# Coordinate Geometry For Class 10

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 $A (x_{1}, y_{1})$ and $B(x_{2}, y_{2})$, then the distance of these points is given by-

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

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 $A (x_{1}, y_{1})$ and $B(x_{2}, y_{2})$,

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 $(x_{1}, y_{1}), (x_{2}, y_{2}) \;\; and \;\; (x_{3}, y_{3})$, then the area of a triangle is given as-

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

#### Practise This Question

To construct a triangle similar to a given triangle with a scale factor 35, 8 points are to be marked on a ray drawn at an acute angle to one of the sides.