Saw this problem through a re-tweet from John Golden:

This is a pretty challenging geometry problem. I asked me older son about it and his first observation was about the inscribed and circumscribed circle of triangle BCD. I really didn’t expect him to have that idea, so I thought it might be fun to try to walk through the problem with him.

As I thought about how to do that, I decided to have my younger son join in as well since we are studying angles right now.

I did a lot of the heavy lifting here, but it was still a really fun project.

First up, showing the kids the problem. The main thing here was to have my older son give his idea about the circumscribed and inscribed circles:

[a quick note on the thumbnails. I seem to be having the same thumbnail troubles that started yesterday. Sorry if some of the video previews are all grey. The videos are uploaded and published and should play fine. I don’t know why the thumbnails are getting goofed up. ]

Before diving into the calculations, we built some pentagons / pentagrams out of our Zometool set. Our construction allowed us to explore some similar triangles that will help us solve the problem:

Next we moved to the whiteboard to look more carefully at some of the angles in the problem. I really enjoyed the conversation with my younger son here and I think he was able to get a little bit better of an understanding about angles in triangles from this part of the project.

Next we moved on to discuss some of the similar triangles that we saw in our Zometool set. The relationship between these triangles will help us determine the (surprising) relationship between the long blue pieces and the medium blue pieces. This video has a quadratic formula calculation that has fun result:

The next part is included only for completeness. Here we walk through the long calculation to arrive at the solution to the problem. Well, a special case of the solution anyway. I think that my older son isn’t quite ready to tackle calculations like this one, but I was hoping that seeing how to work through something like this would be instructive for him. One nice surprise was that my younger son seemed very interested in every step.

Finally, instead of spending more time on the calculation from the last video, we just go to Wolfram Alpha to finish the job. This last step allows the boys to see the number giving the final relationship without seeing another messing square root calculation:

So, a problem that is obviously over both kid’s heads at this point, but nonetheless has a couple of fun parts that are accessible to them. Finding the golden ratio relationship in the blue struts was a pretty fun surprise for them, too. Fun little exercise.