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The problem statement translates to:

4. In the given diagram, \(ABCD\) is a parallelogram. \(BE \perp CE\), and intersects \(AD\) and \(CD\) at points \(F\) and \(G\) respectively. Given \(CE = 9\) and \(BF = 12\), find the area of the parallelogram \(ABCD\).

To solve this problem, we will follow these steps:

1. **Identify the relationships**: Given \(BE \perp CE\), triangle \(BCE\) is a right triangle with \(BE\) as the height and \(CE\) as the base.
2. **Calculate area of \( \triangle BCE\)**: The area of a triangle is given by \(\frac{1}{2} \times \text{base} \times \text{height}\). Here, \(\text{base} = CE = 9\) and \(\text{height} = BE\).
3. **Find \(BE\)**: Using the given \(BF = 12\) and knowing that \(BF\) is the perpendicular distance from \(B\) to line \(AD\), we realize that \(BE = BF = 12\) because both are heights from the same vertex to a parallel side (by the properties of a parallelogram).
4. **Calculate the area of \(\triangle BCE\)**:
   \[
   \text{Area of } \triangle BCE = \frac{1}{2} \times 9 \times 12 = 54
   \]
5. **Determine the area of the parallelogram**: Since \(BE\) is the height from \(B\) to \(AD\), and the area of parallelogram \(ABCD\) is twice the area of \(\triangle BCE\):
   \[
   \text{Area of } ABCD = 2 \times \text{Area of } \triangle BCE = 2 \times 54 = 108
   \]

Thus, the area of the parallelogram \(ABCD\) is \(108\) square units.

If you have any questions or need further details, please let me know.

Here are 5 questions you might want to ask next:
1. How do you find the area of a parallelogram?
2. What properties of a parallelogram are used in calculating its area?
3. How does knowing the perpendicular height help in determining the area of a parallelogram?
4. Can the area of a parallelogram be found using its diagonals?
5. What are some common mistakes to avoid when calculating the area of a parallelogram?

**Tip:** When working with geometric shapes, always consider their properties and the relationships between different parts of the shape to simplify your calculations.

Learn how to calculate the area of parallelogram ABCD with a detailed step-by-step solution. This problem involves finding the area using given dimensions for the heights and bases of triangles and parallelograms. Suitable for students in Grades 9-12.

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