Coordinate Geometry is used to represent a point on a plane. The distance of any given point from y-axis is called as its **‘x-coordinate’ or ‘abscissa,’** whereas the distance from the x-axis is called as its **‘y-coordinate’ or ‘ordinate.’**

**Distance Formula-**

Consider a line having two point \(A (x_{1}, y_{1})\)

\(AB = \sqrt{(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2} }\)

The above formula is said to be distance formula.

**Section Formula-**

Section formula is used to divide any line into two parts which are in the ratio **m:n.**

Let us consider a line AB whose coordinates are given as \(A (x_{1}, y_{1})\)

then the coordinate of the point which divides a line in the given ratio of **m:n **is given as:

\(\left ( \frac{mx_{2} + nx_{1}}{m + n} , \frac{my_{2} + ny_{1}}{m + n} \right )\)

Alternatively, to ease the method of section formula consider the ratio **m:n = k**,

thus the new ratio becomes **‘k:1’**

The section formula is then given as-

\(\left ( \frac{kx_{2} + x_{1}}{k + 1} , \frac{ky_{2} + y_{1}}{k + 1} \right )\)

**Mid-Point Theorem- **As the name suggest, if a line segment is divided in the ratio 1:1, then the point of division is called as the midpoint of the line segment.

The coordinate of the mid-point is given as-

\(\left ( \frac{x_{2} + x_{1}}{2} , \frac{y_{2} + y_{1}}{2} \right )\)

**Area of a triangle-**

Consider the triangle formed by the points \((x_{1}, y_{1}), (x_{2}, y_{2}) \;\; and \;\; (x_{3}, y_{3})\)

\(A = \frac{1}{2}\left [x_{1}(y_{2} – y_{3}) + x_{2} (y_{3} – y_{1}) + x_{3} (y_{1} – y_{3}) \right ]\)