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When dealing with points and lines in a coordinate plane, the distance formula is essential. It's used to calculate the shortest distance from a point to a line. This formula comes from the principles of coordinate geometry, aimed at relating algebra and geometry using a system of coordinates. The generic formula for the distance from a point, \(x_1, y_1\), to a line, \(Ax + By + C = 0\), is:\[ D = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]This works by creating a perpendicular from the point to the line, giving us the shortest distance.

• The top part of the formula, \(|Ax_1 + By_1 + C|\), finds the perpendicular distance component.
• The bottom part, \(\sqrt{A^2 + B^2}\), normalizes the line's direction vectors.

Let's apply this to the problem:\( (-2,1) \) is our point, and \(x - y - 2 = 0\) identifies \(A=1\), \(B=-1\), and \(C=-2\). Substituting into our formula, we calculate the distance.

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This method enables geometric problems to be solved numerically and algebraically. By assigning coordinates to points, we streamline the process of calculating distances and angles. This system uses:

• X-axis and Y-axis: These are the two perpendicular lines that establish the plane on which we plot points.
• Coordinates: Any point's location is indicated by its distance from the two axes, written as (x, y).
• Lines and Curves Representation: Represented algebraically with equations such as \(Ax + By + C = 0\).

In our example, the line \(x - y - 2 = 0\) forms a straight line from which we calculate distance. By placing a point \((-2, 1)\) in this coordinate plane, we can utilize formulas rooted in coordinate geometry to find precise distances or slopes. This translates the visual problem into a solvable equation, demonstrating the power of coordinate geometry.

Rationalizing denominators is a process applied in mathematics to make expressions easier to work with by eliminating radicals from the denominator of a fraction. In essence, we aim to convert a fraction with a square root in the denominator into a form without one. Here's how it works:

• Identify the Radical: Look for the square root in the denominator, such as in \(\frac{5}{\sqrt{2}}\).
• Multiply by a Conjugate: Multiply both the numerator and the denominator by the radical found in the denominator (e.g. \(\sqrt{2}\)).
• Simplify: This yields a rational denominator, such as \(\frac{5 \cdot \sqrt{2}}{2}\), turning it into a fraction with an integer or a simple radical in the numerator.

For the given problem, \(\frac{5}{\sqrt{2}}\) is rationalized to become \(\frac{5\sqrt{2}}{2}\). This makes computations involving the fraction simpler and is a standard process in mathematical presentations.