At the beginning of the year in geometry we talked about how .999999 repeating = 1. There are a lot of fun proofs for this out there. I like having students put 11/99 = .111111 repeating then 56/99 = .56565656 repeating and eventually asking them what 99/99 should be.

In my mind one of the ultimate reasons .999999999999999999999999 repeating = 1 is because there is no distance between these numbers.

Fast forward to 6th period geometry today.

We are talking about isosceles triangles…

*S1: “I was thinking last night; this might be a crazy question with a simple answer but I want to ask it.”*

**T: “Fire away”**

*S1: “Remember how .99999 repeating = 1? Well would it be possible to create an isosceles right triangle?”*

**T: “Oh?”**

*S1: “Well, the measurement of the two base angles could be 89.99999 repeating, which is the same as 90 degrees. So the two base angles would both be 90”*

**T: “Interesting…. Is there a maximum measurement for the base angles?”**

**S2: “60 degrees, they all have to be able to add to 180” (my mind jumped here too)**

**S1: “If they are both 60 degrees all you have to do is increase the height of that triangle and it increases the base angles”**

**T: “Can we have base angles 89 degrees? Would that leave room for the vertex angle?”**

**S1: “Yes, it would leave 2 degrees for the third angle”**

**T: “What would that isosceles triangle look like?”**

**S3: “Very very very very tall”**

**T: “All of this makes sense to me. What we have talked about convinces me we can have base angles of 89.999 repeating , I want to think about it more over the weekend, because I don’t have an answer yet. You do the same thing and we will talk more on Monday.”**

This conversation reads very quickly but was spread out over the span of about 8 minutes. I am wrestling with this question still, it would be an easy fix to say the base angles have to less than 90 degrees but that isn’t much fun, it definitely can’t be obtuse though. After looking at several definitions of isosceles I haven’t come across any resources to help with this question. Right now I want to say the triangle would have an infinite height, so would the shape even classify as a triangle? Would it be a closed figure?

These sort of conversations are incredibly valuable. I was able to model the sort of thinking and patience I expect from students and in turn continue to grow as a mathematician. I don’t think students realize how much their questions help me continue to grow.

Let me know your thought on this question!

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Completely agree that it is great for your students to see you intrigued and stumped when confronted with a new idea. Beyond the fact that you get to model good behaviors, it may make some other good teaching habits easier. Like not forcing them into a particular answer or a specific perspective.

Perhaps others will have a deeper contribution about the specific question, but I think you are starting this the right way. If you want to get to 90 degree angles as the limit of 89, 89.9, 89.99, etc, then it makes sense to think of the resulting figure as the limit of the resulting triangles. Let’s say that you are fixing the starting base length, then we can also recognize the ending figure will be two parallel rays perpendicular to a line segment with the original, fixed length. That could lead us to say that the isoceles right triangle has infinite height and vertex at infinity (you know, that point over there, no a bit beyond that one, … etc)

The next level of depth is when you ask: what would you like to do with this isosceles right triangle? Compute area or perimeter? Looking at the limits of the sequence of triangles that got us there, easy to see that those are both infinite (unbounded) as well. Well, maybe you aren’t so happy about that. What if, instead of fixing the base length, we instead fix the perimeter (or the area)? What happens to our limiting figure then?

Alternatively, we might be interested in some other property of the triangle, say the length of an altitude from one of the base vertices. What is the limit of that length as we increase our angle through this sequence?

Finally, you’ll probably come to realize that there is a standard place where 90degree isosceles triangles appear: the traditional unit circle diagram for trig functions. Can we talk about sin(90)? Of course, and from the diagram, we see that our answer comes from taking right triangles and gradually increasing the measure of one of the acute angles.

This leads to another series of questions: how does our isosceles 90degree triangle differ when it is the limit of a sequence of isosceles triangles for which we are increasing the base angle vs when it comes about from taking a series of right triangles for which we are increasing one of the acute angles and keeping a fixed hypotenuse length?

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