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.

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secondary math revision notes

 

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.

cartesian plane, x and y axis

The Cartesian Coordinate System

Points on the cartesian plane are represented using coordinates (x1, y1), where x1 corresponds to a number on the x-axis and y1 corresponds to a number on the y-axis.

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.

how to plot coordinates

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. 

coordinate geometry formula, gradient

Gradient of line = rise/run

= (y2 − y1)/(x2 – x1)

Coordinate Geometry Formula: Length of Line Segment

Consider Figure 1 of the cartesian plane shown below.

coordinate geometry formula, length

Length of AB = √ ( (y2 – y1)2 + (x2 – x1)) 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))

= √ ( 32 + 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 − y1 = m(x − x1).

**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)

coordinate geometry formulas, equation of line

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)

parallel lines, coordinate geometry formulas

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

y = –2x + 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.

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 = (y2 – y1)/(x2 – x1)

tan θ = (y2 – y1)/(x2 – x1)

∴ tan θ = gradient of AB

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

midpoint, coordinate geometry formulas

The midpoint of AB, M(j ,k) = ( (x1 + x2)/2, (y2 + y1)/2 )

 

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 L1 and L2 have gradients m1 and m2 respectively, then m1 × m2 = −1.

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

3y + 1.5x = 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(xA. yA), B(xB. yB), C(xC. yC), …, N(xN. yN) form a polygon, where A, B, C, … and N are the vertices of the polygon in an anticlockwise sequence, then

shoelace method, coordinate geometry formulas

 

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.

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