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:
We use this to find the distance between any two points (x1, y1) and (x2, y2) 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.
The point B (x2, y1) is at the right angle. We can see that:
- The distance between the points A(x1, y1) and B(x2, y1) is simply x2 − x1 and
- The distance between the points C(x2, y2) and B(x2, y1) is simply y2 − y1.
Distance from (x1, y1) to (x2, y2).
Using Pythagoras' Theorem we can develop a formula for the distance d.
The distance between (x1, y1) and (x2, y2) is given by:
Note: Don't worry about which point you choose for (x1, y1) (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.
Find the distance between the points (3, −4) and (5, 7).
Find the distance between the points (3, −1) and (−2, 5).
What is the distance between (−1, 3) and (−8, −4)?
Find k if the distance between (k,0) and (0, 2k) is 10 units.