G is the centroid of triangle ABC and D and E are the midpoints of line AB and line AC, respectively. line AG and line DE intersect at M. Find GM/GA.

Guest Feb 9, 2020

#2**0 **

The centroid of a triangle is found 2/3^{rds} of the way from the vertex to the opposite side.

Since D and E are midpoints of two sides of a triangle, they create a line parallel to the base of the triangle (BC) and cut all transversals at their midpoints. Therefore, AM is 1/2 the way from the vertex to the opposite side.

This means that MG is (2/3 - 1/2) = 1/6^{th} the distance.

GM/GA = (1/6) / (2/3) = 1/4

geno3141 Feb 10, 2020