Coordinate Geometry

Core Concepts & Formulas

The distance d between two points P(x₁, y₁) and Q(x₂, y₂) is given by:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

The coordinates of point P(x, y) dividing the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m:n are:

(x, y) = ((mx₂ + nx₁) / (m + n), (my₂ + ny₁) / (m + n))

The midpoint of a line segment joining A(x₁, y₁) and B(x₂, y₂) divides it in the ratio 1:1:

(x, y) = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

The area of a triangle formed by points A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃) is:

Area = ½ | x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) |

Collinearity Condition: If three points A, B, and C are collinear, the area of the triangle formed by them is 0.

Distance from Origin: The distance of a point P(x, y) from the origin O(0, 0) is given simply by √(x² + y²).

Find the distance between two given points.
Find the coordinates of a point dividing the line segment joining two given points in the ratio 2:3.
Show that three given points are collinear using the triangle area formula.

d = √((x₂ - x₁)² + (y₂ - y₁)²)

((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n))

((x₁ + x₂)/2, (y₁ + y₂)/2)

½ | x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂) |