This problem from the 2015 AMC 10 b gave my son quite a bit of difficulty today:
The AMC 10B on Art of Problem Solving’s website – scroll down for problem 19
This is a really wonderful problem and I didn’t really appreciate how great it is until we started talking through it tonight.
First was a quick introduction to the problem which was, unfortunately, interrupted by a phone call. My son continued to think about the problem while I talked on the phone:
After a 10 minute call we returned to talking about the problem. In the mean time my son had spent some time thinking about the center of the circle, which gave us a nice idea to talk through. One amazing thing about this problem is that some elementary ideas from geometry (though not necessarily obvious ideas) can help us find the center of the circle in this problem.
Once we knew where the center of the circle was, we redrew the picture and tried to find some more properties that were hidden in the diagram. It took a few minutes to find the right idea, but my son did find it – one of the line segments in our picture has length equal to the sum of the two sides of the original right triangle! That idea plus the fact that the center of the circle is in the middle of the hypotenuse allow us to label some side lengths and do some algebra.
The work and discussion here is, I think, a great example of what a kid learning math can look like.
Finally, we wrap up the problem by working through the two equations we found in the last section. There are some clever ideas that reduce these two equations down to something pretty simple, and these ideas show us that the original triangle was isosceles!
So, a really challenging problem, but we made it all the way to the end! Although this one would probably have been too much for my son to tackle all by himself, I think it is a great example of what learning math can look like.