Locating points on a plane and measuring with coordinates
Coordinate geometry places every point on a plane using an ordered pair (x, y): its distances from the y-axis and the x-axis. The two axes divide the plane into four quadrants, and the signs of x and y tell you which quadrant a point lies in. The distance between P(x₁, y₁) and Q(x₂, y₂) comes straight from Pythagoras' theorem: PQ = √[(x₂ − x₁)² + (y₂ − y₁)²]. For example, the distance from (1, 2) to (4, 6) is √(9 + 16) = 5.
The section formula gives the point that divides the segment joining (x₁, y₁) and (x₂, y₂) internally in the ratio m : n, namely ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). With m = n it becomes the midpoint formula. The area of a triangle with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃) is ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. If this area works out to zero, the three points are collinear, which is a quick test for whether they lie on one straight line.
Why it matters
Coordinates turn geometry into algebra, so shapes can be measured and checked by calculation instead of careful drawing. Maps, GPS positions, computer screens and game graphics all use coordinate systems to locate points. In Class 11 the same tools lead to equations of straight lines, circles and conic sections, and later to vectors and three-dimensional geometry.
Key terms
- Ordered pair
- A pair (x, y) in which the order matters, giving a point's position on the plane.
- Distance formula
- The rule √[(x₂ − x₁)² + (y₂ − y₁)²] for the length between two points.
- Section formula
- The coordinates of the point dividing a line segment in a given ratio m : n.
- Collinear points
- Three or more points that lie on the same straight line.