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Coordinates and midpoints

Tutor: Toby

# 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 $(5,8)$:

You can split the coordinate grid into quadrants:

1. the positive $x$ positive $y$ quadrant (the top right) is quadrant one;
2. the negative $x$ positive $y$ quadrant (the top left) is quadrant two;
3. the negative $x$ negative $y$ quadrant (the bottom left) is quadrant three;
4. 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:

## 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 $\underline{(30,3)}$.

## FAQs - Frequently Asked Questions

### How are the quadrants of a coordinate grid numbered?

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