Distance from Pythagoras

We have a problem to solve, and – don’t worry – it’s an easy one. All we want to do in this post is show the relationship between two mathematical rules. It’s more than a relationship actually; they are exactly the same things. So, we have been asked to find the measure of the line:


Okay, we know this isn’t going to be too difficult. As long as we know the formula for the distance of the line δ = √[(Δx)2+(Δy)2] we’ll be fine. Some of you reading this may know this equation better in the form δ = √[(x2-x1)2+(y2-y1)2]. It’s no biggie, don’t sweat it. They are both the same thing.

So we want to use this formula to find the measure of the |AB|, where A is (-4,-1) and B is (4,3).  Let’s go ahead and do that:

δ = √[(Δx)2+(Δy)2]
δ = √[(x2-x1)2+(y2-y1)2]

δ = √[(4- -4)2+(3- -1)2]
δ = √[(8)2+(4)2]
δ = √[64+16]
δ = √[80]
∴ δ = 4√[5]

Yea, I know. I overdid the step-by-step simplification here. It’s a demonstration. You can skip a few steps when you’re doing it. Stop moaning. Well, anyway, this gets us to the answer, but what I want to show here is that this δ = √[(Δx)2+(Δy)2] equation is essentially the same as the much more simple Pythagorean formula h2 = x2+y2 we’re all used to. We can see this better if we draw it out:


When we plum a line vertically down from B and horizontally through A until both intersect what we end up with is a right triangle where the |AB| becomes the hypotenuse. Isn’t that interesting? Okay, maybe not that interesting. But it is useful. What we can see is that the length of the horizontal is 8 and that of the vertical is 4. This should ring a bell from a line from the working above: δ = √[(8)2+(4)2], or just δ = √[82+42]. Actually what we are looking at here is Pythagoras’ theorem h2 = x2+y2. Let’s explain:

h2 = x2+y2
∴ h = √[x2+y2]
h = √[82+42]
h = √[64+16]
h = √[80]
h = 4√[5]

What we discover at the end of all that is that h is 4√[5] and δ is also 4√[5], therefore h=δ. this means that these two equations are in fact the same equation. The simpler form of the Pythagorean formula may prove easier to memorise and quicker to use. I hope that you found this as much fun as I did, but – as always – if you have any comments or queries please just drop them into the Thoughts and Questions section below. Thanks again.


Thoughts and Questions

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