Use the following diagram for questions 21-24 in which G is the centroid of AABC, FC=35, AG=42, BF=57, and DG=14. Round to the nearest tenth, if necessary.
Find AC
Find BG
Find GC
Find AE

AC = 70

BG = 38

GC = 28

AE = 63

Step-by-step explanation:

The given parameters are;

The centroid of ΔABC = G

The length of FC = 35

The length of AG = 42

The length of BF = 57

The length of DG = 14

Given that the centroid is the intersection of the medians, we have;

FA = FC = 35 Definition of median line FB, (point F is the midpoint of AC)

AC = FA + FC = 35 + 35 = 70 by segment addition postulate

AC = 70

BG = 2/3 × BF = 2/3 × 57 = 38 Definition of median line BF

BG = 38

DG = 14 = 1/3 × DC Definition of median line

∴ DC = 3 × DG = 3 × 14 = 42

GC = 2/3 × DC = 2/3 × 42 = 28 Definition of median line DC

GC = 28

AG = 42 = 2/3 × AE Definition of median line AE

∴ AE = 3/2 × 42 = 63

AE = 63.