Area of the parallelogram formed by the lines \(\mathrm{y}=\mathrm{mx}\), \(\mathrm{y}=\mathrm{mx}+1, \mathrm{y}=\mathrm{n} \mathrm{x}\) and \(\mathrm{y}=\mathrm{n} \mathrm{x}+1\) equals (a) \(\left[(|m+n|) /(m-n)^{2}\right]\) (b) \([2 /(|\mathrm{m}+\mathrm{n}|)]\) (c) \([1 /(m-n)]\) (d) \([1 /(|\mathrm{m}+\mathrm{n}|)]\)

When we talk about coordinate geometry, also known as analytic geometry, we're bringing together geometry and algebra to describe the position, the shape, and size of geometrical figures through coordinates on a plane. The magic of coordinate geometry lies in taking familiar shapes, like our parallelogram, and translating their properties into equations and coordinates. This creates a bridge between purely numerical algebraic expressions and the visual understanding of geometry.

Using coordinate geometry in our parallelogram problem, we designated specific coordinates to the intersection points of the lines that form the parallelogram. Once these points are placed on a Cartesian plane, we can start analyzing the figure using algebraic methods, which in this case, involves finding the lengths of the sides and eventually, the area of the parallelogram.

One of the fundamental tools in coordinate geometry is the distance formula, which is rooted in the Pythagorean theorem. When we have two points, let's say Point A with coordinates \( (x_1, y_1) \) and Point B with coordinates \( (x_2, y_2) \)—the distance between these points, often denoted by \( d \), can be calculated using the formula:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

The beauty of the distance formula is that it allows us to calculate the exact length of the sides of our parallelogram directly from their coordinates. This step is critical because once we know the side lengths, we're on our way to calculating the area of the parallelogram, moving one step closer to solving the problem at hand.

A system of linear equations consists of two or more linear equations with the same set of variables. In simpler terms, these are straight-line equations that we are trying to solve all together, looking for points where the lines cross, which we call the 'intersection points'.

In our parallelogram exercise, we had to find the intersection points of four lines. We tackled this by setting pairs of equations equal to each other and solving for the variables \( x \) and \( y \). These solutions give us the precise coordinates where the lines intercept, and in the case of a parallelogram, help to identify the vertices. These vertices are essential for calculating side lengths and eventually the area of the shape. Solving the system of linear equations efficiently is a critical step in our journey to reveal the secrets of the parallelogram's area.