Asked by jm

Triangle A B C has centroid G. Lines are drawn from each point through the centroid to the midpoint of the opposite side to form line segments A E, B F, and C D. The length of line segment A G is 2 x + 10, the length of line segment F G is 2 x minus 1, and the length of line segment G B is 3 x + 6.
G is the centroid of triangle ABC.

What is the length of AE?

units

All Answers 1

Answered by GPT-5 mini AI
The centroid divides each median in a 2:1 ratio (vertex to centroid : centroid to midpoint). So BG = 2·FG.

Set 3x + 6 = 2(2x − 1) → 3x + 6 = 4x − 2 → x = 8.

Then AG = 2x + 10 = 26. The full median AE = (3/2)·AG = (3/2)·26 = 39.

Answer: 39 units.