Coordinate Geometry
Overview
In this chapter, coordinate geometry is used as an algebraic method for studying geometric figures. Understanding how to express points and distances in a plane forms the foundational knowledge necessary for advanced geometry.
Key Points
This section covers:
1. Understanding Coordinates: Every point in the plane can be represented by a pair of coordinates (x, y), where x is the distance to the y-axis (abscissa) and y is the distance to the x-axis (ordinate).
2. Distance Formula: The distance between two points P(x1, y1) and Q(x2, y2) in the Cartesian plane is given by the formula:
$$ PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$
This formula is derived from the Pythagorean theorem and allows calculating the distance between any two points, whether they lie on the same axis or not.
3. Finding Collinearity: This section also discusses methods to determine if points are collinear using distance calculations.
4. Section Formula: The section formula is introduced, allowing the calculation of coordinates of a point that divides a line segment in a given ratio. The internal division formula is:
$$ P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) $$
where P is the dividing point, and A(x1, y1) and B(x2, y2) are endpoints with ratios m and n.
5. Applications: These concepts have practical applications in fields like physics, engineering, and art.