ABCD is a parallelogram. L is a point on BC which divides BC in the ratio `1:2`. AL intersects BD at P.M is a point on DC which divides DC in the ratio `1 : 2` and AM intersects BD in Q.
Point P divides AL in the ratio

Options

A. `1:2`
B. `1:3`
C. `3:1`
D. `2:1`

Explanation

To solve the problem step by step, let's break down the information given and apply the properties of parallelograms and similar triangles.

### Step 1: Understand the Geometry
We have a parallelogram ABCD with points L and M dividing sides BC and DC in the ratios 1:2 respectively. This means:
- \( BL : LC = 1 : 2 \)
- \( DM : MC = 1 : 2 \)

### Step 2: Set Up Coordinates
Assume the coordinates of the points:
- \( A(0, 0) \)
- \( B(a, 0) \)
- \( C(a + b, c) \)
- \( D(b, c) \)

From this, we can find the coordinates of points L and M:
- Point L divides BC in the ratio 1:2, so using the section formula:
\[ L = \left( \frac{2a + (a + b)}{3}, \frac{2 \cdot 0 + c}{3} \right) = \left( \frac{3a + b}{3}, \frac{c}{3} \right) \]

- Point M divides DC in the ratio 1:2, so:
\[ M = \left( \frac{2b + (a + b)}{3}, \frac{2c + c}{3} \right) = \left( \frac{a + 3b}{3}, \frac{3c}{3} \right) = \left( \frac{a + 3b}{3}, c \right) \]

### Step 3: Find Intersection Points
Next, we need to find the intersection points P and Q:
- The line AL can be represented by the equation derived from points A and L.
- The line AM can be represented by the equation derived from points A and M.
- The line BD can be represented by the equation derived from points B and D.

### Step 4: Use Similar Triangles
Since triangles APD and LPB are formed, we can use the property of similar triangles:
- By the properties of similar triangles:
\[ \frac{AP}{LP} = \frac{AD}{BL} \]

### Step 5: Calculate Ratios
From the ratios given:
- \( BL : LC = 1 : 2 \) implies \( BL = \frac{1}{3} BC \) and \( LC = \frac{2}{3} BC \).
- Therefore, \( AD = BC \) (since opposite sides of a parallelogram are equal).

Substituting these values into the ratio:
\[ \frac{AP}{LP} = \frac{BC}{BL} = \frac{BC}{\frac{1}{3}BC} = 3 \]

### Step 6: Final Ratio
Thus, we can conclude that:
\[ AP : LP = 3 : 1 \]

### Conclusion
The point P divides AL in the ratio \( 3 : 1 \).