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