In parallelogram ABCD, diagonal BD has length 36 cm. P and Q are the centroids of triangles ABC and ADC respectively. What is the length (in cm) of segment PQ joining these two centroids?

Introduction / Context:
This question uses vector or coordinate ideas about centroids in triangles inside a parallelogram. The key fact is that the centroid of a triangle is the average of its vertex position vectors. By writing the centroids of triangles ABC and ADC in terms of the vertices of the parallelogram, we can express the vector PQ in terms of the diagonal BD and then find its length using the given diagonal length.

Given Data / Assumptions:

• ABCD is a parallelogram.
• Diagonal BD has length 36 cm.
• P is the centroid of triangle ABC.
• Q is the centroid of triangle ADC.
• We must find the length of PQ.

Concept / Approach:
In vector form, if a triangle has vertices X, Y and Z, its centroid is at (X + Y + Z) / 3. For parallelogram ABCD, we can introduce vectors for the vertices and write P and Q using the centroid formula. When we take the difference Q − P, many terms cancel and the result can be expressed as a simple fraction of the diagonal vector from B to D. The magnitude of PQ then becomes a fixed fraction of the length of BD, independent of the specific shape of the parallelogram.

Step-by-Step Solution:
Step 1: Let the position vectors of A, B, C and D be a, b, c and d respectively. Step 2: For parallelogram ABCD, we have c = b + d − a, but we only need consistent labelling for centroid formulas. Step 3: Centroid of triangle ABC is P = (a + b + c) / 3. Step 4: Centroid of triangle ADC is Q = (a + d + c) / 3. Step 5: Compute vector PQ = Q − P = [(a + d + c) − (a + b + c)] / 3 = (d − b) / 3. Step 6: Note that vector from B to D is BD = d − b, so PQ = (1 / 3) * BD. Step 7: Therefore length of PQ = (1 / 3) * length of BD = (1 / 3) * 36 cm = 12 cm.

Verification / Alternative check:
We can choose a specific parallelogram to check the ratio. For example, let B at (0, 0) and D at (36, 0) and choose A and C appropriately to form a parallelogram. After computing centroids of triangles ABC and ADC using coordinates, the distance between the centroids will be 12 units, exactly one third of the distance between B and D. Trying several different parallelogram shapes with diagonal BD of length 36 confirms that PQ always equals one third of BD, confirming the general vector result.

Why Other Options Are Wrong:
If PQ were 6 cm, 9 cm, 18 cm or 24 cm, the ratio PQ : BD would be 1 : 6, 1 : 4, 1 : 2 or 2 : 3 respectively, none of which match the one third factor derived from the centroid formulas. These values conflict with the algebraic relation PQ = (1 / 3) BD and would not hold across all parallelograms. Only 12 cm maintains the correct fixed fraction for any parallelogram with diagonal BD of length 36 cm.

Common Pitfalls:
A common mistake is to assume that centroids lie on diagonals of the parallelogram in a simple way without calculation, or to confuse midpoints with centroids. Another error is to treat P and Q as midpoints of segments, which leads to incorrect fractions of BD. Using vector or coordinate notation to write centroids explicitly and then subtracting them is the cleanest route and prevents these confusions.

Final Answer:
The length of PQ is 12 cm.