In triangle PQR, C is the centroid. The side lengths are PQ = 30 cm, QR = 36 cm and PR = 50 cm. If D is the midpoint of side QR, what is the length, in centimetres, of segment CD?

Introduction / Context:
This question tests understanding of centroids and medians in a triangle. The centroid is the point where the three medians intersect, and it divides each median in a fixed ratio. To find the distance from the centroid to a midpoint on a side, we first calculate the length of the median and then apply the centroid ratio property. This is a standard concept in coordinate geometry and vector geometry that often appears in advanced aptitude problems.

Given Data / Assumptions:

• Triangle PQR has side lengths PQ = 30 cm, QR = 36 cm and PR = 50 cm.
• C is the centroid of triangle PQR.
• D is the midpoint of side QR.
• We must find the length of segment CD, in centimetres.
• The triangle is non degenerate and standard Euclidean properties of medians and centroids apply.

Concept / Approach:
A median in a triangle is a segment that joins a vertex to the midpoint of the opposite side. Here, PD is the median from vertex P to side QR. The centroid C lies on PD and divides it in the ratio 2 : 1, with the longer segment from the vertex to the centroid. This means: PC : CD = 2 : 1 Hence: CD = (1/3) * PD Therefore, our main task is to find the length of median PD. There is a formula for the length of a median in a triangle in terms of its sides (Apollonius theorem): m_a^2 = (2b^2 + 2c^2 − a^2) / 4 where m_a is the median from the vertex opposite side a, and b and c are the other two sides.

Step-by-Step Solution:
Step 1: Identify the sides relative to the median PD. Median PD is drawn from P to midpoint of QR, so it is the median to side QR, which has length 36 cm. Thus, in the formula, a = QR = 36, and the other sides are PQ = 30 and PR = 50. Step 2: Apply the median length formula. Let m be the length of median PD. m^2 = (2 * PQ^2 + 2 * PR^2 − QR^2) / 4 = (2 * 30^2 + 2 * 50^2 − 36^2) / 4 = (2 * 900 + 2 * 2500 − 1296) / 4 = (1800 + 5000 − 1296) / 4 = 5504 / 4 = 1376 Thus, m^2 = 1376, so m = √1376. Step 3: Simplify √1376 if possible. 1376 = 16 * 86 (since 16 * 80 = 1280 and 16 * 6 = 96, total 1376). So m = √(16 * 86) = 4√86. Therefore, PD = 4√86 cm. Step 4: Use the centroid ratio to find CD. The centroid C divides PD in the ratio 2 : 1, with PC : CD = 2 : 1. Hence, CD = (1/3) * PD = (1/3) * 4√86 = (4√86) / 3 cm.

Verification / Alternative check:
We can approximate the value numerically to confirm. Approximate √86 ≈ 9.2736. Then: PD ≈ 4 * 9.2736 ≈ 37.09 cm CD ≈ PD / 3 ≈ 37.09 / 3 ≈ 12.36 cm Option (4√86)/3 gives the same numerical value, confirming that the simplified surd form is correct. This is also consistent with the centroid property that CD is shorter than the full median PD.

Why Other Options Are Wrong:
Option 2: (2√86)/3 cm is exactly half of the correct value for CD and would not correspond to the centroid division ratio. Option 3: (5√86)/3 cm is larger than one third of the median and would imply an incorrect division ratio along PD. Option 4: (5√86)/2 cm is even larger and does not relate to any standard centroid proportion. Option 5: None of these is incorrect because (4√86)/3 cm matches the correct derived length for CD.

Common Pitfalls:
A common error is to use the wrong ratio for the centroid and median, such as 1 : 2 instead of 2 : 1 from vertex to centroid. Some students mistakenly think CD equals half of the median instead of one third. Another pitfall is misapplying the median length formula, either by mixing up which side is a or by forgetting the division by 4. Errors in arithmetic when squaring the sides or simplifying the root can also lead to incorrect surd expressions. Writing each step clearly and remembering that the centroid divides each median in a 2 : 1 ratio from the vertex helps avoid these mistakes.

Final Answer:
The length of segment CD is (4√86)/3 cm.