## INTRODUCTION:-

We had study two blogs of coordinate geometry in which we learned COORDINATE GEOMETRY: How to find the distance between two points by Distance Formula? & COORDINATE GEOMETRY: How to find the area of a triangle in the Cartesian plane?

Now we will discuss how to find coordinates & ratio in the cartesian plane by section formula? In some situation, we have coordinates of endpoints & ratio of the line segment.

And we need to find coordinates of the point of intersection (it may be a midpoint in some cases) of the line segment.

In some cases, we have coordinates of the point of intersection & endpoints of the line segment. Here we need to find the ratio of the given line segment.

## Terminology:-

*Line segment*

*It is a part of the line which has two endpoints.*

*Midpoint*

*It is a point of a line segment which divide it into two equal parts.*

*Ratio*

*It is a term in a Cartesian plane which shows partition or division of any line segment.*

## Section Formula:-

It is a formula which is helpful to find coordinates & ratio of the line segment in the cartesian plane.

For coordinate of x, we use

**x = (m₁x₂+m₂x₁)÷(m₁+m₂)**For coordinate of y, we use

**y =****(m₁y₂+m₂y₁)÷(m₁+m₂)**

All details are shown in the below picture.

Where-

- PQ is a line segment with endpoints P & Q.
- (x₁,y₁) are the coordinates of point P.
- (x₂,y₂) are the coordinates of point Q.
- m₁ & m₂ are the ratios of line segment PQ.
- O is the point of intersection of line segment PQ.
- Coordinates of O are (x, y).

### 1.Question

**Find the coordinate of point O which divide the line segment PQ joining P(-3,-2) & Q(2,-2) in the ratio 3:2.**

### Answer

*(All details as shown in the above picture)*

*We have formulas to find the coordinates of the point of intersection O whose coordinates are (x , y).*

* ***x = (m₁x₂+m₂x₁)÷(m₁+m₂)**

**x = (m₁x₂+m₂x₁)÷(m₁+m₂)**

**y =****(m₁y₂+m₂y₁)÷(m₁+m₂)**

*Put all the given values in formulas-*

**For the value of x:-***x = (m₁x₂+m₂x₁)÷(m₁+m₂)*

*x = {(3×2)+(2×-3)}÷(3+2)*

*x = {6-6}÷5*

*x = 0÷5*

*x=0*

**For the value of y:-***y = (m₁y₂+m₂y₁)÷(m₁+m₂)*

*y = {(3×-2)+(2×-2)}÷(3+2)*

*y = {-6-4}÷5*

*y = -10÷5*

*y = -2*

*So here the coordinates of point O are (0,-2).*

### 2.Question

**Find the ratio in which the line segment joining the points (– 3, 10) and (6, – 8) is divided by (– 1, 6).**

### Answer

*Here we have endpoints of a line segment (x₁,y₁)=(-3,10), (x₂, y₂)=(6,-8)*

*& Point of intersection are (x , y) =(-1,6).*

*To find ratios m₁:m₂.*

*Put all the given values in the formula of x or y coordinate to determine the ratios-*

* ***x = (m₁x₂+m₂x₁)÷(m₁+m₂)**

**x = (m₁x₂+m₂x₁)÷(m₁+m₂)**

*-1 = {(m₁×6)+(m₂×-3)}÷(m₁+m₂)*

*-1 = {6m₁-3m₂}÷(m₁+m₂)*

*-1(m₁+m₂) =(6m₁-3m₂)*

*-m₁-m₂ =*

*6m₁-3m₂*

*-m₁-6m₁=-3m₂+m₂*

*-7m₁ = -2m₂*

*m₁÷m₂ = -2÷(-7)*

**So we have m₁:m₂ = (-2):(-7)**

## CONCLUSION:-

In these three parts of coordinate geometry, we learned about Distance Formula, Area of triangle & section formula. We may also find the area of a parallelogram, rhombus, square, etc. in the cartesian plane.