Basic Coordinate Geometry (Edexcel IGCSE Further Pure Maths): Revision Note

Basic Coordinate Geometry

What are Cartesian coordinates?

• Cartesian coordinates are the coordinates used in 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

   

What can I do with coordinates?

• If you have two points with coordinates $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ then you should be able to find

  • The distance between the two points

  • The point that divides the line between the two points in a given ratio

  • The gradient of the line joining the two points

How do I find the distance between two points?

• The distance $d$ between two points with coordinates $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ can be found using the formula

  • $d= \sqrt{{\left(x_1- x_2\right)}^2+{\left(y_1- y_2\right)}^2}$

• $d$ is the hypotenuse of a right-angled triangle

  • The differences between the $x$-coordinates and the $y$-coordinates give the other two side lengths

  • So by Pythagoras’ Theorem  $d^2={\left(x_1-x_2\right)}^2+{\left(y_1-y_2\right)}^2$

How do I divide the line between two points in a given ratio?

• The coordinates of the point dividing the line joining $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ in the ratio $m:n$ are given by

  • $\left(\frac{n{x}_1+m{x}_2}{m+n},\frac{n{y}_1+m{y}_2}{m+n}\right)$

• If $\mathit{m}\mathbf{>}\mathit{n}$ then the point is closer to $\left(x_2,y_2\right)$

• If $\mathit{m}\mathbf{<}\mathit{n}$ then the point is closer to $\left(x_1,y_1\right)$
  

How do I find the midpoint between two points?

• The midpoint is the point halfway between the two points

  • This is just 'dividing in a given ratio' where the ratio is $1:1$

• The coordinates of the midpoint will be

  • $\left(\frac{x_1+ x_2}{2} , \frac{y_1+ y_2}{2} \right)$

How do I find the gradient of a straight line joining two points?

• The gradient $m$ of a line between two points with coordinates $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ can be found using the formula

  • $m= \frac{y_2- y_1}{x_2- x_1}$

• This is also known as  $m=\frac{\mathrm{rise}}{\mathrm{run}}$