Triangle A B C is shown. Lines are drawn from each point to to the opposite side and intersect at point G. Line segments A D, B E, and C F are created.

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.

Answer 1

To determine whether point G can be the centroid of triangle ABC, we need to recall the property of centroids in triangles. The centroid divides each median into two segments, with the ratio of the length from the vertex to the centroid being 2:1 compared to the length from the centroid to the midpoint of the opposite side.

In this case:

• Segment FG = 6 cm (the segment from G to the midpoint of side AC)
• Segment FC = 18 cm (the segment from F to C)

We can denote:

• FG as the segment from G to the midpoint.
• GC as the segment from G to point C.

From the data given:

• FC (which includes FG) = 18 cm
• FG = 6 cm Thus, FC = FG + GC = 6 cm + GC. Therefore, GC = 18 cm - 6 cm = 12 cm.

Now let's look at the ratios. For G to be the centroid:

• The segment from the vertex (C) to the centroid (G) should be twice the length of the segment from the centroid (G) to the midpoint (F).

Hence, we want:

• CG : GF = 2 : 1 This translates to:
• 12 cm : 6 cm = 2 : 1, which is satisfied.

Thus, the appropriate explanation is:

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