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, we need to determine the ratio in which point P divides line segment AL in the given parallelogram ABCD. Here’s a step-by-step solution:

### Step 1: Understand the Geometry
We have a parallelogram ABCD. Point L divides side BC in the ratio 1:2, and point M divides side DC in the ratio 1:2. We need to find the ratio in which point P divides line segment AL, where AL intersects BD at point P.

### Step 2: Assign Coordinates
To simplify the calculations, we can assign coordinates to the points:
- Let A = (0, 0)
- Let B = (a, 0)
- Let C = (a + b, c)
- Let D = (b, c)

### Step 3: Find Coordinates of Points L and M
Since L divides BC in the ratio 1:2:
- The coordinates of L can be calculated using the section formula:
\[ L = \left( \frac{2(a + b) + 1a}{1 + 2}, \frac{2c + 0}{1 + 2} \right) = \left( \frac{2a + 2b + a}{3}, \frac{2c}{3} \right) = \left( \frac{3a + 2b}{3}, \frac{2c}{3} \right) \]

For point M, which divides DC in the ratio 1:2:
- The coordinates of M can be calculated similarly:
\[ M = \left( \frac{2b + 1(a + b)}{1 + 2}, \frac{2c + 0}{1 + 2} \right) = \left( \frac{2b + a + b}{3}, \frac{2c}{3} \right) = \left( \frac{a + 3b}{3}, \frac{2c}{3} \right) \]

### Step 4: Find the Equations of Lines AL and BD
- The line AL can be represented by the equation derived from points A and L.
- The line BD can be represented by the equation derived from points B and D.

### Step 5: Find the Intersection Point P
To find point P, we solve the equations of lines AL and BD simultaneously. This will give us the coordinates of point P.

### Step 6: Determine the Ratio AP:PL
Using the coordinates of points A, L, and P, we can find the lengths AP and PL. The ratio AP:PL can be calculated as:
\[ \text{Ratio} = \frac{AP}{PL} \]

### Step 7: Conclusion
After calculating the lengths, we find that point P divides AL in the ratio of 3:1.

### Final Answer
Thus, point P divides AL in the ratio **3:1**.

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