Coordinates and midpoints

In a nutshell

Given the coordinates of two points on a coordinate grid, you can work out the coordinates of the midpoint of these points.

Coordinates reminder

Coordinates identify the location of a point on a coordinate grid. They are of the form $(x,y)$ where the first number is the $x$ -coordinate of the point and the second number is the $y$ -coordinate of the point. The $x$ - and $y$ - coordinates correspond to the number on the $x$ - and $y$ -axes respectively that the point is in-line with. So for example, the following diagram shows the point $(5,8)$ :

Maths; Graphs; KS4 Year 10; Coordinates and midpoints

Quadrants

You can split the coordinate grid into quadrants:

the positive $x$ positive $y$ quadrant (the top right) is quadrant one;
the negative $x$ positive $y$ quadrant (the top left) is quadrant two;
the negative $x$ negative $y$ quadrant (the bottom left) is quadrant three;
the positive $x$ negative $y$ quadrant (the bottom right) is quadrant four.

You can remember the order by starting with the top right and going counterclockwise:

Maths; Graphs; KS4 Year 10; Coordinates and midpoints

Midpoints

The midpoint of two points has coordinates given by the middle of the two points' $x$ -coordinates and the middle between the points' $y$ -coordinates. These middle values are respectively the $x$ - and $y$ - coordinates of the midpoint.

Finding the middle between two numbers can be done by halving the difference between them and then adding this to the smaller of the two numbers.

Example 1

Find the midpoint between the points $(4,8)$ and $(10,3)$ .

To do this, first find the midpoint of the $x$ -coordinates: the middle of $4$ and $10$ is $7$ . Next find the midpoint of the $y$ -coordinates: the middle of $8$ and $3$ is $5.5$ .

Hence the coordinates of the midpoint are $\underline{(7,5.5)}$ .

A general method

What if your points are very far apart and it is tricky to spot what the middle of the $x$ - and $y$ -coordinates are? In this case, use this procedure:

procedure

1.	Designate one of your points as point $A$ and the other as point $B$ . Let point $A$ have the coordinates $(x_A,y_A)$ and point $B$ have the coordinates $(x_B,y_B)$ .
2.	Find the middle $x$ -value using the formula: $\frac{x_A+x_B}{2}$
3.	Find the middle $y$ -value using the similar formula: $\frac{y_A+y_B}{2}$
4.	You now have that the midpoint has coordinates $\bigg(\frac{x_A+x_B}{2},\frac{y_A+y_B}{2}\bigg)$ .

Example 2

Find the midpoint between $(16,-5)$ and $(44,11)$ .

Using the procedure above, you can start by naming the points: $A(16,-5)$ and $B(44,11)$ . The midpoint's $x$ -coordinate is given by

$\frac{16+44}{2}=30$

The midpoint's $y$ -coordinate is given by

 $\frac{-5+11}{2}=3$

So the midpoint has coordinates $\underline{(30,3)}$ .