The length of the diagonal BD of the parallelogram ABCD is 18 cm. If `P and Q` are the centroid of the `DeltaABC and Delta ADC` respectively then the length of the line segment `PQ` is

Options

A. 4 cm
B. 6 cm
C. 9 cm
D. 12 cm

Answer

To find the length of the line segment \( PQ \) where \( P \) and \( Q \) are the centroids of triangles \( ABC \) and \( ADC \) respectively in the parallelogram \( ABCD \) with diagonal \( BD = 18 \) cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Parallelogram and Diagonal**:
- We have a parallelogram \( ABCD \) with diagonal \( BD \) measuring \( 18 \) cm.

2. **Centroids of Triangles**:
- The centroid \( P \) of triangle \( ABC \) divides the median \( AC \) in the ratio \( 2:1 \).
- The centroid \( Q \) of triangle \( ADC \) also divides the median \( AC \) in the ratio \( 2:1 \).

3. **Length of the Diagonal**:
- The entire length of diagonal \( BD \) is \( 18 \) cm.

4. **Understanding the Ratio**:
- The centroid divides the median into two segments, with the longer segment being twice the length of the shorter segment. Therefore, for the entire diagonal \( BD \), we can express it in terms of units:
- The total length \( BD \) can be divided into \( 3 \) equal parts (since \( 2 + 1 = 3 \)).
- Each part is \( \frac{18 \text{ cm}}{3} = 6 \text{ cm} \).

5. **Finding Length of Segment \( PQ \)**:
- The distance \( PQ \) is the distance between the two centroids \( P \) and \( Q \).
- Since both centroids are \( \frac{2}{3} \) of the way from \( A \) to \( C \) along the diagonal \( AC \), the distance \( PQ \) can be calculated as:
- \( PQ = 2 \times \text{length of one part} = 2 \times 6 \text{ cm} = 12 \text{ cm} \).

### Final Answer:
Thus, the length of the line segment \( PQ \) is \( 12 \) cm.