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In triangle TRS, TZ = (3x) inches and WZ = (2x - 3) inches.

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.

[Figure may not be drawn to scale]

What is WZ?

1 answer

Given that centroid Z divides the median TW in a 2:1 ratio, we can express the relationship between the lengths TZ and WZ.

  1. Centroid Division: The centroid divides each median into two segments, one of which is twice the length of the other. Hence, if we let \( WZ \) be one segment and \( TZ \) be the other segment: \[ TZ = 2 \times WZ \]

  2. Inserting Given Lengths: According to the problem, we have:

    • \( TZ = 3x \)
    • \( WZ = 2x - 3 \)

    From the relationship established: \[ 3x = 2 \times (2x - 3) \]

  3. Solving the Equation: \[ 3x = 4x - 6 \] Rearranging gives: \[ 3x - 4x = -6 \] \[ -x = -6 \] \[ x = 6 \]

  4. Finding WZ: Now that we have \( x \), we can find \( WZ \): \[ WZ = 2x - 3 = 2(6) - 3 = 12 - 3 = 9 \text{ inches} \]

Thus, the length of \( WZ \) is \( \boxed{9} \) inches.