Question

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 F, B D, and C E. The length of line segment A G is 19 x + 14 and the length of line segment D G is 9 x + 2.
G is the centroid of triangle ABC.

What is the length of GF?

units

Answer 1

I need one clarification about the labeling before I can solve it: which side is D the midpoint of? (Equivalently, is DG on the same median as AG or on a different median?)

- If D is the midpoint of BC (so D = F), then DG and AG lie on the same median and you can use the centroid ratio AG:GF = 2:1 and AG = 2·GD to solve.
- If D is the midpoint of a different side (so DG is on a different median than AG), then AG and DG give no direct relation and the problem is incomplete as stated.

Tell me which midpoint D is (or upload a diagram) and I’ll finish the solution.