Coordinate Geometry (AQA GCSE Further Maths): Revision Note

• As we are squaring the difference in and  in the formula, it does not matter if they are positive or negative

  • 3² is the same as (-3)², this may help to speed up your working

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

Calculate the distance of the line segment AB.

Using the formula for the distance between two points, $d=\sqrt{{\left(x_1-x_2\right)}^2+{\left(y_1-y_2\right)}^2}$ 

Substituting in the two given coordinates:

$d=\sqrt{{\left(3--5\right)}^2+{\left(-4-2\right)}^2}$

Simplify: 

$d=\sqrt{{\left(8\right)}^2+{\left(-6\right)}^2}=\sqrt{64+36}=\sqrt{100}=10$

Answer = 10 units

• Unlike when finding the distance between two points, when finding a midpoint we are finding a location, so the negative signs definitely do matter here

  • Be very careful to include any negative signs, and check your calculations

The coordinates of A are (−4, 3) and the coordinates of B are (8, −12).

Find M, the midpoint of AB.

The midpoint can be found using M = $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Fill in the values of x and y  from each coordinate

$\left(\frac{-4+8}{2},\frac{3+-12}{2}\right)=\left(\frac{4}{2},\frac{-9}{2}\right)$

Simplify

M = (2, −4.5)

The coordinates of A are (−4, 3) and the coordinates of B are (8, −12).

A point N divides AB in the ratio 1 : 2.

Find the coordinates of N.

Calculate the difference between the x coordinate of A and the x coordinate of B

$\begin{array}{rcl}8-\left(-4\right)& =& 12\end{array}$

Divide this difference by 3 (as there are 1 + 2 = 3 parts in the ratio 1 : 2)

$\begin{array}{rcl}12÷3& =& 4\end{array}$

Add 4 to the x coordinate of A

$\begin{array}{rcl}x\mathrm{coordinate}\mathrm{of}N& =& -4+4=0\end{array}$

Repeat for y

$\begin{array}{rcl}-12-3& =& -15\\ \left(-15\right)÷3& =& -5\\ y\mathrm{coordinate}\mathrm{of}N& =& 3+\left(-5\right)=-2\end{array}$

Alternatively, you may find it helpful to sketch the coordinates A and B and find N intuitively. Note that in the sketch, the coordinates do not need to be placed in the correct orientation to one another, simply along a straight line:

Write the final answer as a coordinate point

N = (0, −2)