If ABCD is a parallelogram in which P and Q are the centroids of `DeltaABD and DeltaBCD`. then, PQ equals :

Options

A. AQ
B. AP
C. BP
D. DQ

Answer

To find the length of segment \( PQ \) in the parallelogram \( ABCD \), where \( P \) and \( Q \) are the centroids of triangles \( ABD \) and \( BCD \) respectively, we can follow these steps:

### Step 1: Understand the properties of centroids
The centroid of a triangle divides each median in the ratio \( 2:1 \). Therefore, if we consider triangle \( ABD \), the centroid \( P \) divides the median \( AO \) (where \( O \) is the midpoint of \( BD \)) into segments \( AP \) and \( PO \) such that:
\[ AP:PO = 2:1 \]
This means \( AP = 2x \) and \( PO = x \) for some length \( x \).

### Step 2: Analyze triangle \( BCD \)
Similarly, for triangle \( BCD \), the centroid \( Q \) divides the median \( CO \) (where \( O \) is the midpoint of \( AD \)) into segments \( CQ \) and \( OQ \) such that:
\[ CQ:OQ = 2:1 \]
This means \( CQ = 2y \) and \( OQ = y \) for some length \( y \).

### Step 3: Relate the segments
Since \( O \) is the midpoint of both diagonals \( AC \) and \( BD \) in the parallelogram, we have:
\[ AO = OC \quad \text{and} \quad BO = OD \]
Thus, \( AO = OC \) implies that \( AP + PO = OQ + CQ \).

### Step 4: Set up the equations
From the previous steps, we have:
\[ AP + PO = 2x + x = 3x \]
\[ OQ + CQ = y + 2y = 3y \]
Since \( AO = OC \), we can equate these two:
\[ 3x = 3y \implies x = y \]

### Step 5: Find \( PQ \)
Now, we can express \( PQ \) in terms of \( x \):
\[ PQ = PO + OQ = x + y = x + x = 2x \]
Since \( AP = 2x \), we can conclude that:
\[ PQ = AP \]

### Conclusion
Thus, the length of segment \( PQ \) is equal to the length of segment \( AP \).

### Final Answer
\[ PQ = AP \]