3.1.1 Coordinate Geometry

Basic Coordinate Geometry

Cartesian coordinates are basically the x-y coordinate system
- They allow us to label where things are in a two-dimensional plane
In the 2D cartesian system, the horizontal axis is labelled x and the vertical axis is labelled y

If we have two points with coordinates (x₁ , y₁) and (x₂ , y₂) then we should be able to find
- The midpoint of the two points
- The distance between the two points
- The gradient of the line between them

The midpoint is the average (middle) point
- It can be found by finding the middle of the x-coordinates and the middle of the y-coordinates
The coordinates of the midpoint will be

$(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$

- This is given in the formula booklet under the prior learning section at the beginning

Basic Coordinate Geometry Notes Diagram 5

The distance between two points with coordinates (x₁ , y₁) and (x₂ , y₂) can be found using the formula

$d = \sqrt{{(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2}}$

- This is given in the formula booklet in the prior learning section at the beginning

Pythagoras’ Theorem $a^{2} = b^{2} + c^{2}$ is used to find the length of a line between two coordinates
If the coordinates are labelled A and B then the line segment between them is written with the notation [AB]

Basic Coordinate Geometry Notes Diagram 2

The gradient of a line between two points with coordinates (x₁ , y₁) and (x₂ , y₂) can be found using the formula

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

- This is given in the formula booklet under section 2.1 Gradient formula
- This is usually known as $m = \frac{rise}{run}$

Point A has coordinates (3, -4) and point B has coordinates (-5, 2).

Calculate the distance of the line segment AB.

ai-sl-3-1-1-basic-cg-we-i

ii)

Find the gradient of the line connecting points A and B.

ai-sl-3-1-1-basic-cg-we-ii

iii)

Find the midpoint of [AB ] .

ai-sl-3-1-1-basic-cg-we-iii