Length from centroid to midpoint on a median:\nIn triangle PQR, centroid C. Given PQ = 30 cm, QR = 36 cm, PR = 50 cm. If D is the midpoint of QR, find the exact length (in cm) of CD.

Introduction / Context:
The centroid divides each median in a 2:1 ratio measured from the vertex to the midpoint. We first compute the full median length using the formula for a median in terms of side lengths, then take one-third of that to get CD.

Given Data / Assumptions:

• PQ = 30 cm, QR = 36 cm, PR = 50 cm.
• D is the midpoint of QR.
• Median from P to D has length m_p with m_p^2 = (2PQ^2 + 2PR^2 − QR^2)/4.
• CD = (1/3) * m_p (since centroid divides median in ratio 2:1).

Concept / Approach:
Compute the median length algebraically and scale by 1/3 to obtain the centroid-to-midpoint segment.

Step-by-Step Solution:

Verification / Alternative check:
Ratio property check: PC = (2/3)m_p and CD = (1/3)m_p; their sum is the median length m_p.

Why Other Options Are Wrong:
Other expressions do not equal one-third of 4√86.

Common Pitfalls:
Taking 2/3 instead of 1/3 for CD or miscomputing the median formula.

Final Answer:
(4√86)/3