COORDINATE GEOMETRY: How to find the coordinates & ratios by section formula?

UTPAL SHARMA

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.

COORDINATE GEOMETRY: How to find coordinates & ratio by section formula?

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)

 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)

-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.

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