'When will I use Pythagoras (or Ratios) in real life?'

[ICRqVNmDrU8FDi]

Every so often I will have a student ask me this, but rather than respond sarcastically, or even not at all, I say...

I don't have much space on my desk at home, so when my monitor went kaput a few years back I had to be very careful about the size of it's replacement. Monitors are always advertised according to the length of their hypotenuse,

https://i.imgur.com/YY8YHIn.jpg

so if a monitor says it is a 22-inch monitor, it is saying it's hypotenuse is 22 inches long. This isn't very useful for me though, I need to know how wide it is to know whether it'll fit onto my desk, and to convert the hypotenuse to a width, I have to use Pythagoras.

The sides of monitors aren't often the same length though, they are usually wider than they are tall. A normal monitor (and any monitor I want to buy) will have dimensions in the ratio 16:9, i.e. for every 16 centimeters they are long horizontally, they are 9 centimeters tall vertically. In simpler terms, we can label the width of our monitor 16x and the height of it 9x, like so:

https://i.imgur.com/JxdFyLR.jpg

Image not to scale.

We know (thanks to Pythagoras) that a^2 + b^2 = c^2 , but how does this translate to our beautiful real life problem? Well it translates as (16x)^2 + (9x)^2 = (our hypotenuse)^2. (16x)^2 is 256x^2 and (9x)^2 is 81x^2 so together, c2=(81+256)x^2 or 337x^2. To get c alone we then need to square root 337x^2 , we receive 18.36x.

https://i.imgur.com/92LN2UR.jpg

Final step now. My desk only has space for a monitor 17.5 inches wide, what is the hypotenuse of said monitor (assuming it is in the ratio 16:9)? Well, if 17.5 inches = 16x, then x = (17.5/16) inches. If our hypotenuse is 18.36x, then in inches it is 18.36 times (17.5/16), (or put simply) 20.08 inches. So what size monitor am I going to buy? I'm going to buy a 20 inch monitor.

Extension Question: How tall is my monitor going to be?

[oblib]

I'll offer this life experience:

I got into a debate with some fellow backpackers a couple years ago about how to measure total distance traveled when hiking in mountainous areas. They all said they could measure the total distance using the Pythagoras theorem but they were only measuring the distances of the angled sides of a triangle, not both the forward and up/down, which you have to traverse in the real world.

When I explained that they were only measuring the distance in a 2D world but since we live in a 3D world we need to measure both forward and up and down to get total distance traveled they were all in agreement that I couldn't be more wrong.

When I tried pointing out that this is an exercise in the physics of spatial dimensions (3D), and not a 2D flat plane, they couldn't (or more aptly put, refused) to grasp the difference.

I think this approach might be another good way to illustrate using it because while they all understood how it worked they didn't have a clue about when to apply it, which your example perfectly illustrates.

[vitaminbandit]

I'm sure they understood exactly what you meant and could apply it. Nobody measures distance the way you described it.

[jefecoon]

Ask your student(s) to look at a map, or better yet, open up Google Maps and flip to 'Earth' view.

All that satellite imagery? Every single pixel, from every single satellite fly-over / observation, all interpreted / post-processed / calculated into actual meaningful data via Pythagorean theorem.

E.g. every data point processed for "incident angle" of the observation platform above parcel/pixel observed. Basics @

https://en.wikipedia.org/wiki/Angle_of_incidence_(optics)

http://www.crisp.nus.edu.sg/~research/tutorial/freqpol.htm

Math FTW.

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[wmoser]

They are also used quite a lot in building trades or wood working or machining. Pitch of roofs are often given as a ratio, tapers on machinery shafts, making sure your pipes have sufficient slope to drain properly. Another construction example would be how long of a board do you need for the stringers (the long boards that are notched for the stairs if it has to go up 10ft over a fifteen foot distance, etc. whether they might end up the designer or the constructor of things it's probably one of the more practical math concepts.

[joeclark77]

And add to that that the 3-4-5 triangle is still a useful way to establish a right angle without any tools (apart from a piece of string).