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

*. It’s no biggie, don’t sweat it. They are both the same thing.*

**δ = √[(x**_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}]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 = √[(x

_{2}-x

_{1})

^{2}+(y

_{2}-y

_{1})

^{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

**we’re all used to. We can see this better if we draw it out:**

*h*^{2}= x^{2}+y^{2}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

**. Actually what we are looking at here is Pythagoras’ theorem**

*δ = √[8*^{2}+4^{2}]**. Let’s explain:**

*h*^{2}= x^{2}+y^{2}*h ^{2} = x^{2}+y^{2}*

*∴ h = √[x*

^{2}+y^{2}]*h = √[8*

^{2}+4^{2}]*h =*

*√[64+16]*

h = √[80]

h = 4√[5]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.