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?”
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!