Talking about angle bisectors

My younger son is working his way through Art of Problem Solving’s Introduction to Geometry book. He’s been doing almost all of the work on his own – I just check in every now and then.

Today we had a little extra time so we did a project on the section he’s currently studying -> angle bisectors.

We started with some of the basic properties -> why is the intersection point of the angle bisectors the same distance away from every side:

Next we moved on to a slightly harder problem -> what is that distance?

This problem gave him a little trouble. BUT, after a hint to think about how the 1/2 base * height formula for the area of a triangle might help, he made some nice progress:

Finally, I had him work put the ideas we talked about to work in a specific triangle. Here he finds the radius of the inscribed circle in a 3-4-5 triangle:

It was interesting to see him pull some old ideas from geometry in to help understand some of the new ideas he’s learning here. I haven’t looked ahead in the book, but assume that the angle bisector theorem is coming soon. That theorem was really difficult for my older son to grasp, so I’m going to try to work a little more carefully with my younger son in the coming weeks to help him with any difficulty he might have there.