Length from centroid to midpoint on a median: In 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.
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A(4√86)/3
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B(2√86)/3
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C(5√86)/3
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D(5√86)/2
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E√86
Answer
Correct Answer: (4√86)/3
Explanation
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:
m_p^2 = (2*30^2 + 2*50^2 − 36^2)/4 = (1800 + 5000 − 1296)/4 = 5504/4 = 1376.m_p = √1376 = √(16*86) = 4√86.CD = (1/3) * m_p = (4√86)/3.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