Naming Convention

To name a triangle by its three side lengths, we need its three side lengths. Of course, this should make sense. To know your friend’s full name, you would need their three names: first, middle (if they have one), and last. For triangles, their names would be ordered shortest, middle, and longest.

Let’s meet a few triangles and start naming them!

This one should be easy to name

(Answer: In a straightforward manner, just going from shortest to longest. 3 is obviously the shortest side and 5 is obviously the longest, so we should already have 3 - ? - 5 mentally, and then 4 is all that is remaining, whereby we get a 3-4-5 triangle)

How about this one?

(Answer: 3-3-3 So, it might be confusing to think of something as being both the “shortest” and the “longest” at the same time, but if there is only one side length, it is the shortest, the middle, and the longest, all at once. The work of sorting is already done for you! This would be like someone with the name John John John, which is definitely not the most creative name)

Name Uniqueness

Now, unlike with humans, if any two triangles have the same name, they are congruent triangles (remember, that means “the same”). So, sadly, there cannot exist sixteen different Arnold Alois Schwarzeneggers in Triangle World
 

So, in case you’ve forgotten about congruence, it means you can overlap one triangle on top of another. Let’s take two 3-4-5 triangles, for example.
 

We want to be absolutely sure that these two triangles with the same name overlap, right? So let’s rotate and move them around until they overlap:
 

Well, I guess they don’t overlap and trigonometry doesn’t work!

Nah, I’m just kidding. We have to rotate the left triangle using the third dimension to get these two triangles to overlap.
 

Now, this is only one example with the 3-4-5 triangles, and I haven’t shown why this works (since it would take about twenty pages of proof), but this is an important rule to remember! In more fancy terms, this is called the side-side-side (or SSS) postulate.

Impossible Triplets

 Not all side length names can give rise to a triangle. For example, 1-2-5 and 2-5-7 are names that couldn’t give rise to a triangle. To think about how unusual this would be in the human world, imagine there were laws banning names like “Maria Walburga Christina” or “John von Neumann.” But because triangles must always close an area with three sides, certain triangles are impossible.

Let’s take 1-2-5, for example. If I place the 5 edge down, then I just need to find a way to use the 1 and the 2 to close the rest.
 

No matter how I try to complete this triangle, it’s impossible: there’s just not enough “slack,” so to speak.
 

Sadly, triangle sides are not like rubber bands: we can’t stretch them in length, and they just have to stay as they are.

And somehow, it’s even worse for the 2-5-7. The three edges get so close to making a triangle, but it’s not enough

The closest the sides get is smooshing into a line segment, which is obviously not what we think of as a triangle.

The rule that all triangle names have to follow is this: the smallest side and the middle side have to sum up to more than the biggest side. So again, smallest + middle > biggest, or else, we don’t have a triangle. If you think about this carefully, this will make sense, since there has to be enough room for the smallest and middle sides not to smoosh into a line.

Homework/Activities:

1. Using thin strips of paper (be careful not to get paper cuts!), we can see why we need the smallest-middle-longest inequality. Come up with an impossible triplet of side lengths (either longest is greater than or equal to smallest + middle), and cut out thin strips of paper with those lengths. You will see that it is impossible to arrange them into a perfectly closed triangle.
2. To see why side length names are unique, we can also use scissors and glue to cut out 3 side lengths that satisfy the smallest-middle-longest inequality. We need two copies of each smallest, middle, and longest side length strip. Fiddle around separately with each set of 3 side lengths, until they form a triangle. Use some glue to make the triangles stick (be careful not to adjust the angles). They should overlap.
3. Why is a convention, like the shortest-middle-longest convention, necessary for giving a triangle a name? Could there exist other conventions? (We want to give triangles one name and one name only. If there were no conventions, I could call the same triangle 5-6-7 or 6-5-7, which is a waste. There can exist other conventions like longest-middle-shortest or non-ordered ones like shortest-longest-middle, but shortest-middle-longest is most common.)