice the length of GE, which is the remaining 1/3.

so

BE = 2*GE

-4x - 2 = 2(31 + 2x)

-4x - 2 = 62 + 4x

-64 = 8x

x = -8

DG = -3x + 6 = -3(-8) + 6 = 30