# 1. Distance Formula

We have a right-angled triangle with hypotenuse length *c*, as shown:

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Recall Pythagoras' Theorem, which tells us the length of the longest side (the hypotenuse) of a right triangle:

`c=sqrt(a^2+b^2)`

We use this to find the distance between any two points (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) on the cartesian (*x*-*y*) plane:

### The Cartesian Plane

The cartesian plane was named after Rene Descartes.

See more about Descartes in Functions and Graphs.

_{1}, y

_{1})

_{2}, y

_{1})

_{2}, y

_{2})

*d*

The point *B* (*x*_{2}, *y*_{1}) is at the right angle. We can see that:

- The distance between the points
*A*(*x*_{1},*y*_{1}) and*B*(*x*_{2},*y*_{1}) is simply*x*_{2}−*x*_{1}and - The distance between the points
*C*(*x*_{2},*y*_{2}) and*B*(*x*_{2},*y*_{1}) is simply*y*_{2}−*y*_{1}.

_{1}, y

_{1})

_{2}, y

_{1})

_{2}, y

_{2})

_{2}− x

_{1}

_{2}− y

_{1}

*d*

Distance from (*x*_{1}, *y*_{1}) to (*x*_{2}, *y*_{2}).

Using Pythagoras' Theorem we can develop a formula for the distance *d*.

## Distance Formula

The distance between (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) is given by:

`d=sqrt((x_2-x_1)^2+(y_2-y_1)^2`

**Note:** Don't worry about which point you choose for (*x*_{1}, *y*_{1}) (it can be the first or second point given), because the answer works out the same.

## Interactive Graph - Distance Formula

You can explore the concept of distance formula in the following JSXGraph (it's not a fixed image).

**Drag** either point A or point B to investigate how the distance formula works.

You can move the graph up-down, left-right if you hold down the "Shift" key and then drag the graph.

Sometimes the explanation boxes overlap. It can't be helped (especially on a small screen)!

If you get lost, you can always refresh the page.

### Example 1

Find the distance between the points (3, −4) and (5, 7).

### Example 2

Find the distance between the points (3, −1) and (−2, 5).

### Example 3

What is the distance between (−1, 3) and (−8, −4)?

### Example 4

Find *k* if the distance between (*k*,0) and (0, 2*k*) is 10 units.

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