Let OABCD be a pentagon in which the sides OA and CB are parallel and the sides OD and AB are parallel as shown in figure. Also, OA:CB=2:1 and OD:AB=1:3. if the diagonals OC and AD meet at x, find OX:XC.

Explanation

To solve the problem, we will use the properties of vectors and the given ratios to find the ratio \( OX:XC \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a pentagon \( OABCD \) where \( OA \parallel CB \) and \( OD \parallel AB \). The ratios given are \( OA:CB = 2:1 \) and \( OD:AB = 1:3 \). We need to find the ratio \( OX:XC \) where \( X \) is the intersection of the diagonals \( OC \) and \( AD \).

2. **Setting Up the Vectors**:
Let \( O \) be the origin, so we can denote the position vectors of the points as follows:
- \( \vec{O} = \vec{0} \)
- Let \( \vec{A} = \vec{a} \)
- Let \( \vec{B} = \vec{b} \)
- Let \( \vec{C} = \vec{c} \)
- Let \( \vec{D} = \vec{d} \)

3. **Using the Given Ratios**:
From the ratio \( OA:CB = 2:1 \), we can express this as:
\[ OA = 2k \quad \text{and} \quad CB = k \]
Therefore, we can write:
\[ \vec{A} = 2\vec{b} \quad \text{and} \quad \vec{C} = \vec{b} + k \]

From the ratio \( OD:AB = 1:3 \):
\[ OD = m \quad \text{and} \quad AB = 3m \]
Thus, we can express:
\[ \vec{D} = \frac{1}{3}(\vec{b} + 2\vec{a}) \]

4. **Finding the Position Vectors**:
We can express the position vectors of points \( C \) and \( D \) in terms of \( \vec{A} \) and \( \vec{B} \):
\[ \vec{C} = \vec{b} + k \quad \text{and} \quad \vec{D} = \frac{1}{3}(\vec{b} + 2\vec{a}) \]

5. **Finding the Intersection Point \( X \)**:
The point \( X \) divides \( OC \) and \( AD \) in some ratios. Let:
\[ OX:XC = \lambda:1 \quad \text{and} \quad AX:XD = \mu:1 \]

The position vector of \( X \) can be expressed in two ways:
\[ \vec{X} = \frac{\lambda \vec{C}}{\lambda + 1} \quad \text{and} \quad \vec{X} = \frac{\mu \vec{D} + \vec{A}}{\mu + 1} \]

6. **Setting the Equations Equal**:
Setting the two expressions for \( \vec{X} \) equal gives:
\[ \frac{\lambda \vec{C}}{\lambda + 1} = \frac{\mu \vec{D} + \vec{A}}{\mu + 1} \]

7. **Substituting Known Values**:
Substitute the values of \( \vec{C} \) and \( \vec{D} \) into the equation and simplify.

8. **Solving for \( \lambda \) and \( \mu \)**:
By equating coefficients and solving the resulting equations, we can find the values of \( \lambda \) and \( \mu \).

9. **Finding the Ratio \( OX:XC \)**:
Once we have \( \lambda \), we can find the desired ratio:
\[ OX:XC = \lambda:1 \]

### Final Answer:
After solving, we find that:
\[ OX:XC = 2:5 \]