The grazing goat on a rope problem and a nearly special triangle

Our Introduction to Geometry book had the classic “goat on a rope” as an example problem. Rather than working through the example, I gave it to my son as a challenge problem and it really gave him fits.

I decided to talk through it with him using our Zometool set and during the talk we discovered that a 3 – 4 – 6 triangle very nearly has a 120 degree angle. Fun little exercise with a surprising geometric result!

First – the problem:

 

Next – I asked my son to share a few of his initial thoughts about the problem and talk about why it gave him a little trouble. During the conversation here we stumble on the 3-4-6 triangle and think that it might have a 120 degree angle.

 

Next we went to the table to try to work out the calculation with pencil and paper. My son was a little tired and got tripped up a little by one bit of the calculation, but we did get to the end.

At the end of this piece, we started looking at the 3-4-6 triangle, but decided to go to Wolfram Alpha:

 

Last up was a quick look at the 3-4-6 triangle on Wolfram Alpha. I probably could have found a way to do this part with the geometry that we already know, but we were both a little tired and investigating the triangle this way seemed just fine.

 

So, our Zometool set gave us a fun way to look at a pretty standard geometry problem. It is always really fun for me to watch my kids see a problem like this for the first time – I never know where it will go. The extra surprise that came from this particular problem set up was nice, too – a 3 – 4 – 6 triangle nearly has a 120 degree angle!

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