ABCD is a parallelogram of area 162 sq. Cm P is a point on AB such that AP : PB = 1 :2
Calculate
The ratio of PA : DC.

Answer

To solve the problem, we need to find the ratio of \( PA \) to \( DC \) in the parallelogram \( ABCD \) given that \( AP : PB = 1 : 2 \).

### Step-by-Step Solution:

1. **Define the segments**:
Let \( AP = x \) and \( PB = 2x \). This is based on the given ratio \( AP : PB = 1 : 2 \).

2. **Calculate the length of \( AB \)**:
Since \( AB = AP + PB \), we can write:
\[ AB = x + 2x = 3x \]

3. **Use properties of parallelograms**:
In a parallelogram, opposite sides are equal. Therefore, we have:
\[ DC = AB \]
Thus,
\[ DC = 3x \]

4. **Find the ratio \( PA : DC \)**:
We know \( PA = AP = x \) and \( DC = 3x \). Therefore, the ratio can be expressed as:
\[ \frac{PA}{DC} = \frac{x}{3x} \]

5. **Simplify the ratio**:
Simplifying the fraction gives:
\[ \frac{PA}{DC} = \frac{1}{3} \]

6. **Express in ratio form**:
Thus, the ratio of \( PA : DC \) is:
\[ PA : DC = 1 : 3 \]

### Final Answer:
The ratio of \( PA : DC \) is \( 1 : 3 \).