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 distance from the vertex to the centroid is twice the distance from the centroid to the mid - point of the opposite side. If FG is the distance from the centroid to the mid - point of a side and FC is the distance from the vertex to a point on the side, and G is the centroid, then \(CG = FC - FG\). Given \(FG = 6\) cm and \(FC=18\) cm, so \(CG=18 - 6=12\) cm.

Step2: Check the ratio

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

Answer:

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