# Coordinate Geometry

The position of a point in a plane is fixed w.r.t. to its distances from two axes of reference, which are usually drawn by the two graduated number lines XOX' and YOY', at right angles to each other at O.

Any point (x, 0) lies on x-axis.

Any point (0, y) lies on y-axis.

(x, y) and (y, x) do not represent the same point when x ≠ y.

Co-ordinates of origin are (0, 0).

### Distance Formula

Distance between two points A(x1, y1) and B(x2, y2)

$$AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Three points A, B and C are collinear, if AB + BC = AC

Parallelogram: If length of opposite sides are equal.

Rectangle: If opposite sides are equal and diagonals are equal.

Square: If all 4 sides are equal, diagonals are also equal.

Rhombus: If all 4 sides are equal.

Parallelogram but Not rectangle: Opposite sides are equal but diagonals are not equal.

Rhombus but not square: All sides are equal but diagonals are not equal.

### Section Formula

$$x = \frac{mx_2 + nx_1}{m + n}$$

$$y = \frac{my_2 + ny_1}{m + n}$$

Midpoint

$$x = \frac{x_2 + x_1}{2}$$

$$y = \frac{y_2 + y_1}{2}$$

### Centroid of Triangle

The centroid of a triangle is the point of concurrency of its medians and divides each median in the ratio of 2:1.

$$x = \frac{x_1 + x_2 + x_3}{3}$$

$$y = \frac{y_1 + y_2 + y_3}{3}$$