# Coordinate Geometry Formulas

In this Coordinate Geometry Formula list, you will learn all the coordinate geometry formulas and must-know concepts for O-level Math Exams.

Coordinate Geometry is the study of geometric figures when they are plotted in the cartesian plane.

**Before you read on, you might want to download this entire revision notes in PDF format to print it out, or to read it later. **

**This will be delivered to your email inbox.**

## The Cartesian Plane

The cartesian plane is a coordinate system used to represent points and graphically illustrate relationships between two variables. The Cartesian plane consists of two perpendicular number lines, usually labeled the x-axis and the y-axis, intersecting at a point called the origin.

## The Cartesian Coordinate System

Points on the cartesian plane are represented using coordinates (*x _{1}, y_{1}*), where

*x*corresponds to a number on the x-axis and

_{1}*y*corresponds to a number on the y-axis.

_{1}**Example of Cartesian Coordinate**

The point *A* (2, 5) means 2 units to the right of the origin and 5 units up.

The point *B* (−2, 3) means 2 units to the left of the origin and 3 units up.

The point *C* (3, −2) means 3 units to the right of the origin and 2 units down.

The point *D* (−3, −4) means 3 units the left of the origin and 4 units down.

## Coordinate Geometry Formula: Gradient

The gradient of a line (formed by two points) is a measure of the steepness of the line, and it is represented by a real number.

The gradient of a line (formed by two points) is also the ratio of the vertical change (*rise*) between the points to the horizontal change (*run*) between the two points.

Gradient of line = rise/run

= (*y _{2} − y_{1}*)/(

*x*)

_{2}– x_{1}## Coordinate Geometry Formula: Length of Line Segment

Consider Figure 1 of the cartesian plane shown below.

Length of* AB* = √ ( (*y _{2} – y_{1})*

^{2}+ (

*x*

_{2}– x_{1})^{2 }) units

**Example of Length of Line Segment**

Given that *A* (1, 6) and *B* (4, 8), find the length of the line segment *AB*.

AB = √ ( (4 – 1*)*^{2} + (8* – 6)*^{2 })

= √ ( 3^{2} + 2^{2 })

= √13 (*ans*)

## Coordinate Geometry Formula: Equation of Line

For a line with gradient m and passing through the point (1, 1), the equation of the line is given by: y − y_{1} = m(x − x_{1}).

**You may still use *y = mx + c* and substitute (1, 1) into the equation to find the value of c.

**Example of Equation of Line**

Find the equation of the straight lines joining two points (−4, −1) and (4, 5)

gradient = ( 5 – (–1) )/( 4– (–4) )

= 6/8

=3/4

The equation of a straight line is y – 5 = 3/4 (*x* – 4)

y = 3/4 *x* + 2 (*ans*)

## Parallel Lines

When two lines are parallel, then the two lines must have the **same gradient**.

Conversely, when two lines have the **same gradient**, then the two lines must be **parallel.**

**Example of Parallel Lines**

Find the equation of the straight line which is parallel to the given straight line below and passing through a point (2, 3)

The equation of a straight line passing through (2, 3) is y – 3 = –2(*x – 2)*

y = –2*x* + 7 (*ans*)

## Collinear Points

If two line segments have the **same gradient** and there is a **common point** between the two line segments, then the line segments must be **collinear**.

**Example of Collinear Points**

Prove that A, B, and C are collinear points.

The gradient of AC = (3 – (–3))/ (–6 – 4)

= – 3/5

The gradient of AB = (3 – 0)/(–6 – (–1) )

= –3/5

The gradient of BC = (0 – (– 3))/(–1 – 4)

= –3/5

Observe that gradient of *AC* = gradient of *AB* = gradient of *BC*.

Since they share a common point, the 3 points are collinear.

## Coordinate Geometry Formula: Angle of Inclination

tan ∠*BAC* = (*y _{2} – y_{1}*)/(

*x*)

_{2}– x_{1}tan θ = (*y _{2} – y_{1}*)/(

*x*)

_{2}– x_{1}∴ tan θ = gradient of *AB*

## Coordinate Geometry Formula: Midpoint of a Line Segment (For A-Math)

The midpoint of *AB, M*(*j ,k*) = ( (*x _{1} + x_{2}*)/2, (

*y*)/2 )

_{2}+ y_{1}

**Example of Midpoint**

Find the coordinates of the midpoint, M, of (1, −1) and (−1, −5).

Midpoint = ( (1 + (–1) )/2, ((–1) + (–5))/2 )

= (0, –3)

## Coordinate Geometry Formula: Perpendicular Lines (For A-Math)

If two lines *L _{1}* and

*L*have gradients

_{2}*m*and

_{1}*m*respectively, then

_{2}*m*×

_{1 }*m*= −1.

_{2 }Find the equation of the line through *B*(0,6) and perpendicular to the line 3*y* + 1.5*x* = 2.

3*y* + 1.5*x* = 2

*y* = 2/3 *x*− 0.5

Gradient of the line through *B*(0,6) = −3/2

Sub (0,6) into = −3/2 *x* + *c*

*c* = 6

Equation of the line *y* = −3/2 *x* + 6

## Coordinate Geometry Formula: Area of Polygons (The “Shoelace” Method) (For A-Math)

If *A*(*x _{A}. y_{A}*),

*B*(

*x*),

_{B}. y_{B}*C*(

*x*), …,

_{C}. y_{C}*N*(

*x*) form a polygon, where

_{N}. y_{N}*A, B, C,*… and

*N*are the vertices of the polygon in an

**anticlockwise**sequence, then

### Example of Shoelace Method

The vertices of the triangle is given as (3, 5), (−2, 4), and (−2, −3). Find the area of triangle ABC.

## Last Minute Revision for O Level Math?

## Check out our exam guide on other topics here!

**Before you go, you might want to download this entire revision notes in PDF format to print it out, or to read it later. **

**This will be delivered to your email inbox.**

**Does your child need help in his or her studies?**

**1) Live Zoom Lessons at Grade Solution Learning Centre**

At Grade Solution Learning Centre, we are a team of dedicated educators whose mission is to guide your child to academic success. Here are the services we provide:

– Live Zoom lessons

– EdaptIQ™, a smart learning platform that tracks your child’s progress, strengths and weaknesses through personalised digital worksheets.

– 24/7 Homework Helper Service

We provide all these services above at a very affordable monthly fee to allow as many students as possible to access such learning opportunities.

We specialise in English, Math, and Science subjects.

You can see our fees and schedules here >>

**2) Pre-recorded Online courses on Jimmymaths.com**

If you are looking for something that fits your budget, or prefer your child learn at his or her own pace, you can join our pre-recorded online Math courses.

Your child can:

– Learn from recorded videos

– Get access to lots of common exam questions to ensure sufficient practice

– Get unlimited support and homework help

You can see the available courses here >>