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:

013

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:

014

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.

Advertisements

Thoughts and Questions

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s