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The problem in the image asks to compute the area of a parallelogram using simple plane geometry in terms of Cartesian coordinates. Let's analyze the steps needed to address each part of the question based on the provided information.

### Summary of the Problem:
1. **Parallelogram Setup:**
   - A parallelogram \( P \) with vertices at \( (a, 0) \), \( (b, c) \), \( (0, d) \), and \( (e, f) \).
   - Goal: Use plane geometry to compute the area of this parallelogram.

2. **Questions in the Problem:**
   - **(a)** Prove that the area of \( P \) is equal to base times height by dividing the parallelogram into triangles or using a transformation (e.g., shear transformation).
   - **(b)** Derive a formula for the area of the parallelogram in terms of \( a, b, c, d, e, f \) and prove why some formulas are true while others are easier to compute.
   - **(c)** Specific scenario with given conditions for the vertices and simplifications for area computations.

### Step-by-Step Analysis:

#### (a) Proving the Area of a Parallelogram as Base Times Height:
The task is to use geometric methods to show that the area \( A \) of the parallelogram equals base times height. A useful approach might be to:
- **Divide the Parallelogram**: Cut the parallelogram into two triangles by drawing a diagonal.
- **Transformations**: Apply geometric transformations to manipulate the parallelogram into a rectangle without changing its area.
- Use the formula for the area of a triangle and add the areas of two triangles.

#### (b) Deriving a Formula for the Area in Terms of Coordinates:
- You need to express

In this problem, you will explain using simple plane geometry how to compute the area of a parallelogram in terms of Cartesian coordinates. Let P be a parallelogram with vertices at (a,0), (b,c), (0,d), and (e,f). Prove that the area of P is equal to base times height. Derive a formula for the area of the parallelogram in terms of a, b, c, d, e, f.

Learn how to calculate the area of a parallelogram using simple plane geometry and Cartesian coordinates. This problem demonstrates the use of base times height formula and derivation of a formula based on vertices. Suitable for high school students.

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