Question

in triangle abc, the segments drawn from the vertices intersect at point g. segment fg measures 6 cm, and segment fc measures 18 cm. which best explains whether point g can be the centroid? point g cannot be the centroid because 18:6 does not equal 2:1. point g cannot be the centroid because fg should be longer than cg. point g can be the centroid because 12:6 equals 2:1. point g can be the centroid because fc is longer than fg.

Explanation:

Step1: Recall centroid property

The centroid of a triangle divides each median in a ratio of 2:1, where the longer segment is from the vertex to the centroid and the shorter segment is from the centroid to the mid - point of the opposite side.

Step2: Calculate segment lengths

Given \(FG = 6\mathrm{cm}\) and \(FC=18\mathrm{cm}\), then \(CG=FC - FG=18 - 6 = 12\mathrm{cm}\).

Step3: Check the ratio

The ratio of \(CG:FG\) is \(12:6 = 2:1\).

Answer:

Point G can be the centroid because 12:6 equals 2:1.