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# Coordinates

### Coordinates

The coordinate system allows us to visualise functions in 2D space.

Cartesian coordinates (or just coordinates) are a pair of numbers that define a point in the plane (as shown). What is the right way to write these coordinates?

The axes divide the plane into 4 quadrants

Each quadrant is numbered as shown in the diagram.

The horizontal or $x$ axis is first and moving right is positive. The vertical or $y$ axis is second and moving up is positive. What type of coordinate sits in the 3rd quadrant?

We can use the same principle to work out the other quadrants.

Each quadrant has a unique combination of positive and negative values for the coordinates.

Where the axes cross, where you see a $0$, is called the origin. What are the coordinates of the origin?

What is the coordinate of the point $A$ shown here?

What is the coordinate of point $B$?

Point $C$ is the mid-point between points $A=(2,5)$ and $B=(-4,3)$.

We can see it on the graph, but how can we calculate the coordinates?

To find the mid-point between two numbers we add them together and divide by $2$. Based on that, what will the $x$ coordinate of the mid-point be?

Using the same process, what is the $y$ coordinate of the mid-point?

Putting the two together our mid-point has coordinates $(-1,4)$.

We can see this matches what we have on the graph.

This gives us a formula for finding the mid-point between two points.

For any two points $(x_1,y_1)$ and $(x_2,y_2)$ we can use this formula to find the mid-point between them.

What is the mid-point between the two points $(-4,5)$ and $(3,-1)$?

We may also want to find the distance between two points.

By completing a right angled triangle, we can find this distance.

What is the horizontal distance in this triangle?

What is the vertical distance in this triangle?

Now we have a right angled triangle and we know the distance along two sides.

We can use Pythagoras' Theorem to find the third side. It is $a^2 + b^2 = c^2$

Using Pythagoras' Theorem ($a^2 + b^2 = c^2$), what is the distance between the two points? Give your answer to one decimal place.

We can summarise these steps in one formula.

For any two points $(x_1,y_1)$ and $(x_2,y_2)$ we can use this formula to find the distance between them.

What is the distance between the points $(-4,5)$ and $(3,-1)$? Give your answer to 1 decimal place

In summary, Cartesian coordinates show us where a point is in the plane.

Cartesian coordinates are written $(x,y)$.

The point where the axes cross is called the *origin*.

The coordinates are the origin are $(0,0)$.

There are four quadrants in the plane.

Each quadrant has a unique combination of positive and negative values for the coordinates.

The mid-point between two lines can be found using this formula.

The mid-point between $(-8,6)$ and $(3,4)$ is $(\frac{-8+3}{2},\frac{6+4}{2})=(-2.5,5)$.

We can find the distance between two points using Pythagoras's Theorem.

The distance between $(-8,6)$ and $(3,4)$ is $\sqrt{(-8-3)^2+(6-4)^2}=\sqrt{121+4}=11.2$ to one decimal place. Distance between two points $=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$