Coordinate geometry is one of the most exciting ideas of mathematics that provides a connection between algebra and geometry through graphs of lines and curves. This enables geometric problems to be solved algebraically and provides geometric insights into algebra. You must have studied the basics of coordinate geometry in your earlier classes and these have been summarized as follows. And after which we move on to explain the main concepts of this article, which are Distance Formula and Section Formula.

 

Basics of Coordinate Geometry

 

The number plane (Cartesian plane) is divided into four quadrants by two perpendicular axes called the x-axis (horizontal line) and the y-axis (vertical line). These axes intersect at a point O called the origin. The position of any point in the plane can be represented by an ordered pair of numbers (x, y). Wherein:

 

x � is the x-coordinate or abscissa which is the distance of a point from the y-axis

 

y - is the y-coordinate or ordinate which is the distance of a point from the x-axis

 

 

Let us now study about two important applications of coordinate geometry which are Distance Formula and Section Formula

 

Distance Formula

 

Let us now find the distance between any two points A (x₁, y₁) and B (x₂, y₂).

 

 

Draw AC and BD perpendicular to the x-axis.

 

A perpendicular from the point A on BD is drawn to meet it at the point E.

 

Then,

 

OC = x₁,

 

OD = x₂

 

So, CD = x₂ � x₁ = AE

 

Also

 

DB = y₂,

 

DE = AC = y₁

 

So, BE = y₂ � y₁

 

Now, applying the Pythagoras theorem in ?AEB, we get

 

AB² = AE² + BE²

 

       = (x₂ � x₁)² + (y₂ � y₁)²

 

Therefore, AB =

 

 

 

Note that since distance is always non-negative, we take only the positive square root.

 

So, the distance between the points A (x₁, y₁) and B (x₂, y₂) is

 

 

which is called the Distance Formula.

 

 

Distance Formula Examples

 

Let us try to understand it better through an example:

 

Example: Find the distance between the points A (2, -5) and B (5, -1).

 

Solution:

 

The co-ordinates of point A and B are (2, -5) and (5, -1).

 

Here,

 

x₁ = 2, x₂ = 5

 

And y₁ = -5, y₂ = -1

 

So, the distance between points A (2, -5) and B (5, -1) is given by

 

 

Therefore, the distance between points A (2, -5) and B (5, -1) is 5 units.

 

Section Formula

 

Let us try to derive the section formula now:

 

Consider any two points A (x₁, y₁) and B (x₂, y₂) and assume that P (x, y) divides AB internally in the ratio m₁: m₂, i.e.,

 

 

Draw AR, PS and BT perpendicular to the x-axis.

 

Draw AQ and PC parallel to the x-axis.

 

Then, ?PAQ ~ ?BPC (AA similarity criterion)

 

Therefore,

 

 

Now,

 

AQ = RS = OS � OR = x � x₁

 

PC = ST = OT � OS = x₂ � x

 

PQ = PS � QS = PS � AR = y � y₁

 

BC = BT� CT = BT � PS = y₂ � y

 

Substituting these values in (i), we get

 

 

Taking,

 

 

So, the coordinates of the point P (x, y) which divides the line segment joining the points A (x₁, y₁) and B (x₂, y₂), internally, in the ratio m₁ : m₂ are

 

 

This is the Section Formula.

 

 

Section Formula Examples

 

Let us try to understand it better through an example:

 

Example: In what ratio does the point (� 4, 6) divide the line segment joining the points A (� 6, 10) and B (3, � 8)?

 

Solution:

 

The co-ordinates of point A and B are (� 6, 10) and (3, � 8).

 

Let (� 4, 6) divide AB internally in the ratio m₁ : m₂.

 

Here,

 

x = -4 and y = 6

 

x₁ = -6, x₂ = 3,

 

And y₁ = 10, y₂ = -8

 

Using the section formula,

 

 

Recall that if (x, y) = (a, b) then x = a and y = b.

 

 

Now, taking

 

 

= 7m₁ = 2m₂

 

= m₁: m₂ = 2: 7

 

We should verify that the ratio satisfies the y-coordinate also.

 

Now,

 

Dividing throughout by m₂,

 

 

Therefore, the point (� 4, 6) divides the line segment joining the points A (� 6, 10) and B (3, � 8) in the ratio 2: 7.

 

Read More- What is Circle - Definition, Formula, Properties, & Examples