Triangle T R S has centroid Z. Lines are drawn from each point to the midpoint of the opposite side to form line segments T W, R V, and S U.

Question

Triangle T R S has centroid Z. Lines are drawn from each point to the midpoint of the opposite side to form line segments T W, R V, and S U.
In triangle TRS, VZ = 6 inches. What is RZ?

3 inches
6 inches
12 inches
18 inches

GPT-4o mini · Answer

In triangle $ TRS $, the centroid $ Z $ divides each median into two segments, where the segment connecting a vertex to the centroid is twice the length of the segment connecting the centroid to the midpoint of the opposite side. Given that $ VZ $ (the segment from the centroid $ Z $ to the midpoint $ V $ of side $ TS $) is 6 inches, we can determine the length of $ RZ $ as follows: 1. The centroid $ Z $ divides median $ RV $ in a ratio of $ 2:1 $. 2. This means that $ RZ $ is twice the length of $ VZ $. Since $ VZ = 6 $ inches, we can find $ RZ $: $$ RZ = 2 \times VZ = 2 \times 6 \text{ inches} = 12 \text{ inches}. $$ Thus, the value of $ RZ $ is $ \boxed{12} $ inches.