Centroid segment length – Triangle XYZ with sides 11, 14, 7 cm: G is centroid and M is the midpoint of XY. Given XY = 11 cm, YZ = 14 cm, XZ = 7 cm. Find GM (in cm).

Introduction / Context:The centroid divides each median in the ratio 2:1 from vertex to midpoint. Thus, GM equals one-third of the median length from the opposite vertex to side XY.

Given Data / Assumptions:

• Let median from Z meet XY at M (midpoint of XY).
• Sides: XY = 11, XZ = 7, YZ = 14.

Concept / Approach:Median to side a (opposite vertex) has length m = (1/2) * √(2b^2 + 2c^2 − a^2). Here a = XY, b = XZ, c = YZ. Then GM = (1/3) * m.

Step-by-Step Solution:

Verification / Alternative check:Using exact radical form confirms GM ≈ 3.206 cm; the nearest provided value is 3.

Common Pitfalls:Using 2/3 instead of 1/3 for the segment from centroid to midpoint; mixing the side labels in the median formula.

Final Answer:3