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Coordinates and midpoints

Coordinates and midpoints

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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)(x,y) where the first number is the xx-coordinate of the point and the second number is the yy-coordinate of the point. The xx- and yy- coordinates correspond to the number on the xx- and yy-axes respectively that the point is in-line with. So for example, the following diagram shows the point (5,8)(5,8):

Maths; Graphs; KS4 Year 10; Coordinates and midpoints


You can split the coordinate grid into quadrants: 

  1. the positive xx positive yy quadrant (the top right) is quadrant one;
  2. the negative xx positive yy quadrant (the top left) is quadrant two;
  3. the negative xx negative yy quadrant (the bottom left) is quadrant three;
  4. the positive xx negative yy 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


The midpoint of two points has coordinates given by the middle of the two points' xx-coordinates and the middle between the points' yy-coordinates. These middle values are respectively the xx​- and yy- 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)(4,8) and (10,3)(10,3).


To do this, first find the midpoint of the xx-coordinates: the middle of 44 and 1010 is 77. Next find the midpoint of the yy-coordinates: the middle of 88 and 33 is 5.55.5

Hence the coordinates of the midpoint are (7,5.5)\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 xx- and yy-coordinates are? In this case, use this procedure:


Designate one of your points as point AA​ and the other as point BB​. Let point AA have the coordinates (xA,yA)(x_A,y_A)​ and point BB have the coordinates (xB,yB)(x_B,y_B).​
Find the middle xx-value using the formula:
Find the middle yy-value using the similar formula:
You now have that the midpoint has coordinates (xA+xB2,yA+yB2)\bigg(\frac{x_A+x_B}{2},\frac{y_A+y_B}{2}\bigg).​


Example 2

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

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


The midpoint's yy-coordinate is given by 


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

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FAQs - Frequently Asked Questions

How do you find the middle of two numbers?

How do you find the midpoint of two points?

How are the quadrants of a coordinate grid numbered?


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