Coordinate Geometry: Basics, Distance & Section Formula With Examples

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

Cartesian Plane

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).

Coordinate Geometry

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 ΔAEB, we get

AB2 = AE2 + BE2

       = (x2 – x1)2 + (y2 – y1)2

Therefore, AB =

Coordinate Geometry

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

Coordinate Geometry formula

which is called the Distance Formula.

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

distance formula examples

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.,

section formula

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

Draw AQ and PC parallel to the x-axis.

Then, ΔPAQ ~ ΔBPC (AA similarity criterion)

Therefore,

section formula

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

coordinate geometry equations

Taking,

coordinate geometry equations

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

coordinate geometry equations

This is the Section Formula.

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,

section formula examples

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

section formula examples

Now, taking

section formula examples

= 7m1 = 2m2

= m1: m2 = 2: 7

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

Now,

Dividing throughout by m2,

section formula examples

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

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